Ray Class Fields of Global Function Fields with Many Rational Places

Vom Fachbereich Mathematik der Carl-von-Ossietzky-Universität Oldenburg

zur Erlangung des Grades eines Doktors der Naturwissenschaften

angenommene Dissertation, vorgelegt von

Roland Auer

aus

Lindau/Bodensee Erstreferent: Prof. Dr. H.-G. Quebbemann Korreferent: Prof. Dr. H. Stichtenoth

Tag der mündlichen Prüfung: 16. Juli 1999.

Contents

Introduction 4

I Global Function Fields 8

1 Preliminaries 8 2 S-Units 14 3 Ramification Groups 19

II Ray Class Fields 26

4 Class Field Theory 26 5 Definition and Properties 30 6 Computation of Degrees 36

III Many Rational Places 46

7 Upper Bounds 46 8 Rational Function Field 51 9 Tables of Examples 59 List of Symbols 73 References 76

Introduction

The search for algebraic curves over finite fields 𝔽_{q} with many rational points or, what amounts to the same, for global function fields with many rational places was motivated by Goppa’s algebro-geometric construction of errorcorrecting codes and initiated by Serre [S2, S3] in the early eighties. Serre was the first to utilize class field theory, among other methods, to exhibit such curves, and he also provided explicit formulas for the maximal numbers N_{q}(1) and N_{q}(2) of rational points on curves over 𝔽_{q} of genus g = 1 and 2, respectively.

Until today, no such formula is known for N_{q}(g) when g ≥ 3. Instead, the work of many mathematicians has led to large tables, such as in [GV7], documenting the state of the art by giving lower and upper bounds for the numbers N_{q}(g). The lower bounds are obtained by more or less explicit constructions of curves with many points. Hansen and Stichtenoth [Han, HS, St1], Wirtz [Wi], van der Geer and van der Vlugt [GV1– GV6], Niederreiter and Xing [NX1, NX2, NX5, NX6, NX7] and Shabat [Sh] provided defining equations for such curves, mostly of Artin-Schreier or Kummer type. The present work, which performs class field constructions, follows the theme of research by Schoof [Sf], Lauter [La1, La3] and Niederreiter and Xing [XN, NX2, NX3, NX4, NX7, NX8, NX9]. In particular, it has been influenced by the paper [XN], the methods of which are refined upon in Section 6. Making use of algorithmic number theory, we succeed in improving the mentioned tables in numerous places.

When dealing with class field theory, instead of an algebraic curve X (smooth, projective, absolutely irreducible) defined over a finite field 𝔽_{q}, it is convenient to consider its field K = 𝔽_{q}(X) of algebraic functions, a global function field with full constant field 𝔽_{q}. The genus of K is that of X, and coverings of X correspond to field extensions of K, the degree of the covering being the degree of the extension. To a point P ∈ X with minimal field of definition over 𝔽_{q} say 𝔽_{q^{d}} we associate the set 𝔭 ⊆ K of all functions vanishing in P and call 𝔭 a place of K. This 𝔭 is the maximal ideal in the discrete valuation ring R consisting of those functions which are regular at P, it corresponds in fact to the whole Frobenius (q-th power map) orbit of P, which contains d points, and it has (residue field) degree deg 𝔭 := [R/𝔭 : 𝔽_{q}] = d. In particular, the rational places, i.e. the places of degree 1, of K|𝔽_{q} are in 1–1 correspondence with the 𝔽_{q}-rational points on X. In this thesis we shall work with a purely field theoretic definition of a global function field K|𝔽_{q} and forget about curves completely. The necessary background is given in Section 1.

The central objects in this work are the ray class fields K_{S}^{𝔪}, where S is a non-empty set of places of K and 𝔪 is an S-cycle, i.e. a finite formal sum ∑ m_{𝔭}𝔭 of places 𝔭 outside S together with non-negative integer multiplicities m_{𝔭}. The extension K_{S}^{𝔪}|K can be characterized as the largest abelian extension L|K of conductor 𝔣(L|K) ≤ 𝔪 such that every place 𝔭 ∈ S splits completely in L (i.e. there are [L : K] places of L lying above 𝔭). Roughly speaking, the conductor measures the “wildness” of ramification at each place of K, thus the S-cycle 𝔪 confines this wildness. For example, if 𝔪 = 𝔬, no ramification is allowed at all, and we obtain the special case of a Hilbert class field K_{S}^{𝔬}, whose degree over K equals the S-class number h_{S}.

The extension K_{S}^{𝔪}|K is always finite. This would no longer be the case if we dropped the splitting condition in the characterization of a ray class field, i.e. if we allowed S to be empty, because the maximal constant field extension 𝔽̄_{q}K|K is abelian of infinite degree and unramified everywhere. Thinking of ray class fields of number fields, the role of S is usually played by the archimedean primes. For function fields, however, there is no canonical choice of S.

The K_{S}^{𝔪} provide a rather fine classification of the finite abelian extensions of K and are canonical and quite beautiful objects in themselves. Already because of this, they deserve being studied more intimately. We utilize them mainly to produce curves with many points. For example, if S consists of rational places only, then K_{S}^{𝔪} has at least [K_{S}^{𝔪} : K] |S| rational places. For suitable choices of S and 𝔪, this number tends to be large compared with the genus of K_{S}^{𝔪}, which remains to be calculated.

There are various ways of obtaining the ray class fields. If S consists of exactly one place of K, then K_{S}^{𝔪} can be constructed by adjoining the 𝔪torsion of a rank 1 Drinfel’d module to the Hilbert class field K_{S}^{𝔬}. This gives us a so-called narrow ray class field, whose Galois group over K_{S}^{𝔬} naturally contains 𝔽_{q}^{∗}, and K_{S}^{𝔪} is the subfield fixed by 𝔽_{q}^{∗}. In the general situation, one singles out a place 𝔮 ∈ S and gains K_{S}^{𝔪} as the subfield of K_{{𝔮}}^{𝔪} fixed by the Frobenius automorphisms of all the other places in S (cf. [XN, NX4]).

Our approach is by class field theory: we define a suitable open subgroup C_{S}^{𝔪} of finite index in the idèle class group C_{K} of K and obtain K_{S}^{𝔪} directly as the class field associated to this subgroup. The Galois group G(K_{S}^{𝔪}|K) is isomorphic to C_{K}/C_{S}^{𝔪}. To obtain its order, this time one has to factor out S-units, i.e. algebraic functions having all their poles and zeros in S, instead of Frobenius automorphisms. For the convenience of the reader, we include a brief introduction to class field theory in Section 4 also indicating ideas for the proofs of its main theorems.

It should be mentioned that defining equations for the extensions K_{S}^{𝔪}|K are only known in very few cases (Propositions 8.4 and 8.9). Surely, one could gain such equations in each single case by calculating the minimal polynomial of a suitable norm from the narrow ray class field in the Drinfel’d module construction. But the necessary calculations as well as the resulting polynomials tend to get very lengthy. The proper method seems to consist in determining (the coefficients of) the equations for small successive Kummer and Artin-Schreier extensions in the fashion of [GV6], and thereby make up the whole of K_{S}^{𝔪}|K step by step.

As was mentioned above, it is important to know the genus of K_{S}^{𝔪}. By means of a remarkable genus formula (Theorem 5.8 and Corollary 5.9), the problem is reduced to the determination of the degrees [K_{S}^{𝔪′} : K] for certain 𝔪′ ≤ 𝔪. This formula is deduced in Sections 3 and 5 from Hilbert’s Different Formula, the Hasse-Arf Theorem and the connection between upper ramification groups and higher unit groups known from local class field theory. Another proof not included in this thesis applies Möbius inversion to the Conductor Discriminant Product Formula and was found by Cohen et al. [CDO].

Unfortunately we only have explicit formulas for the degrees [K_{S}^{𝔪} : K] in very special cases (Theorems 8.1 and 8.5). Therefore we content ourselves with giving algorithms for their computation under the additional assumption that S is finite. The case of mixed S-cycles 𝔪 is only treated for tame ramification, i.e. for 𝔪 of the form ∑_{𝔭∈R} 𝔭 with a finite set R of places of K. Instead, from page 40 onwards, we concentrate on ramification at a single place 𝔭 of K. In this situation we introduce the concept of the S-description at 𝔭, a polynomial δ(t) with non-negative integer coefficients satisfying δ(1) − δ(0) = |S| − 1. Together with the S-class number h_{S} it carries the information about the degrees of the extensions K_{S}^{m𝔭}|K for all m ∈ ℕ_{0} (see Theorem 6.4 and the discussion following it).

An algorithm for computing these S-descriptions was implemented in KASH, the shell programming language of KANT, a powerful algorithmic number theory tool developed by the Berlin KANT group [K] under the guidance of M. Pohst. By searching through a large number of choices for K, S and 𝔭, in Section 9 we filter out those S-descriptions which lead to ray class fields having many rational places compared with their genera.

Acknowledgements

First of all, I wish to thank F. Heß, who supplied me with a special KASH version capable of dealing with residue fields, who kept supporting me in the programming process by giving valuable advice whenever necessary, and who discussed with me the problem of finding fundamental S-units, thereby contributing considerably to the progress of Section 2. I am also indebted to C. Dement, P. Kunkel and B. Wittje for a number of error corrections, to P. Harmand for seeing me through my LaTeX and TeX troubles, and especially to K. Roegner for her excellent and thorough proof reading. The thesis benefits from elucidating discussions I enjoyed with N. Manolache, W. Schmale, V. Shabat, P. Stevenhagen, H. Stichtenoth, U. Vetter, B. Wittje and certainly many others. Thank you all for that. Finally, I want to express my gratitude to my supervisor Prof. Dr. H.-G. Quebbemann for his patience and permanent support. He spent hours and hours listening and discussing problems with me and accompanied me through the ups and downs of my research work.

Dependence of Sections

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9]

Part I

Global Function Fields

This first part treats certain topics which manage without class field theory. The first section is a very concise introduction to global function fields, and the second discusses the problem of finding S-units. In Section 3 we turn to Hilbert’s ramification theory, thereby preparing ourselves for entering Part II.

1 Preliminaries

This section aims at familiarizing the reader with the notation and some basic facts from the theory of algebraic function fields used in this thesis. Apart from some results about completions (for which we shall give instant proof), its whole content is covered by Stichtenoth’s book [St2].

The group of a Galois extension L|K is denoted G(L|K). For the notions of a Dedekind domain, a discrete valuation (here always assumed to be additive and normalized with values in ℤ ∪ {∞}) and a discrete valuation ring we refer the reader to e.g. [FJ, pp. 12ff]. If P is a prime element in a factorial domain or a maximal ideal in a Dedekind domain, then we write v_{P} for the associated discrete valuation on the field of fractions.

Throughout the text, K is a global function field, i.e. K is finitely generated and of transcendence degree 1 over a finite field 𝔽_{q}, where q is a power of the characteristic p. We always require that 𝔽_{q} is the full constant field of K, i.e. is algebraically closed in K, and express this circumstance by writing K|𝔽_{q}.

A place of K is the maximal ideal 𝔭 in some discrete valuation ring R_{𝔭} of K. We say that 𝔭 is a zero or a pole of the function z ∈ K if z ∈ 𝔭 or z ∉ R_{𝔭}, respectively. The constant field 𝔽_{q} is naturally contained in the (finite) residue field 𝔽_{𝔭} := R_{𝔭}/𝔭 of 𝔭. The degree of 𝔭 is defined as deg 𝔭 := [𝔽_{𝔭} : 𝔽_{q}]. A place of degree 1 is called a rational place. The map 𝔭 ↦ v_{𝔭} is a 1–1 correspondence between the places and the (discrete) valuations of K. We denote the set of all places of K by ℙ_{K} and define ℙ_{K}^{d} := {𝔭 ∈ ℙ_{K} | deg 𝔭 = d} for d ∈ ℕ.

The free abelian group D_{K} on the set ℙ_{K} is called the divisor group of K. The map deg : ℙ_{K} → ℤ is extended to D_{K} by linearity. Its kernel is denoted by D_{K}^{0}. A divisor 𝔪 ∈ D_{K} is written as a formal sum 𝔪 = ∑_{𝔭∈ℙ_{K}} m_{𝔭}𝔭 with integer coefficients m_{𝔭} and finite support supp 𝔪 := {𝔭 ∈ ℙ_{K} | m_{𝔭} ≠ 0}. If 𝔫 = ∑_{𝔭} n_{𝔭}𝔭 ∈ D_{K} is another divisor and m_{𝔭} ≤ n_{𝔭} for all 𝔭 ∈ ℙ_{K}, then we write 𝔪 ≤ 𝔫. The zero element in D_{K} is denoted 𝔬, and a divisor 𝔪 ≥ 𝔬 is called effective.

Each function z ∈ K^{∗} has only finitely many zeros and poles, hence one can define its principal divisor (z) = (z)_{K} := ∑_{𝔭∈ℙ_{K}} v_{𝔭}(z)𝔭. Breaking up (z) into the zero divisor (z)_{0} ≥ 𝔬 and the pole divisor (z)_{∞} ≥ 𝔬 of z such that (z) = (z)_{0} − (z)_{∞}, one has the following

1.1. Theorem. Let z ∈ K \ 𝔽_{q}. Then deg(z)_{0} = deg(z)_{∞} = [K : 𝔽_{q}(z)].

In particular, deg(z)_{K} = 0 for all z ∈ K^{∗}. The map ( ) = ( )_{K} : K^{∗} → D_{K}^{0}, z ↦ (z) is a group morphism with kernel 𝔽_{q}^{∗}, hence (K^{∗}) ≃ K^{∗}/𝔽_{q}^{∗}. The quotient D_{K}^{0}/(K^{∗}) is called the (divisor) class group of K. Its order, the (divisor) class number h_{K}, is finite.

For 𝔪 ∈ D_{K}, the Riemann-Roch space L(𝔪) := {z ∈ K^{∗} | (z) + 𝔪 ≥ 𝔬} ∪ {0} is a finite 𝔽_{q}-space, and we set dim 𝔪 := dim_{𝔽_{q}} L(𝔪). The genus of K can be defined as the non-negative integer g_{K} := max{deg 𝔪−dim 𝔪+1 | 𝔪 ∈ D_{K}}.

The power series

Z_{K}(t) := ∑_{n=0}^{∞} A_{n}t^{n} ∈ ℤ[[t]] ⊆ ℂ((t))

with A_{n} = |{𝔪 ∈ D_{K} | 𝔪 ≥ 𝔬, deg 𝔪 = n}| is called the zeta function of K. It is actually a rational function, namely Z_{K}(t) = \frac{P_{K}(t)}{(1−t)(1−qt)} with numerator polynomial P_{K}(t) ∈ ℤ[t] of degree 2g_{K}.

1.2. Theorem. The numerator polynomial of the zeta function of K satisfies the functional equation

P_{K}(t) = q^{g_{K}}t^{2g_{K}}P_{K}(\frac{1}{qt}).

Moreover, P_{K}(1) = h_{K} is the class number of K.

The simplest example of a global function field is that of a rational function field 𝔽_{q}(x) with an indeterminate x over 𝔽_{q}. Extending the degree map deg : 𝔽_{q}[x] → ℕ_{0} ∪ {−∞} to the whole of 𝔽_{q}(x) yields a discrete valuation v_{∞} := − deg on 𝔽_{q}(x). The place ∞ of 𝔽_{q}(x) corresponding to v_{∞} has degree 1 and is the (only) pole of x. All (discrete) valuations on 𝔽_{q}(x) are given by the set

{v_{P} | P ∈ 𝔽_{q}[x] monic, irreducible} ∪ {v_{∞}}.

Hence there is a 1–1 correspondence P ↔ 𝔭 with v_{P} = v_{𝔭} between the monic irreducibles P ∈ 𝔽_{q}[x] and the places 𝔭 ≠ ∞ of 𝔽_{q}(x). In this correspondence, 𝔭 is the maximal ideal in the localization of 𝔽_{q}[x] at P, and the principal divisor of P is (P) = 𝔭 − (deg P)∞, thus deg 𝔭 = deg P. A rational function field has genus 0 and trivial divisor class group. Conversely, each global function field of genus 0 is a rational function field.

Let L|𝔽_{q^{d}} be a finite separable extension of K|𝔽_{q}. In other words, L is a global function field containing K with full constant field of degree d ∈ ℕ over that of K, and L^{p} does not contain K. Let 𝔮 be a place of L; then 𝔮 ∩ K =: 𝔭 is a place of K. We say that 𝔮 lies above 𝔭 and write 𝔮|𝔭. The canonical inclusion 𝔽_{𝔭} ⊆ 𝔽_{𝔮} is compatible with the inclusions 𝔽_{q} ⊆ 𝔽_{𝔭} and 𝔽_{q} ⊆ 𝔽_{q^{d}} ⊆ 𝔽_{𝔮}, and [𝔽_{𝔮} : 𝔽_{𝔭}] is called the inertia degree of 𝔮|𝔭. Note that, by definition, [𝔽_{𝔮} : 𝔽_{𝔭}] deg 𝔭 = d deg 𝔮. The restriction v_{𝔮}|_{K} equals e · v_{𝔭} for some positive integer e =: e(𝔮|𝔭), called the ramification index of 𝔮|𝔭. These numbers satisfy the following

1.3. Fundamental Equality. Let 𝔭 ∈ ℙ_{K}, then

∑_{𝔮|𝔭}e(𝔮|𝔭)[𝔽_{𝔮} : 𝔽_{𝔭}] = [L : K],

where the sum is over all places 𝔮 of L lying above 𝔭.

We say that 𝔮|𝔭 is ramified if e(𝔮|𝔭) > 1, otherwise 𝔮|𝔭 is called unramified. If e(𝔮|𝔭) = [L : K], then (𝔮 is the only place of L above 𝔭 and) we say that 𝔮|𝔭 (or 𝔭) is totally ramified (in L). We call 𝔭 unramified in L if 𝔮|𝔭 is unramified for all 𝔮 ∈ ℙ_{L} above 𝔭. The extension L|K itself is called unramified if all places of K are unramified in L. A place 𝔭 of K is said to split completely in L if there are [L : K] places of L above 𝔭 (necessarily each with ramification index and inertia degree equal to 1).

The different exponent d(𝔮|𝔭) of 𝔮|𝔭 (for a definition see [St2, p. 82]) is always ≥ e(𝔮|𝔭) − 1 and = 0 iff 𝔮|𝔭 is unramified. Since only finitely many places of K ramify in L, one can define the different ∑_{𝔮∈ℙ_{L}} d(𝔮|𝔮 ∩ K)𝔮 ∈ D_{L} of L|K. There is a norm map for divisors

N_{L|K} : \begin{matrix}D_{L} & → & D_{K} \\ ∑_{𝔮∈ℙ_{L}} m_{𝔮}𝔮 & ↦ & ∑_{𝔭∈ℙ_{K}} ∑_{𝔮|𝔭} m_{𝔮}[𝔽_{𝔮} : 𝔽_{𝔭}]𝔭. \end{matrix}

We note that deg N_{L|K}𝔪 = d deg 𝔪 for 𝔪 ∈ D_{L}. The norm behaves nicely on principal divisors in that N_{L|K}(z)_{L} = (N_{L|K}z)_{K} for any z ∈ L, where N_{L|K} on the right hand side denotes the usual norm of elements (see [CF, p. 75]).

Defining the discriminant 𝔡(L|K) ∈ D_{K} as the norm of the different of L|K, the genus formula in [St2, p. 88] reads:

1.4. Hurwitz Genus Formula. In the above situation, the genera of K and L satisfy the relation

d · (g_{L} − 1) = [L : K](g_{K} − 1) + \frac{1}{2} deg 𝔡(L|K).

The special case of L = K(𝔽_{q^{d}}) = 𝔽_{q^{d}}K is called the constant field extension of degree d over K. It is Galois with group G(L|K) ≃ G(𝔽_{q^{d}}|𝔽_{q}). For any place 𝔭 of K and 𝔮 of L above 𝔭, one has e(𝔮|𝔭) = 1 and 𝔽_{𝔮} = 𝔽_{q^{d}}𝔽_{𝔭}, i.e. there are exactly r := gcd(d, deg 𝔭) places 𝔮 of L lying above 𝔭, each of which has degree deg 𝔮 = deg 𝔭/r. By the Hurwitz Genus Formula, L has the same genus as K.

If 𝔭 is any place of K, then the absolute value map K → ℝ_{+}, z ↦ |z|_{𝔭} := q^{−(deg 𝔭)v_{𝔭}(z)} defines a metric on K. (Two functions z, z′ ∈ K are close to each other if v_{𝔭}(z − z′) is large.) In this way, K is made into a metrical field, i.e. addition, multiplication and the inversion z ↦ z^{−1} are continuous w.r.t. the defined topology. By the usual construction of “Cauchy sequences modulo zero sequences”, we obtain a complete field K_{𝔭} naturally containing K as a dense subfield, and by these properties it is uniquely determined up to Kisomorphism. K_{𝔭} is called the completion of K at 𝔭. The topological closure 𝔭̄ of 𝔭 in K_{𝔭} is the maximal ideal in the discrete valuation ring R𝔭̄ := R̄_{𝔭}, the topological closure of R_{𝔭} in K_{𝔭}. The residue field 𝔽𝔭̄ = R𝔭̄/𝔭̄ is naturally isomorphic with 𝔽_{𝔭} and v𝔭̄|_{K} = v_{𝔭}. Thus we can identify 𝔽𝔭̄ with 𝔽_{𝔭}, and there is no ambiguity in writing v_{𝔭} instead of v𝔭̄. Now and then we shall need the following form of the

1.5. Weak Approximation Theorem. Let S ⊆ ℙ_{K} be finite, and, for each 𝔭 ∈ S, let a (positive) integer m_{𝔭} and an element z_{𝔭} ∈ K_{𝔭} be given. Then there exists z ∈ K such that v_{𝔭}(z − z_{𝔭}) ≥ m_{𝔭} for all 𝔭 ∈ S.

This theorem is readily derived from its non-complete variant in [St2, p. 11] using the fact that K is dense in K_{𝔭}. An application of this theorem will usually be referred to in the text as “weak approximation”.

We return to the situation of a finite separable extension L|K. Let 𝔮 ∈ ℙ_{L} lie above 𝔭 ∈ ℙ_{K}. The natural inclusion K_{𝔭} ⊆ L_{𝔮} is compatible with the inclusions K ⊆ K_{𝔭} and K ⊆ L ⊆ L_{𝔮}, and one has L_{𝔮} = K_{𝔭}L. The “local extension” L_{𝔮}|K_{𝔭} has degree

[L_{𝔮} : K_{𝔭}] = e(𝔮|𝔭)[𝔽_{𝔮} : 𝔽_{𝔭}].

In particular, considering the constant field extension L = 𝔽_{q^{d}}K of degree d := deg 𝔭 over K, we see that 𝔽_{𝔭} = 𝔽_{𝔮} = 𝔽_{q^{d}} ⊆ L ⊆ L_{𝔮} = K_{𝔭}. This inclusion obviously identifies each residue class α ∈ 𝔽_{𝔭} = 𝔽𝔭̄ with one of its representatives in R𝔭̄ ⊆ K_{𝔭} and is therefore independent of the choice of 𝔮.

Given a sequence of elements z_{n} ∈ K_{𝔭}, n ∈ ℕ, the corresponding sequence (s_{m})_{m∈ℕ_{0}} of partial sums s_{m} := ∑_{n=1}^{m} z_{n} is Cauchy (and hence converges in K_{𝔭}) iff lim_{n→∞} z_{n} = 0, i.e. iff v_{𝔭}(z_{n}) tends to ∞. The limit of the sequence (s_{m})_{m∈ℕ_{0}} is denoted ∑_{n=1}^{∞} z_{n}. An element π ∈ K_{𝔭} satisfying v_{𝔭}(π) = 1, i.e. 𝔭̄ = R𝔭̄π, is called a uniformizer at 𝔭. The structure of K_{𝔭} is as follows (cf. [H, p. 155]).

1.6. Proposition. Let 𝔭 be a place of K and π ∈ K_{𝔭} a uniformizer at 𝔭. Then

K_{𝔭} = 𝔽_{𝔭}((π)) = {∑_{n=m}^{∞} α_{n}π^{n} | m ∈ ℤ and α_{n} ∈ 𝔽_{𝔭} for all n}.

For each z ∈ K_{𝔭}, the coefficients α_{n} ∈ 𝔽_{𝔭} of the expansion z = ∑_{n} α_{n}π^{n} at 𝔭 in π are uniquely determined by z; moreover, α_{n} = 0 for all n < m := v_{𝔭}(z), and α_{m} ∈ 𝔽_{𝔭}^{∗} is the residue class of z/π^{m} if z ≠ 0.

Proof. The proposition is probably better known in the case of 𝔭 ∈ ℙ_{K}^{1}, for which a proof is given in [St2, p. 143]. (Although there the proposition is only formulated with the uniformizer chosen inside K, the argument given works for any choice of π ∈ K_{𝔭}.) We reduce the general case of 𝔭 ∈ ℙ_{K}^{d} with d ∈ ℕ to the former by considering the constant field extension L := 𝔽_{q^{d}}K of degree d over K. Choose a place 𝔮 of L above 𝔭. Because e(𝔮|𝔭) = [𝔽_{𝔮} : 𝔽_{𝔭}] = 1, we have v_{𝔭} = v_{𝔮} on K_{𝔭} = L_{𝔮}, and π is a uniformizer at 𝔮 ∈ ℙ_{L}^{1} as well. Now, everything follows from the first case.

Expansions can in many cases be determined very easily from the defining equation of a function field as the following examples show.

1.7. Example. (a) Consider the function field K := 𝔽_{2}(x, y) where x is an indeterminate over 𝔽_{2} and y^{2} + y = x^{3}(x + 1)^{2}. By [St2, p. 115], the pole ∞ of x is the only place of 𝔽_{2}(x) ramified in K, the different exponent being d(𝔭_{0}|∞) = 6, where 𝔭_{0} ∈ ℙ_{K}^{1} is the place above ∞. Therefore g_{K} = 2 according to the Hurwitz Genus Formula 1.4. The other rational places of K are 𝔭_{1} := (0, 1), 𝔭_{2} := (1, 0), 𝔭_{3} := (1, 1) and 𝔭_{4} := (0, 0), where (α, β) with α, β ∈ 𝔽_{2} denotes the common zero of x − α and y − β. Since ℙ_{K}^{2} is empty, Z_{K}(t) = 1 + 5t + 15t^{2} + · · ·. Hence P_{K}(t) = (1 − 3t + 2t^{2})Z_{K}(t) = 1 + 2t + 2t^{2} + 4t^{3} + 4t^{4} and h_{K} = P_{K}(1) = 13 according to Theorem 1.2. For later purposes we need expansions at 𝔭_{4}. Let us take x as a uniformizer. Using the defining equation for K, we obtain

y = x^{3} + x^{5} + y^{2} = x^{3} + x^{5} + x^{6} + x^{10} + x^{12} + x^{20} + x^{24} + x^{40} + ⋯,

from which we can compute the expansion of any z ∈ K = 𝔽_{2}(x) ⊕ 𝔽_{2}(x)y at 𝔭_{4} in x.

(b) In the rational function field K = 𝔽_{8}(x), consider the zero 𝔭 ∈ ℙ_{K}^{2} of π := x^{2} + x + 1. Write 𝔽_{8}^{∗} = ⟨α⟩ with α^{3} = α + 1. Then the residue class ω := (αx + α^{5}) + 𝔭 ∈ 𝔽_{𝔭} satisfies ω^{9} = α, ω^{21} = x + 𝔭 and 𝔽_{𝔭}^{∗} = ⟨ω⟩. From x + ω^{21} = π + (x + ω^{21})^{2} ∈ 𝔭̄ ⊆ K_{𝔭} we derive the expansion

x = ω^{21} + π + π^{2} + π^{4} + π^{8} + π^{16} + π^{32} + π^{64} + π^{128} + ⋯

of x at 𝔭.

In the course of this work we shall perform several calculations within global function fields K|𝔽_{q}. Each of these is obtainable from a rational function field 𝔽_{q}(x) by at most two successive extensions of Kummer or Artin-Schreier type, whence the genus and the places of K can be determined according to the corresponding theorems in [St2]. However, since these are not the only calculations we have to do, we shall make use of KANT/KASH, a powerful number theory tool (for details see [K]), also capable of dealing with global function fields. It requires a global function field to be input in the form K = 𝔽_{q}(x, y), where y must be integral over the polynomial ring 𝔽_{q}[x]. It will then decompose places in K|𝔽_{q}(x) and compute the genus of K.

Let 𝔭 ∈ ℙ_{K} and let π ∈ K_{𝔭} be a uniformizer at 𝔭. There are certain subgroups of K_{𝔭}^{∗} which will play an important role in our computations concerning ray class fields, viz. the unit group

U_{𝔭}^{(0)} = U_{𝔭} := R∗ ; 𝔭̄ = {α_{0} + α_{1}π + α_{2}π^{2} + ⋯ | α_{0} ∈ 𝔽_{𝔭}^{∗}, α_{1}, α_{2}, … ∈ 𝔽_{𝔭}}

and, for any postive integer n, the n-th one-unit group at 𝔭,

U_{𝔭}^{(n)} := 1 + 𝔭̄^{n} = {1 + α_{n}π^{n} + α_{n+1}π^{n+1} + ⋯ | α_{n}, α_{n+1}, … ∈ 𝔽_{𝔭}}.

The first one-unit group U_{𝔭}^{(1)} is simply called the one-unit group. We note that we have a canonical isomorphism 𝔽_{𝔭}^{∗} ≃ U_{𝔭}/U_{𝔭}^{(1)}, α ↦ αU_{𝔭}^{(1)} and that, for positive integers m ≤ n, the factor group

U_{𝔭}^{(m)}/U_{𝔭}^{(n)} = {(1 + α_{m}π^{m} + ⋯ + α_{n−1}π^{n−1})U_{𝔭}^{(n)} | α_{m}, …, α_{n−1} ∈ 𝔽_{𝔭}}

has q^{(n−m) deg 𝔭} elements.

2 S-Units

Let K|𝔽_{q} be as before. We are concerned here with objects depending on a non-empty set S ⊆ ℙ_{K} of places of K. The S-units will play an important role in our treatment of the ray class fields. Some tools to compute them are developed in the following.

Let D_{S} be the group of divisors of K with support in S, and D_{S}^{0} := D_{S} ∩D_{K}^{0}. We write deg S for the (positive) greatest common divisor of the degrees of the places in S; hence deg(D_{S}) = ℤ deg S.

Denote by O_{S} = ⋂_{𝔭∉S} R_{𝔭} the ring of S-integral functions in K, i.e. those functions which have poles only in S. The group O_{S}^{∗} of invertible elements in O_{S} is called the group of S-units. By [FJ, p. 22] and [St2, p. 69] the map S ↦ O_{S} is a 1–1 correspondence between the non-empty subsets of ℙ_{K} and the Dedekind domains with field of fractions K. Accordingly, O_{S} is a Dedekind domain. Its ideal class group, Cℓ(O_{S}), which will be referred to as the S-class group, is, as usual, defined as the quotient of the group of non-zero fractional ideals of O_{S} by its subgroup of principal ideals zO_{S} with z ∈ K^{∗}. It has finite order h_{S} := |Cℓ(O_{S})|, called the S-class number.

The following proposition provides the connection between the S-class group and the group of S-units (cf. [Ro1]).

2.1. Proposition. The S-class group Cℓ(O_{S}) is naturally isomorphic to D_{K}/(D_{S} + (K^{∗})), and the sequence

0 → D_{S}^{0}/(O_{S}^{∗}) → D_{K}^{0}/(K^{∗}) → D_{K}/(D_{S} + (K^{∗})) →^{deg} ℤ/ℤ deg S → 0

is exact.

Proof. The canonical map D_{K} → Cℓ(O_{S}) taking a divisor ∑ m_{𝔭}𝔭 to the class of the fractional ideal ∏_{𝔭∉S}(𝔭∩O_{S})^{m_{𝔭}} is surjective with kernel D_{S} +(K^{∗}). The exactness of the sequence follows (from left to right) from the facts (O_{S}^{∗}) = D_{S}^{0}∩ (K^{∗}), D_{S}^{0} +(K^{∗}) = D_{K}^{0} ∩(D_{S} +(K^{∗})), D_{K}^{0} +D_{S} = {𝔪 ∈ D_{K} | deg 𝔪 ∈ ℤ deg S} and deg ℙ_{K} = 1 (F. K. Schmidt’s Theorem, see [FJ, p. 32] or [St2, p. 164]).

The index (D_{S}^{0} : (O_{S}^{∗})) is called the S-regulator and is denoted by reg_{S}. More generally, for any subgroup U of finite index in O_{S}^{∗} we define its regulator reg U as the index (D_{S}^{0} : (U)); hence reg_{S} = reg O_{S}^{∗}.

2.2. Corollary. We have reg_{S} h_{S} = h_{K} deg S, and (O_{S}^{∗}) ≃ O_{S}^{∗}/𝔽_{q}^{∗} is a free abelian group of rank |S| − 1 (Dirichlet’s Unit Theorem).

2 S-Units 15

We now turn to the practical computation of S-units. It is reasonable to restrict to the case of S being finite (and non-empty).

We call a family of functions u_{i} ∈ K^{∗}, i ∈ I, (multiplicatively) independent (over ℤ) if they are linearly independent in the ℤ-module K^{∗}. Obviously, (u_{i})_{i∈I} are independent if and only if the corresponding family of principal divisors (u_{i}) ∈ D_{K}^{0} are ℤ-linearly independent.

In all the examples that we shall treat in this thesis, it is not hard to find s := |S| − 1 independent S-units u_{1}, …, u_{s} simply by guessing. (A general algorithm, which does this automatically, is currently being implemented in KASH.) However, the principal divisors (u_{1}), …, (u_{s}) found in this way sometimes do not generate all of (O_{S}^{∗}). In order to check if they do and, if they do not, to see how to produce other S-units generating larger subgroups of O_{S}^{∗}, it is useful to determine the regulator.

2.3. Lemma. Let s := |S|−1 ∈ ℕ_{0} and write S = {𝔭_{0}, …, 𝔭_{s}}. Let u_{1}, …, u_{s} be independent S-units, U the subgroup of O_{S}^{∗} generated by them, and A_{0} := (v_{𝔭_{j}}(u_{i}))_{1≤i,j≤s} ∈ ℤ^{s×s}. Then

reg U = \frac{deg S}{deg 𝔭_{0}} |det A_{0}| .

This is a slight improvement compared to a result by Schörnig [Sg, p. 53], who only made use of the fact that reg U divides | det A_{0}|. Note that Schörnig’s definition of the regulator differs from ours by the factor ∏_{j=0}^{s} deg 𝔭_{j}/ deg S.

Proof. Set d_{j} := deg{𝔭_{0}, …, 𝔭_{j}} and b_{j} := d_{j−1}/d_{j}. Applying the submodule basis construction from the proof in [Wa, p. 3] of the well-known theorem on free modules over a principal ideal domain shows that there exists an upper triangular matrix B ∈ ℤ^{s×s} having b_{1}, …, b_{s} on its diagonal such that

(𝔟_{1}, …, 𝔟_{s}) := (𝔭_{0}, …, 𝔭_{s}) \binom{∗}{B}

is a ℤ-basis for D_{S}^{0}. Since Φ : D_{S}^{0} → ℤ^{s}, ∑_{i=0}^{s} c_{i}𝔭_{i} ↦ (c_{1}, …, c_{s}) is injective, we can conclude that

(D_{S}^{0} : (U)) = (ΦD_{S}^{0} : Φ(U)) = |\frac{det A_{0}}{det B} | = \frac{d_{s}}{d_{0}} |det A_{0}| .

2.4. Remark. In the situation of the lemma, let S′ be a non-empty subset of S. Then a ℤ-basis for U ∩ O_{S′} can be obtained as follows. Assume w.l.o.g. that S′ = {𝔭_{0}, …, 𝔭_{s′}} where s′ = |S′| − 1. Since ℤ is euclidean, we can compute T = (t_{ij})_{1≤i,j≤s} ∈ GL_{s}(ℤ) such that T A_{0} is lower triangular. Setting u′_{i} := ∏_{j=1}^{s} u_{j}^{t_{ij}}, it is easily seen that u′_{1}, …, u′_{s′} form a basis for U ∩ O_{S′}.

A basis for O_{S}^{∗}/𝔽_{q}^{∗} can now be constructed from u_{1}, …, u_{s} by applying the following proposition to each prime number l dividing reg U.

2.5. Proposition. In the situation of the previous lemma we set A := (v_{𝔭_{j}}(u_{i}))_{{1≤i≤s \atop 0≤j≤s}} ∈ ℤ^{s×(s+1)} and write ¯ to denote reduction modulo a given prime number l ∈ ℕ. Consider the group isomorphism

𝔲 : ℤ^{s} → (U) ≃ U, (m_{1}, …, m_{s}) ↦ (u_{1}^{m_{1}} ⋯ u_{s}^{m_{s}}).

The pre-images M := 𝔲^{−1}((U) ∩ (K^{∗}^{l})) and N := 𝔲^{−1}((U) ∩ lD_{K}) satisfy lℤ^{s} ⊆ M ⊆ N, hence M̄ := M/lℤ^{s} ⊆ N̄ := N/lℤ^{s} ⊆ 𝔽_{l}^{s}.

(a) N̄ is the kernel of 𝔽_{l}^{s} → 𝔽_{l}^{s+1}, ν ↦ νĀ.

(b) M̄ = 0 ⟺ l |/ ((O_{S}^{∗}) : (U)).

(c) Let m ∈ N and z ∈ K^{∗}. Then 𝔲(m) = (z^{l}) iff z ∈ L(−𝔲(m)/l) \ {0}.

(d) The dimension of N̄ equals the l-rank of D_{S}^{0}/(U), which in turn is bounded by v_{l}(reg U).

Proof. Note that lℤ^{s} is the 𝔲-pre-image of (U^{l}) ⊆ (U) ∩ (K^{∗}^{l}).

(a) For m ∈ ℤ^{s}, by definition, mA ∈ ℤ^{s+1} is the row vector whose components are the coefficients of the divisor 𝔲(m). Reduction modulo l shows the assertion.

(b) l|((O_{S}^{∗}) : (U)) means that there exists z ∈ K^{∗} satisfying (z^{l}) ∈ (U) and (z) ∉ (U), i.e. 𝔲^{−1}(l(z)) ∈ M \ lℤ^{s}.

(c) This is clear from the definition of Riemann-Roch space and because 𝔲 has values in D_{K}^{0}.

(d) Recall that the l-rank of an abelian group G is defined as the 𝔽_{l}-dimension of G/lG, and that it is ≤ v_{l}(|G|) if G is finite. Now 𝔲 induces a surjective 𝔽_{l}- homomorphism 𝔽_{l}^{s} → ((U) + lD_{S}^{0})/lD_{S}^{0}, whose kernel is N̄. Hence

dim_{𝔽_{l}} N̄ = s − dim_{𝔽_{l}}((U) + lD_{S}^{0})/lD_{S}^{0} = dim_{𝔽_{l}} D_{S}^{0}/((U) + lD_{S}^{0}),

but D_{S}^{0}/((U) + lD_{S}^{0}) ≃ \frac{D_{S}^{0}/(U)}{l(D_{S}^{0}/(U))}.

The proposition suggests an algorithm that we want to describe briefly and informally in the following. First we determine N̄ from the matrix A by means of 2.5(a). In order to find out whether l divides ((O_{S}^{∗}) : (U)), by 2.5(b) it suffices to test the condition

𝔲(m) ∈ (K^{∗l}) (1) with m̄ running through a system of representatives for the orbits {𝔽_{l}^{∗}ν | ν ∈ N̄ \ {0}}, since M̄ is a subspace of N̄. These are (|N̄| − 1)/(l − 1) tests, so we have an obvious improvement of the algorithm in [Sg, p. 54], which contains a loop over l^{s} − 1 elements. This simplification has the additional advantage that we can choose the vectors m with at least one component, say m_{i}, equal to 1, so that, in case (1) holds, we simply replace u_{i} by z ∈ L(−𝔲(m)/l) \ {0} and thus obtain a larger subgroup U′ of O_{S}^{∗} satisfying (U′ : U) = l, to which we can apply the described procedure anew.

Due to 2.5(c), each test (1) comes down to determining a Riemann-Roch space (of dimension 0 or 1), a task for which a KANT algorithm has been implemented by Heß [He]. If l equals the characteristic p of K, then (1) can be tested much easier, since taking p-th roots in K is a simple exercise in linear algebra. Recall that the p-th power map in characteristic p is an injective ring homomorphism, i.e. a p-th root is unique (if it exists), and that the p-th root of α ∈ 𝔽_{q} is α^{q/p}.

2.6. Lemma. Let (y_{1}, …, y_{n}) be a basis for the characteristic p field extension K|k. We form the transition matrix W = (w_{ij})_{1≤i,j≤n} ∈ k^{n×n} with y_{i}^{p} = ∑_{j=1}^{n} w_{ij}y_{j}.

(a) The matrix W is invertible if and only if K|k is separable.

(b) Let u = ∑_{i=1}^{n} f_{i}y_{i} and z = ∑_{i=1}^{n} g_{i}y_{i} with f_{1}, …, f_{n}, g_{1}, …, g_{n} ∈ k. Then u = z^{p} is equivalent to (f_{1}, …, f_{n}) = (g_{1}^{p}, …, g_{n}^{p})W.

(c) The p-th power image of 𝔽_{q}(x), x an indeterminate of 𝔽_{q}, is 𝔽_{q}(x^{p}).

Proof. (a) W is invertible iff y_{1}^{p}, …, y_{n}^{p} are linearly independent over k, which amounts to saying that K^{p} and k are linearly disjoint over k^{p} (here the superscript p denotes p-th power); but this means that K|k is separable.

(b) and (c) are immediately verified.

This lemma can readily be applied to any global function field K|𝔽_{q} because, by [St2, p. 128], there always exists x ∈ K \ 𝔽_{q} such that K|𝔽_{q}(x) is separable.

2.7. Example. Let K be the function field 𝔽_{4}(x, y), where x is an indeterminate over 𝔽_{4} and

y^{4} + (x^{2} + x + 1)y^{2} + (x^{2} − x)y = x^{3} − 1.

Because u = (y^{2} − y)/(x^{2} − x) satisfies u^{2} − u = \frac{x^{2}+x+1}{x^{2}(x−1)}, K is obtainable from 𝔽_{4}(x) by two successive Artin-Schreier extensions. Moreover K|𝔽_{4}(x) is Galois. The Galois group is generated by the two automorphisms which take y to y + 1 and y + x, respectively, and hence is abelian of type (2, 2). Choose α ∈ 𝔽_{4} \ 𝔽_{2}. The following facts are easily verified using [St2, p. 115]. Exactly those places of K associated to x, x−1 and the pole ∞ of x are ramified in K, each with ramification index 2 and inertia degree 1, and K has genus 3 and 14 rational places, namely

𝔭_{0} := (0, α), 𝔭_{1} := (0, α^{2}), 𝔭_{2} := (1, 0), 𝔭_{3} := (1, 1),
𝔭_{4} := (α, 0), 𝔭_{5} := (α, 1), 𝔭_{6} := (α, α), 𝔭_{7} := (α, α^{2}),
𝔭_{8} := (α^{2}, 0), 𝔭_{9} := (α^{2}, 1), 𝔭_{10} := (α^{2}, α), 𝔭_{11} := (α^{2}, α^{2}),
𝔭_{12}, 𝔭_{13},

where (α, β) with α, β ∈ 𝔽_{4} denotes the place of K which is the common zero of x−α and y−β, and 𝔭_{12} and 𝔭_{13} are the zeros of y/x and y/x−1, respectively, which lie above ∞. Let S := {𝔭_{0}, …, 𝔭_{13}} and “guess” S-units

(u_{1}, …, u_{13}) = (x, x + α, x + α^{2}, y, y + 1, y + α, y + α^{2}, y + x, y + x + α, y + αx, y + αx + α, y + α^{2}x, y + α^{2}x + α);

then

A := (v_{𝔭_{j}}(u_{i}))_{{1≤i≤13 \atop 0≤j≤13}} = ( \begin{matrix}2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −2 & −2 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & −2 & −2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & −2 & −2 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & −1 & −2 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & −1 & −2 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & −1 & −2 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & −1 & −2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & −2 & −1 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & −2 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & −2 & −2 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & −2 & −2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & −2 & −2 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & −2 & −2 \end{matrix} ) .

Without regard to sign, any 13×13-minor of A (erase one column and compute the determinant) equals 512, which by Lemma 2.3 is the regulator of U := ⟨u_{1}, …, u_{13}⟩. We want to test whether 2 divides ((O_{S}^{∗}) : (U)) by means of Propositon 2.5 and find N̄ = 𝔽_{2}e_{1} ⊕ 𝔽_{2}e_{10} ⊕ 𝔽_{2}e_{12}, where e_{i} denotes the i-th canonical basis vector in 𝔽_{2}^{13}. Hence we have to search for squares in the set

V :={u_{1}^{m_{1}}u_{10}^{m_{10}}u_{12}^{m_{12}} | (0, 0, 0) ≠ (m_{1}, m_{10}, m_{12}) ∈ {0, 1}^{3}}
={x, y + αx, y + α^{2}x, xy + αx^{2}, xy + α^{2}x^{2},
y^{2} + xy + x^{2}, xy^{2} + x^{2}y + x^{3}}.

Applying Lemma 2.6 with (1, y, y^{2}, y^{3}) as a basis for K|𝔽_{4}(x), i.e.

W^{−1} = \frac{1}{x^{2} − x} ( \begin{matrix}x^{2} − x & 0 & 0 & 0 \\ x^{3} − 1 & x^{2} + x + 1 & 1 & 0 \\ 0 & x^{2} − x & 0 & 0 \\ 0 & x^{3} − 1 & x^{2} + x + 1 & 1 \end{matrix} ) ,

shows that in fact no element of V is a square in K and hence proves O_{S}^{∗} = 𝔽_{4}^{∗}U and reg_{S} = 512.

If we have information about the S-class number and the divisor class number, we can determine the regulator reg_{S} directly or estimate it from below by means of 2.2. One such technique is demonstrated in the following example continued from Section 1.

2.8. Example. Let K = 𝔽_{2}(x, y) and 𝔭_{0}, …, 𝔭_{4} as in Example 1.7(a). Suppose S ⊆ ℙ_{K}^{1} contains at least two elements, say 𝔭 ≠ 𝔭′, then 𝔭−𝔭′ ∈ D_{S}^{0} \(K^{∗}) by Theorem 1.1, and consequently reg_{S} > 1. Let 0 ≤ k ≤ 4 and consider the set S_{k} := {𝔭_{0}, …, 𝔭_{k}}. Then reg_{S_{k}} = h_{K} = 13 and h_{S_{k}} = 1 for k ≥ 1 by Corollary 2.2. We “guess” S_{4}-units u_{1} := x, u_{2} := x + 1, u_{3} := y and u_{4} := y + x^{2} and set

A := (v_{𝔭_{j}}(u_{i}))_{{1≤i≤4 \atop 0≤j≤4}} = ( \begin{matrix}−2 & 1 & 0 & 0 & 1 \\ −2 & 0 & 1 & 1 & 0 \\ −5 & 0 & 2 & 0 & 3 \\ −5 & 0 & 0 & 3 & 2 \end{matrix} ) and T := ( \begin{matrix}13 & 6 & −3 & −2 \\ 5 & 3 & −1 & −1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) ∈ GL_{4}(ℤ).

By Lemma 2.3,

T A = ( \begin{matrix}−13 & 13 & 0 & 0 & 0 \\ −6 & 5 & 1 & 0 & 0 \\ −2 & 0 & 1 & 1 & 0 \\ −2 & 1 & 0 & 0 & 1 \end{matrix} )

shows that O_{S_{4}}^{∗} = ⟨u_{1}, u_{2}, u_{3}, u_{4}⟩. Moreover, writing T = (t_{ij})_{1≤i,j≤4} and setting u′_{k} := ∏_{j=1}^{4} u_{j}^{t_{kj}}, we obtain O_{S_{k}}^{∗} = ⟨u′_{1}, …, u′_{k}⟩ according to Remark 2.4. This example is continued in Section 6.

3 Ramification Groups

We shall recall some basics of Hilbert’s ramification theory as found in [Nk, pp. 186ff] or [St2, pp. 118ff] and prove an upper index version of Hilbert’s Different Formula for abelian extensions by means of the Hasse-Arf Theorem. Although we restrict ourselves to the function field case here, everything in this section, except for Example 3.3 and Corollary 3.5, carries over literally to number fields by merely requiring the “places” (i.e. primes) under consideration to be non-archimedean.

Let K|𝔽_{q} be as before, K̄ a fixed algebraic closure of K, L|K a finite Galois extension with group G(L|K), 𝔭 a place of K and 𝔮 a place of L lying above 𝔭. It is well-known that G(L|K) acts transitively on the set of all places of L above 𝔭. For s ∈ [−1, ∞) there is the s-th lower ramification group of 𝔮|𝔭,

G_{s}(𝔮|𝔭) := {σ ∈ G(L|K) | v_{𝔮}(σz − z) ≥ s + 1 ∀z ∈ R_{𝔮}}.

We note that G_{−1}(𝔮|𝔭) = {σ ∈ G | σ𝔮 = 𝔮} is the decomposition group and G_{0}(𝔮|𝔭) is the inertia group of 𝔮|𝔭. See [St2, p. 119] for their specific properties. After renumbering by means of the continuous and strictly isotonous bijection

η = η_{𝔮|𝔭} : \begin{matrix}[−1, ∞) & → & [−1, ∞) \\ s & ↦ & ∫_{0}^{s} \frac{|G_{x}(𝔮|𝔭)|}{|G_{0}(𝔮|𝔭)|}dx, \end{matrix}

we get the upper ramification groups G^{η(s)}(𝔮|𝔭) := G_{s}(𝔮|𝔭). (Note that η is the identity in any case on [−1, 0].) Whereas, by definition, the lower numbering is stable under modification of the lower field K (see [Nk, p. 187]), the upper numbering is compatible with a change of the upper field (see [Nk, p. 190]):

3.1. Proposition. For any Galois subextension L′|K of L|K with 𝔮′ := 𝔮∩L′, the restriction map G(L|K) → G(L′|K) sends G^{s}(𝔮|𝔭) onto G^{s}(𝔮′|𝔭) for all s ∈ [−1, ∞).

Analogously we can define the ramification groups of the local extension L_{𝔮}|K_{𝔭}, which we denote by G_{s}(L_{𝔮}|K_{𝔭}) and G^{s}(L_{𝔮}|K_{𝔭}) for the moment. Clearly, G_{−1}(L_{𝔮}|K_{𝔭}) = G^{−1}(L_{𝔮}|K_{𝔭}) = G(L_{𝔮}|K_{𝔭}). Since the restriction map G(L_{𝔮}|K_{𝔭}) → G(L|K) is injective and sends G_{s}(L_{𝔮}|K_{𝔭}) onto G_{s}(𝔮|𝔭) and G^{s}(L_{𝔮}|K_{𝔭}) onto G^{s}(𝔮|𝔭) for all s ∈ [−1, ∞) (see [CF, p. 41]), we shall identify these groups with one another and drop the notations G_{s}(L_{𝔮}|K_{𝔭}) and G^{s}(L_{𝔮}|K_{𝔭}) again.

The canonical surjection G^{−1}(𝔮|𝔭) ↠ G(𝔽_{𝔮}|𝔽_{𝔭}) has as kernel the inertia group G^{0}(𝔮|𝔭). Hence, in the case that 𝔮|𝔭 is unramified, there is a unique element φ_{𝔮|𝔭} ∈ G^{−1}(𝔮|𝔭) satisfying

φ_{𝔮|𝔭}z ≡ z^{q^{deg 𝔭}} mod 𝔮

for all z ∈ R_{𝔮}, called the Frobenius automorphism of 𝔮|𝔭 (cf. [CF, p. 164]), which will be used in Section 4. Note that φ_{σ𝔮|𝔭} = σφ_{𝔮|𝔭}σ^{−1} for σ ∈ G(L|K).

Since G_{s}(σ𝔮|𝔭) = σG_{s}(𝔮|𝔭)σ^{−1} for any automorphism σ ∈ G(L|K) and s ∈ [−1, ∞), the orders of the (lower and upper) ramification groups do not depend on the specific choice of 𝔮. Therefore, we can define the conductor exponent f(L, 𝔭) of 𝔭 in L as the least integer m ≥ 0 such that G^{m}(𝔮|𝔭) is trivial. The effective divisor

𝔣(L|K) := ∑_{𝔭∈ℙ_{K}} f(L, 𝔭)𝔭

of K is called the conductor of L|K. Because G^{0}(𝔮|𝔭) is the inertia group of 𝔮|𝔭, the places of K occurring in 𝔣(L|K) are exactly those which ramify in L. The Conductor Lemma in Section 5 below shows that in the abelian case 𝔣(L|K) is indeed the conductor of L|K in the sense of class field theory.

We also define the splitting set S(L|K) ⊆ ℙ_{K} of L|K as the set consisting of those places 𝔭 of K which split completely in L. Due to the properties of the decomposition group we have

(2) S(L|K) = {𝔭 ∈ ℙ_{K} | G^{−1}(𝔮|𝔭) = 1 for 𝔮 ∈ ℙ_{L} above 𝔭}.

As a consequence of Proposition 3.1, the splitting set and the conductor enjoy the following properties:

3.2. Corollary. Let L_{1} and L_{2} be two finite Galois extensions of K within K̄.

(a) If L_{1} ⊆ L_{2} then S(L_{1}|K) ⊇ S(L_{2}|K) and 𝔣(L_{1}|K) ≤ 𝔣(L_{2}|K).

(b) Let L := L_{1}L_{2}. Then S(L|K) = S(L_{1}|K) ∩ S(L_{2}|K), and 𝔣(L|K) = max{𝔣(L_{1}|K), 𝔣(L_{2}|K)}, where the maximum is taken coefficientwise.

Proof. Let s ∈ [−1, ∞) and 𝔭 ∈ ℙ_{K}.

(a) Choose 𝔮_{1} ∈ ℙ_{L_{1}} and 𝔮_{2} ∈ ℙ_{L_{2}} such that 𝔮_{2}|𝔮_{1}|𝔭. By 3.1, the triviality of G^{s}(𝔮_{2}|𝔭) implies that of G^{s}(𝔮_{1}|𝔭). Hence f(L_{1}, 𝔭) ≤ f(L_{2}, 𝔭) by definition of the conductor exponent.

(b) Choose a place 𝔮 of L above 𝔭 and let 𝔮_{1} := 𝔮 ∩ L_{1} and 𝔮_{2} := 𝔮 ∩ L_{2}. From 3.1 we see that G^{s}(𝔮|𝔭) is trivial iff G^{s}(𝔮_{1}|𝔭) and G^{s}(𝔮_{2}|𝔭) are trivial. Hence f(L, 𝔭) = max{f(L_{1}, 𝔭), f(L_{2}, 𝔭)}. The assertion on the splitting sets follows with (2).

As an application we give the following

3.3. Example (Artin-Schreier Extensions). Let ℘ : K̄ → K̄, z ↦ z^{p} − z, be the Artin-Schreier operator and define the Artin-Schreier reduced valuation of a function u ∈ K at a place 𝔭 ∈ ℙ_{K} by

v_{𝔭}^{∗}(u) := max{v_{𝔭}(u − ℘t) | t ∈ K} ∈ ℤ ∪ {∞}.

Let U ⊆ K be finite and L := K(℘^{−1}U) ⊆ K̄.

(a) For u ∈ K with v_{𝔭}(u) < 0 we have the equivalence

v_{𝔭}(u) = v_{𝔭}^{∗}(u) ⟺ v_{𝔭}(u) ≢ 0 mod p.

(b) The extension L|K has conductor

𝔣(L|K) = ∑_{m_{𝔭}>0}(m_{𝔭} + 1)𝔭,

where m_{𝔭} := − min_{u∈U} v_{𝔭}^{∗}(u) for 𝔭 ∈ ℙ_{K}.

(c) The splitting set of L|K is S(L|K) = {𝔭 ∈ ℙ_{K} | v_{𝔭}^{∗}(u) > 0 ∀u ∈ U}.

(d) The degree of L|K is [L : K] = (V + ℘K : ℘K), where V := ∑_{u∈U} 𝔽_{p}u is the 𝔽_{p}-subspace of K spanned by U.

Proof. (a) See [St2, p. 114].

(b) Let u ∈ U. We show that K_{u} := K(℘^{−1}u) ⊆ L satisfies

𝔣(K_{u}|K) = ∑_{v_{𝔭}^{∗}(u)<0}(1 − v_{𝔭}^{∗}(u))𝔭. (3)

If u ∈ ℘K, then K_{u} = K and v_{𝔭}^{∗}(u) = ∞ for all 𝔭 ∈ ℙ_{K}, and we are done. Otherwise K_{u}|K is Galois of degree p and we can use Proposition III.7.8 of [St2, p. 115]. Let 𝔭 ∈ ℙ_{K}. If v_{𝔭}^{∗}(u) ≥ 0, then 𝔭 is unramified in L, i.e. f(K_{u}, 𝔭) = 0. In case m := −v_{𝔭}^{∗}(u) > 0, exactly one place 𝔮 ∈ ℙ_{K_{u}} lies over 𝔭 (with e(𝔮|𝔭) = p, i.e. 𝔭 is totally ramified in K_{u}) and the different exponent equals

d(𝔮|𝔭) = (p − 1)(m + 1).

A comparison with Hilbert’s Different Formula (see [St2, p. 124])

d(𝔮|𝔭) = ∑_{n=0}^{∞}(|G_{n}(𝔮|𝔭)| − 1)

yields G(K_{u}|K) = G_{0}(𝔮|𝔭) = ⋯ = G_{m}(𝔮|𝔭) and G_{m+1}(𝔮|𝔭) = 1. Hence the bijection η_{𝔮|𝔭} is the identity on [−1, m] and η_{𝔮|𝔭}(s) = m + \frac{s−m}{p} for s ∈ [m, ∞). We conclude that G^{m}(𝔮|𝔭) = G_{m}(𝔮|𝔭) = G(K_{u}|K) and G^{m+1}(𝔮|𝔭) = G_{m+p}(𝔮|𝔭) = 1; hence, f(K_{u}, 𝔭) = m + 1 = 1 − v_{𝔭}^{∗}(u) by the definition of the conductor exponent. Thus we have established (3) for all u ∈ U, and the formula for 𝔣(L|K) follows from 3.2(b) since L = ∏_{u∈U} K_{u}.

(c) Let 𝔭 ∈ ℙ_{K} and u ∈ U. We write K_{u} = K(℘^{−1}u) = K(y) with y ∈ ℘^{−1}u and show

v_{𝔭}^{∗}(u) > 0 ⟺ 𝔭 ∈ S(K_{u}|K).

Suppose v_{𝔭}^{∗}(u) > 0. Then there exists t ∈ K such that ỹ := y − t satisfies ỹ^{p}−ỹ = u−℘t ≡ 0 mod 𝔭. Hence 𝔭 splits completely in K_{u} = K(ỹ) according to [St2, p. 76]. Conversely, suppose 𝔭 splits completely in K_{u}. Then we can assume v_{𝔭}(u) ≥ 0 by the proof of (b) and, using [St2, pp. 96,76], conclude that u ≡ ℘t mod 𝔭 for some t ∈ R_{𝔭}; hence v_{𝔭}^{∗}(u) ≥ v_{𝔭}(u − ℘t) > 0. The assertion follows with 3.2(b).

(d) This is due to the Kummer theory of the operator ℘ (also called Artin-Schreier theory, see [Nk, p. 294]).

By Čebotarev’s Density Theorem^{1}, S(L|K) and, for [L : K] > 1, also its complement ℙ_{K} \ S(L|K) is an infinite set. Another well-known consequence of this deep theorem is the fact that L is uniquely determined by S(L|K) as a finite Galois extension of K. More precisely, sharpening 3.2(a), we have

3.4. Theorem. Let L_{1} and L_{2} be finite Galois extensions of K within K̄. Then

L_{1} ⊆ L_{2} ⟺ S(L_{1}|K) ⊇ S(L_{2}|K).

Proof. See [CF, p. 362].

From this we can derive the following generalization of F. K. Schmidt’s Theorem:

3.5. Corollary. For d ∈ ℕ, let S_{d} = ⋃_{n=1}^{∞} ℙ_{K}^{nd} be the set of all places of K|𝔽_{q} whose degree is divisible by d, and K_{d} = 𝔽_{q^{d}}K ⊆ K̄ the constant field extension of degree d over K. Then S_{d} = S(K_{d}|K), deg S_{d} = d, and K_{d} is the largest Galois extension L of K within K̄ such that S_{d} ⊆ S(L|K).

Proof. Clearly, 𝔭 ∈ ℙ_{K} splits completely in K_{d}|K if and only if d divides deg 𝔭. To verify the second assertion, set d′ := deg S_{d}. Then S_{d} = S_{d′}, which by Theorem 3.4 implies K_{d} = K_{d′} and hence d = d′. Finally, let L ⊆ K̄ be finite Galois over K with S_{d} = S(K_{d}|K) ⊆ S(L|K). Then, again by Theorem 3.4, L ⊆ K_{d}.

In what follows, L|K will be assumed finite abelian. The Frobenius automorphism and the ramification groups no longer depend on 𝔮, so we can write φ_{L,𝔭} instead of φ_{𝔮|𝔭} and G^{s}(L, 𝔭) instead of G^{s}(𝔮|𝔭). The subfield of L fixed by G^{s}(L, 𝔭) is denoted L^{s}(𝔭) and called the s-th upper ramification field of 𝔭 in L. In particular, L^{−1}(𝔭) is the decomposition field and L^{0}(𝔭) is the inertia field of 𝔭 in L (cf. [St2, p. 118]). As we shall see later, the upper ramification fields have an interpretation in terms of ray class fields. In view of this, it will turn out useful to have an upper index version of Hilbert’s

1We do not want to formulate the theorem itself here. Instead, we state some of its consequences which are not essential for what follows but produce illustrative examples.

3.6. Different Formula. The different exponent of 𝔮|𝔭 satisfies

d(𝔮|𝔭) = ∑_{n=0}^{∞} (|G^{0}(L, 𝔭)| − (G^{0}(L, 𝔭) : G^{n}(L, 𝔭)))

= ∑_{n=0}^{∞} ([L : L^{0}(𝔭)] − [L^{n}(𝔭) : L^{0}(𝔭)]) .

Proof. For convenience, put G^{s} := G^{s}(L, 𝔭), η := η_{𝔮|𝔭} and ψ := η^{−1}. Set t_{0} := −1 and let 0 ≤ t_{1} < ⋯ < t_{r} be the (other) jumps of the filtration (G^{s})_{s≥−1}, i.e. the numbers s ≥ −1 satisfying G^{s+ε} ⊊ G^{s} for all ε > 0. Obviously, all ψ(t_{i}) are integers, and t_{0} = −1 is a jump iff [𝔽_{𝔮} : 𝔽_{𝔭}] > 1. The bijection η is piecewise linear, its slopes being

η′(s) = \frac{1}{(G^{0} : G^{t_{i}})} = \frac{t_{i} − t_{i−1}}{ψ(t_{i}) − ψ(t_{i−1})}

for ψ(t_{i−1}) < s < ψ(t_{i}) and i ∈ {1, …, r}. By the Hasse-Arf Theorem (see [S1, p. 93]^{2}), the t_{i} are integers, too; in particular, t_{r} = f(L, 𝔭) − 1. Substituting the above equation into Hilbert’s (lower index) Different Formula (see [St2, p. 124] or [CF, p. 36]) yields

d(𝔮|𝔭) = ∑_{n=0}^{∞}(|G_{n}(𝔮|𝔭)| − 1) = ∑_{i=1}^{r} ∑_{s=ψ(t_{i−1})+1}^{ψ(t_{i})} (|G^{t_{i}} | − 1)

= ∑_{i=1}^{r}(ψ(t_{i}) − ψ(t_{i−1}))\frac{|G^{0}| − (G^{0} : G^{t_{i}})}{(G^{0} : G^{t_{i}})}

= ∑_{i=1}^{r}(t_{i} − t_{i−1}) (|G^{0}| − (G^{0} : G^{t_{i}}))

= ∑_{i=1}^{r} ∑_{n=t_{i−1}+1}^{t_{i}} (|G^{0}| − (G^{0} : G^{n})) = ∑_{n=0}^{∞} (|G^{0}| − (G^{0} : G^{n})) .

2We prefer Serre’s direct proof to one deduced from local class field theory (see [Nk, p. 372] or [CF, p. 157]), which is merely touched upon in our next section.

As an immediate consequence we obtain the following

3.7. Discriminant Formula. Let 𝔣(L|K) ≤ 𝔪 = ∑_{𝔭} m_{𝔭}𝔭 ∈ D_{K}. Then the discriminant of L|K is given by

𝔡(L|K) = [L : K]𝔪 − ∑_{𝔭∈ℙ_{K}}( ∑_{n=0}^{m_{𝔭}−1}[L^{n}(𝔭) : K])𝔭.

Proof. Since L^{0}(𝔭) is the inertia field of 𝔭 in L, we have ∑_{𝔮|𝔭}[𝔽_{𝔮} : 𝔽_{𝔭}] = [L^{0}(𝔭) : K] for each 𝔭 ∈ ℙ_{K}. The discriminant is the norm of the different. Hence 𝔡(L|K) = ∑_{𝔭} d(L, 𝔭)𝔭 with

d(L, 𝔭) = ∑_{𝔮|𝔭}[𝔽_{𝔮} : 𝔽_{𝔭}]d(𝔮|𝔭)

= [L^{0}(𝔭) : K] ∑_{n=0}^{m_{𝔭}−1} ([L : L^{0}(𝔭)] − [L^{n}(𝔭) : L^{0}(𝔭)])

= [L : K]m_{𝔭} − ∑_{n=0}^{m_{𝔭}−1}[L^{n}(𝔭) : K].

Part II

Ray Class Fields

This part is devoted entirely to the definition and investigation of the central objects of this thesis, the ray class fields K_{S}^{𝔪}, where S is a non-empty set of places and 𝔪 an effective divisor of K with support away from S. Their existence is based on (global) class field theory, of which we shall give a brief survey in Section 4.

The extension K_{S}^{𝔪}|K will be characterized as the largest abelian extension of K with conductor ≤ 𝔪 such that every place in S splits completely. This and other properties of the ray class fields are discussed in Section 5. Here we also see that everything comes down to knowing the degrees [K_{S}^{𝔪} : K], the determination of which is therefore attacked in Section 6.

In the whole of Part II we assume again that K|𝔽_{q} is a global function field. Moreover, we fix an algebraic closure K̄ of K, inside which all algebraic extensions of K are chosen.

4 Class Field Theory

The purpose of this section is to make available the necessary tools from class field theory. A complete exposition of this theory, as presented in [CF] and [AT], would be beyond the scope of this thesis. Nevertheless, we want to give an impression of how the results depend on each other and how the proofs are arranged. A general introduction to topological groups, which play a central role here, is found e.g. in [Lu].

We recall from Section 1 that for each place 𝔭 of K the multiplicative group K_{𝔭}^{∗} is a topological group, in which the subgroups U_{𝔭}^{(n)} with n ∈ ℕ_{0} form a basis of open neighbourhoods of 1. The idèle group I_{K} of K is defined as the restricted topological product of the K_{𝔭}^{∗}, 𝔭 ∈ ℙ_{K}, with respect to their subgroups U_{𝔭} (see [CF, p. 68]). In other words,

I_{K} := {(α_{𝔭})_{𝔭∈ℙ_{K}} ∈ ∏_{𝔭∈ℙ_{K}} K_{𝔭}^{∗} | α_{𝔭} ∈ U_{𝔭} for almost all 𝔭 ∈ ℙ_{K}},

and the topology on I_{K} is the initial topology induced by the canonical projections I_{K} ↠ K_{𝔭}^{∗}, 𝔭 ∈ ℙ_{K}. Therefore, a subset of I_{K} is open iff it is a union of sets of the form ∏_{𝔭∈ℙ_{K}} V_{𝔭} with V_{𝔭} ⊆ K_{𝔭}^{∗} open for all 𝔭 ∈ ℙ_{K} and V_{𝔭} = U_{𝔭} for almost all 𝔭 ∈ ℙ_{K}. It is readily verified that I_{K} is a topological group.

The canonical embedding of K_{𝔭}^{∗} into I_{K} will be expressed by the symbol [ ]_{𝔭}, i.e. [z]_{𝔭} for z ∈ K_{𝔭}^{∗} is the idèle (…, 1, z, 1, …) having z at its 𝔭-th position and 1 elsewhere. Clearly, the morphism [ ]_{𝔭} preserves the topology, i.e. is a homeomorphism onto its image. The diagonal embedding of K^{∗} into I_{K} is considered as inclusion. The factor group C_{K} = I_{K}/K^{∗} is called the idèle class group of K and carries the quotient topology induced from I_{K}, i.e. the canonical projection I_{K} ↠ C_{K} is continuous and open.

Let L be finite separable over K. The inclusion K^{∗} ⊆ L^{∗} is extended to idèles by identifying α = (α_{𝔭})_{𝔭∈ℙ_{K}} ∈ I_{K} with (α_{𝔮})_{𝔮∈ℙ_{L}} ∈ I_{L}, where α_{𝔮} = α_{𝔭} for 𝔮|𝔭. Since L^{∗} ∩I_{K} = K^{∗}, it induces an inclusion C_{K} ⊆ C_{L}. Both inclusions preserve the respective topologies. Also the norm N_{L|K} from L^{∗} to K^{∗} is extended to idèles by the continuous morphism

N_{L|K} : \begin{matrix}I_{L} & → & I_{K} \\ (β_{𝔮})_{𝔮∈ℙ_{L}} & ↦ & (∏_{𝔮|𝔭} N_{L_{𝔮}|K_{𝔭}}β_{𝔮})_{𝔭∈ℙ_{K}} \end{matrix}

(see [CF, p. 75]). Since N_{L|K}L^{∗} ⊆ K^{∗}, it induces a norm map N_{L|K} : C_{L} → C_{K}, which is continuous, too. Finally, the norm is obviously transitive, i.e. if L′ is an intermediate field of L|K, then N_{L|K} = N_{L′|K} ∘ N_{L|L′}.

Until the end of this section we assume L|K to be finite abelian. Consider G(L|K) as a topological group equipped with the discrete topology. Class field theory asserts the existence of a canonical group morphism built up from the Frobenius automorphisms defined in Section 3.

4.1. Reciprocity Law. (a) Let S ⊆ ℙ_{K} be a finite set of places of K including all those which ramify in L. Then there is a unique continuous homomorphism

( , L|K) : I_{K} → G(L|K)

such that (K^{∗}, L|K) = 1 and

(α, L|K) = ∏_{𝔭∈ℙ_{K}\S} φ_{L,𝔭}^{v_{𝔭}(α_{𝔭})}

for each α = (α_{𝔭})_{𝔭∈ℙ_{K}} ∈ I_{K} satisfying α_{𝔭} = 1 for all 𝔭 ∈ S.

(b) The map ( , L|K) is surjective with kernel K^{∗}N_{L|K}I_{L}.

(c) If L′ is an intermediate field of L|K, then (α, L′|K) = (α, L|K)|_{L′} for all α ∈ I_{K}.

The homomorphism ( , L|K) is called the Artin map or the norm residue symbol of L|K.

Remarks on the proof. We can follow [CF, pp. 170–193]; only for the part concerning Artin-Schreier extensions we refer to [AT, pp. 29–37].

The uniqueness of ( , L|K) follows from the continuity by means of weak approximation and shows that the definition of the Artin map does not depend on the exceptional set S. Also, (c) can easily be derived from this uniqueness, noting that φ_{L′,𝔭} = φ_{L,𝔭}|_{L′}. The behaviour of the Frobenius automorphism in towers, by applying weak approximation once more, implies that N_{L|K}I_{L} is contained in the kernel of ( , L|K).

In order to prove (b) and the existence of the Artin map one usually employs Galois cohomology. In the beginning the so-called first inequality, stating that [L : K] divides (C_{K} : N_{L|K}C_{L}), is established for cyclic extensions L|K by determining the Herbrand quotient of C_{L} as a G(L|K)-module. Assuming the existence of an Artin map, one can now derive that it must be onto for any finite abelian extension L|K.

The second inequality, (C_{K} : N_{L|K}C_{L})|[L : K], can be stated more generally for L|K abelian, because it is easily reduced to the situation where L|K is cyclic of prime degree n and K contains the n-th roots of unity. If n = p is the characteristic, L is an Artin-Schreier extension of K and therefore contained in K̃ := K(℘^{−1}K). Thinking of G(K̃|K) as equipped with the Krull topology, one exhibits a homeomorphism^{3} ω : C_{K}/C_{K}^{p} ≃ G(K̃|K) in terms of traces of residues of local differentials and shows, by the aid of the first inequality, that ω(N_{L|K}C_{L}/C_{K}^{p}) = G(K̃|L), whence C_{K}/N_{L|K}C_{L} ≃ G(L|K). The case n ≠ p is somewhat more technical. This time L|K is a Kummer extension, and the proof of the second inequality uses an intricate interplay between Kummer groups and idèle norms, too difficult to present here. Having shown the second inequality, one can immediately conclude that the Artin map (if it exists) must have the kernel K^{∗}N_{L|K}I_{L}.

In order to finally prove the existence of the Artin map, ( , L|K) is defined as the product of all local Artin maps ( , L_{𝔭}|K_{𝔭}) with 𝔭 ∈ ℙ_{K}, as indicated at the end of this section. By this definition ( , L|K) is continuous and also the representation by Frobenius automorphisms as required in (a) verifies trivially. The crucial point is to show that ( , L|K) indeed swallows K^{∗}. This is easy for constant field extensions of K and generalizes via an assertion on the Brauer group of K to arbitrary finite abelian extensions.

Since K^{∗} is contained in its kernel, we can also think of the Artin map ( , L|K) as defined on C_{K}. As such it is continuous, too. Its kernel

3Although not needed for the proof, it should help understanding the Reciprocity Law to verify that ω( )|_{L} = ( , L|K) for all finite subextensions L|K of K̃|K.

N_{L} := N_{L|K}C_{L} = K^{∗}N_{L|K}I_{L}/K^{∗}

is called the norm group of L, and L is called the class field associated to this norm group. We note that the Artin map induces an isomorphism

C_{K}/N_{L} ≃ G(L|K).

From the Reciprocity Law we can derive some consequences on inclusions of class fields and norm groups.

4.2. Corollary. Let L_{1}, L_{2} be finite abelian over K. Then

(a) L_{1} ⊆ L_{2} ⟺ N_{L_{1}} ⊇ N_{L_{2}}.

(b) N_{L_{1}L_{2}} = N_{L_{1}} ∩ N_{L_{2}}.

(c) N_{L_{1}∩L_{2}} = N_{L_{1}}N_{L_{2}}.

Proof. We can follow the arguments in [Nk, p. 321]. The implication ⟹ in (a) and the inclusions ⊆ in (b) and ⊇ in (c) follow from the transitivity of the norm. If in (b) conversely αK^{∗} ∈ N_{L_{1}} ∩ N_{L_{2}} for α ∈ I_{K} then (α, L_{1}L_{2}|K)|_{L_{i}} = (α, L_{i}|K) = 1 for i ∈ {1, 2} by 4.1(c), thus αK^{∗} ∈ N_{L_{1}L_{2}}. Using (b) to prove the opposite implication in (a) we see that N_{L_{1}} ⊇ N_{L_{2}} implies N_{L_{1}L_{2}} = N_{L_{2}}, thus [L_{1}L_{2} : K] = [L_{2} : K] by the Reciprocity Law, whence L_{1} ⊆ L_{2}.

It remains to show that N_{L_{1}∩L_{2}} ⊆ N := N_{L_{1}}N_{L_{2}}. For i ∈ {1, 2}, by Galois theory, (N , L_{i}|K) = G(L_{i}|L′_{i}) for some intermediate field L′_{i} of L_{i}|K, and by the Reciprocity Law, N is the kernel of ( , L′_{i}|K), i.e. equals N_{L′_{i}}. Using (a) we conclude L′_{1} = L′_{2} ⊆ L_{1} ∩ L_{2} and N ⊇ N_{L_{1}∩L_{2}}.

4.3. Existence Theorem. The map L ↦ N_{L} is a 1–1 correspondence between the finite abelian extensions of K (within K̄) and the open subgroups of finite index in C_{K}.

Remarks on the proof. We follow [CF, pp. 201f]. Clearly the kernel N_{L} = N_{L|K}C_{L} of the continuous map ( , L|K) : C_{K} → G(L|K) is open and of finite index in C_{K}. Moreover, N_{L} is uniquely determined by L according to 4.2(a). Let us call a subgroup N of C_{K} normic if it is the norm group N_{L} of some abelian extension L|K. We note that if N contains a normic subgroup N′, say with class field L′, then N is itself normic, for, by the Reciprocity Law, (N , L′|K) is a subgroup of G(L′|K), whose fixed field L ⊆ L′ has norm group N_{L} = N.

It remains to show that any open subgroup N of finite index in C_{K} is normic. If the index n = (C_{K} : N) is a prime number and K contains the n-th roots of unity, this follows from the proof of the Reciprocity Law, or more precisely of the second inequality, where the class field (of a subgroup) of N was constructed. By a lemma on norm lifts in cyclic extensions one can reduce the general case to the former.

To conclude this section, we point out the connection between global and local class field theory. Let 𝔭 ∈ ℙ_{K} and denote by L_{𝔭} the completion of L at any one of the places above 𝔭. (They are all K_{𝔭}-isomorphic.) By [CF, p. 175], the composed morphism

([ ]_{𝔭}, L|K) : K_{𝔭}^{∗} ↪ I_{K} ↠ G(L|K)

has image G^{−1}(L, 𝔭). In the course of the proof of the Reciprocity Law this morphism turned out to be equal to the local norm residue symbol ( , L_{𝔭}|K_{𝔭}), whose definition is given in [CF, p. 140]. By [CF, p. 155] we obtain:

4.4. Theorem. The homomorphism ([ ]_{𝔭}, L|K) : K_{𝔭}^{∗} → G^{−1}(L, 𝔭) is surjective and maps U_{𝔭}^{(n)} onto G^{n}(L, 𝔭) for all n ∈ ℕ_{0}.

In particular, ([U_{𝔭}]_{𝔭}, L|K) = 1 for all but the finitely many places 𝔭 ∈ ℙ_{K} which ramify in L. Hence for each α = (α_{𝔭}) ∈ I_{K}, by continuity,

(α, L|K) = ∏_{𝔭∈ℙ_{K}}([α_{𝔭}]_{𝔭}, L|K),

and the product is finite.

5 Definition and Properties

Pursuing the program we announced in the beginning of this part, our next task is to find a suitable open subgroup of finite index in C_{K} such that its class field has the desired properties. Reflecting on Theorem 4.4 encourages us to give the following definitions.

Let S be a set of places of K. By an S-cycle we mean an effective divisor of K with support disjoint from S. Let 𝔪 = ∑_{𝔭} m_{𝔭}𝔭 be such an S-cycle. Then we define the S-congruence subgroups mod 𝔪,

I_{S}^{𝔪} := (∏_{𝔭∈S}K_{𝔭}^{∗} × ∏_{𝔭∈ℙ_{K}\S} U_{𝔭}^{(m_{𝔭})}) ∩ I_{K} and C_{S}^{𝔪} := K^{∗}I_{S}^{𝔪}/K^{∗},

of I_{K} and C_{K}, respectively. (Here intersection with I_{K} is only necessary, of course, when S is infinite.) For an arbitrary effective divisor 𝔫 = ∑_{𝔭} n_{𝔭}𝔭 of K we set I^{𝔫} := I_{ℙ_{K}\supp 𝔫}^{𝔫} and define

ϕ(𝔫) := ∏_{𝔭∈ℙ_{K}}(U_{𝔭} : U_{𝔭}^{(n_{𝔭})}) = ∏_{𝔭∈supp 𝔫}(q^{deg 𝔭} − 1)q^{(n_{𝔭}−1) deg 𝔭}.

We note some elementary properties of the congruence subgroups.

5.1. Remark. Let S ⊆ ℙ_{K} and 𝔪 = ∑_{𝔭} m_{𝔭}𝔭 be an S-cycle.

(a) If T ⊆ S and 𝔫 ≥ 𝔪 is a T -cycle, then I_{T}^{𝔫} ⊆ I_{S}^{𝔪} and C_{T}^{𝔫} ⊆ C_{S}^{𝔪}.

(b) If T ⊆ ℙ_{K} and 𝔫 is a T -cycle, then I_{S}^{𝔪}I_{T}^{𝔫} = I_{S∪T}^{min{𝔪,𝔫}}, where the minimum is taken coefficientwise.

(c) The congruence subgroups I_{∅}^{𝔫} and C_{∅}^{𝔫} with 𝔫 running through all effective divisors of K form a basis of open neighbourhoods of 1 in I_{K} and C_{K}, respectively.

(d) Both I_{S}^{𝔪} and C_{S}^{𝔪} are open.

(e) We have I_{S}^{𝔬}/I_{S}^{𝔪} ≃ ∏_{𝔭} U_{𝔭}/U_{𝔭}^{(m_{𝔭})} and (I_{S}^{𝔬} : I_{S}^{𝔪}) = ϕ(𝔪).

(f) The index (C_{K} : C_{∅}^{𝔫}) is infinite for every effective divisor 𝔫 of K.

(g) K^{∗}I^{𝔫} = I_{K} for any 𝔫 ≥ 𝔬.

(h) I_{S}^{𝔪} is topologically generated by its subgroups [K_{𝔭}^{∗}]_{𝔭}, 𝔭 ∈ S, and [U_{𝔭}^{(m_{𝔭})}]_{𝔭}, 𝔭 ∈ ℙ_{K} \ S.

Proof. (a)–(e) are obvious.

(f) Choosing a place 𝔭 ∈ ℙ_{K} \ supp 𝔫, the homomorphism K_{𝔭}^{∗} → I_{K}/K^{∗}I_{∅}^{𝔫} induced by the embedding [ ]_{𝔭} has kernel U_{𝔭}. But K_{𝔭}^{∗}/U_{𝔭} ≃ ℤ.

(g) This is merely an idèlic reformulaton of the Weak Approximation Theorem 1.5.

(h) Let α = (α_{𝔭})_{𝔭∈ℙ_{K}} ∈ I_{S}^{𝔪} and 𝔫 = ∑_{𝔭} n_{𝔭}𝔭 be an effective divisor of K. Then the set T := {𝔭 ∈ ℙ_{K} | α_{𝔭} ∉ U_{𝔭}^{(n_{𝔭})}} is finite and α ≡ ∏_{𝔭∈T}[α_{𝔭}]_{𝔭} mod I_{∅}^{𝔫}. Thus the assertion follows by (b).

Because of 5.1(f) we require S to be non-empty from now on. In a moment we shall see that under this assumption the index of C_{S}^{𝔪} in C_{K} is indeed finite. In order to obtain an ideal theoretic interpretation of the corresponding factor group, we introduce the S-class group mod 𝔪, Cℓ^{𝔪}(O_{S}), defined as the quotient of the group of fractional ideals of O_{S} prime to 𝔪 by its subgroup of principal ideals zO_{S} with z ∈ K^{∗} ∩ I^{𝔪}. Note that Cℓ^{𝔬}(O_{S}) = Cℓ(O_{S}) is the usual S-class group defined in Section 2.

5.2. Proposition. Let S ⊆ ℙ_{K} be non-empty and 𝔪 an S-cycle.

(a) Cℓ^{𝔪}(O_{S}) is naturally isomorphic to C_{K}/C_{S}^{𝔪}.

(b) The sequence

O_{S}^{∗} → I_{S}^{𝔬}/I_{S}^{𝔪} → I_{K}/K^{∗}I_{S}^{𝔪} → I_{K}/K^{∗}I_{S}^{𝔬} → 1

is exact.

(c) (C_{K} : C_{S}^{𝔪}) is finite.

Proof. (a) The natural homomorphism I^{𝔪} → Cℓ^{𝔪}(O_{S}) mapping the idèle (α_{𝔭})_{𝔭∈ℙ_{K}} ∈ I^{𝔪} to the class of the fractional O_{S}-ideal ∏_{𝔭∈ℙ_{K}\S}(𝔭 ∩ O_{S})^{v_{𝔭}(α_{𝔭})} is onto with kernel I_{S}^{𝔪}(K^{∗} ∩ I^{𝔪}) = K^{∗}I_{S}^{𝔪} ∩ I^{𝔪}. Consequently, Cℓ^{𝔪}(O_{S}) ≃ K^{∗}I^{𝔪}/K^{∗}I_{S}^{𝔪} = I_{K}/K^{∗}I_{S}^{𝔪} by 5.1(g).

(b) We have O_{S}^{∗} = K^{∗} ∩ I_{S}^{𝔬}, implying O_{S}^{∗}I_{S}^{𝔪}/I_{S}^{𝔪} = (K^{∗}I_{S}^{𝔪} ∩ I_{S}^{𝔬})/I_{S}^{𝔪}, which is the exactness at I_{S}^{𝔬}/I_{S}^{𝔪}. The exactness at the other groups is even simpler.

(c) This follows from (b) since both I_{S}^{𝔬}/I_{S}^{𝔪} and I_{K}/K^{∗}I_{S}^{𝔬} ≃ Cℓ(O_{S}) are finite.

According to the Existence Theorem 4.3, we can now define K_{S}^{𝔪} as the class field associated to the congruence subgroup C_{S}^{𝔪} and call it the S-ray class field mod 𝔪. By 4.1(b) we have an isomorphism

G(K_{S}^{𝔪}|K) ≃ I_{K}/K^{∗}I_{S}^{𝔪} ≃ C_{K}/C_{S}^{𝔪} ≃ Cℓ^{𝔪}(O_{S}).

We remark that K_{S}^{𝔬} is the Hilbert class field of O_{S} in the sense of Rosen [Ro2] (with the slight generalization that we allow S to be infinite). Moreover, in case S consists of exactly one place, an explicit construction of K_{S}^{𝔪} in terms of rank 1 Drinfel’d modules has been carried out by Hayes [Hay]. This special case is treated in an example at the end of this section.

As a consequence of our remark about the congruence subgroups we obtain the following

5.3. Corollary. Let S and T be non-empty subsets of ℙ_{K}, 𝔪 an S-cycle and 𝔫 a T -cycle.

(a) If S ⊇ T and 𝔪 ≤ 𝔫, then K_{S}^{𝔪} ⊆ K_{T}^{𝔫}.

(b) K_{S}^{𝔪} ∩ K_{T}^{𝔫} = K_{S∪T}^{min{𝔪,𝔫}}.

(c) If S ⊇ T, 𝔪 ≤ 𝔫 and 𝔫 is also an S-cycle, then [K_{S}^{𝔫} : K_{S}^{𝔪}] ≤ [K_{T}^{𝔫} : K_{T}^{𝔪}].

Proof. (a) This is clear from 5.1(a) and 4.2(a).

(b) By 5.1(b) we have C_{S}^{𝔪}C_{T}^{𝔫} = C_{S∪T}^{min{𝔪,𝔫}}, so that the assertion follows from 4.2(c) and the definition of the ray class fields.

(c) By 4.1(c), the norm residue symbol ( , K_{S}^{𝔫}|K) induces an isomorphism K^{∗}I_{S}^{𝔪}/K^{∗}I_{S}^{𝔫} ≃ G(K_{S}^{𝔫}|K_{S}^{𝔪}), and likewise for T instead of S. Because of 5.1(b), the natural map K^{∗}I_{T}^{𝔪}/K^{∗}I_{T}^{𝔫} → K^{∗}I_{S}^{𝔪}/K^{∗}I_{S}^{𝔫} is onto.

Next we want to investigate some extremality properties of the ray class fields.

5.4. Conductor Lemma. Let L|K be finite abelian and let S be any nonempty subset of S(L|K). Then 𝔣(L|K) is the smallest S-cycle 𝔪 such that L is contained in K_{S}^{𝔪}.

Proof. Let 𝔪 = ∑_{𝔭} m_{𝔭}𝔭 be an S-cycle. By the definition of the conductor exponent and Theorems 4.4 and 4.1(b), for each 𝔭 ∈ ℙ_{K} we have the equivalence

f(L, 𝔭) ≤ m_{𝔭} ⟺ G^{m_{𝔭}}(L, 𝔭) = 1 ⟺ [U_{𝔭}^{(m_{𝔭})}]_{𝔭} ⊆ K^{∗}N_{L|K}I_{L},

and moreover 𝔭 ∈ S implies G^{−1}(L, 𝔭) = 1, i.e. [K_{𝔭}^{∗}]_{𝔭} ⊆ K^{∗}N_{L|K}I_{L}. Since K^{∗}N_{L|K}I_{L} is open, hence closed, by 5.1(h) and 4.2(a) the above facts can be summarized as

𝔣(L|K) ≤ 𝔪 ⟺ I_{S}^{𝔪} ⊆ K^{∗}N_{L|K}I_{L} ⟺ C_{S}^{𝔪} ⊆ N_{L} ⟺ L ⊆ K_{S}^{𝔪},

which is what we had to show.

We deduce the following characterization of the ray class fields.

5.5. Proposition. Let S be a non-empty set of places of K and 𝔪 an S-cycle. Then K_{S}^{𝔪} is the largest abelian extension L of K such that S ⊆ S(L|K) and 𝔣(L|K) ≤ 𝔪. Moreover, 𝔽_{q^{d}} with d := deg S is the full constant field of K_{S}^{𝔪}.

Proof. Let L|K be finite abelian. By the lemma, S ⊆ S(L|K) and 𝔣(L|K) ≤ 𝔪 imply L ⊆ K_{S}^{𝔪}. Conversely, let L ⊆ K_{S}^{𝔪}. Then 𝔣(L|K) ≤ 𝔪 is also clear from the Conductor Lemma, and it remains to show that S ⊆ S(L|K). Indeed, 𝔭 ∈ S implies [K_{𝔭}^{∗}]_{𝔭} ⊆ I_{S}^{𝔪} ⊆ K^{∗}N_{L|K}I_{L} by 4.2(a); hence, from Theorem 4.4 it follows that G^{−1}(L, 𝔭) = ([K_{𝔭}^{∗}]_{𝔭}, L|K) = 1, i.e. 𝔭 ∈ S(L|K).

As for the second assertion, recall that the constant field extension K_{d} = 𝔽_{q^{d}}K of degree d over K is unramified and S ⊆ S(K_{d}|K). This implies 𝔽_{q^{d}} ⊆ K_{d} ⊆ K_{S}^{𝔬} ⊆ K_{S}^{𝔪}. On the other hand, in order that 𝔽_{q^{d′}} for some d′ ∈ ℕ be contained in K_{S}^{𝔪}, d′ must divide the degree of every place in S(K_{S}^{𝔪}|K) ⊇ S, so that d′|d.

Recall that a rational function field has trivial divisor class group. So by 2.1 we obtain the following

5.6. Corollary. Let K = 𝔽_{q}(x) be a rational function field and S ⊆ ℙ_{K} nonempty. Then h_{S} = deg S =: d and the Hilbert class field K_{S}^{𝔬} = 𝔽_{q^{d}}K = 𝔽_{q^{d}}(x) is again a rational function field.

If we choose S large enough, it may happen that the S-ray class field does not even depend on the modulus 𝔪, as we see in our next

5.7. Corollary. With notation as in Corollary 3.5, we have K_{S_{d}}^{𝔪} = K_{d} and Cℓ^{𝔪}(O_{S_{d}}) ≃ ℤ/dℤ for any S_{d}-cycle 𝔪; in particular, h_{S_{d}} = d.

Proof. The equality K_{S_{d}}^{𝔪} = K_{d} follows from 3.5 and the previous proposition. We conclude that Cℓ^{𝔪}(O_{S_{d}}) ≃ G(K_{S_{d}}^{𝔪}|K) = G(K_{d}|K) ≃ ℤ/dℤ.

Given a place 𝔭 and an effective divisor 𝔪 of K whose 𝔭-th coefficient is m_{𝔭}, we set 𝔪 \ 𝔭 := 𝔪 − m_{𝔭}𝔭. With this notation, we can write down the remarkable fact that the ramification fields of the S-ray class field mod 𝔪 are the S-ray class fields mod 𝔪′ for certain 𝔪′ ≤ 𝔪.

5.8. Theorem. Let S be a non-empty set of places of K, 𝔪 an S-cycle and L an intermediate field of K_{S}^{𝔪}|K. For any place 𝔭 of K and any n ∈ ℕ_{0} we have

L^{n}(𝔭) = L ∩ K_{S}^{𝔪\𝔭+n𝔭} and L^{−1}(𝔭) = L ∩ K_{{𝔭}}^{𝔪\𝔭} = L ∩ K_{S∪{𝔭}}^{𝔪\𝔭}.

In particular, writing 𝔪 = ∑_{𝔭} m_{𝔭}𝔭, the discriminant of K_{S}^{𝔪}|K is

𝔡(K_{S}^{𝔪}|K) = [K_{S}^{𝔪} : K]𝔪 − ∑_{𝔭}(∑_{n=1}^{m_{𝔭}}[K_{S}^{𝔪−n𝔭} : K])𝔭.

Proof. Consider an arbitrary intermediate field L′ of L|K. By Propositions 3.1 and 5.5 we obtain the following equivalence of conditions on L′:

L′ ⊆ L^{n}(𝔭) ⟺ G(L|L′) ⊇ G^{n}(L, 𝔭) ⟺ G^{n}(L′, 𝔭) = 1 ⟺
n ≥ f(L′, 𝔭) ⟺ 𝔪 \ 𝔭 + n𝔭 ≥ 𝔣(L′|K) ⟺ L′ ⊆ K_{S}^{𝔪\𝔭+n𝔭}.

Similarly,

L′ ⊆ L^{−1}(𝔭) ⟺ G^{−1}(L′, 𝔭) = 1 ⟺ 𝔭 ∈ S(L′|K)
⟺ L′ ⊆ K_{{𝔭}}^{𝔪\𝔭} ⟺ L′ ⊆ K_{S∪{𝔭}}^{𝔪\𝔭}.

The last identity follows from the Discriminant Formula 3.7 since 𝔣(K_{S}^{𝔪}|K) ≤ 𝔪.

Together with the Disriminant Formula 3.7, Proposition 5.5 and the Hurwitz Genus Formula 1.4, we deduce the following

5.9. Corollary. Let S ⊆ ℙ_{K} be non-empty, 𝔭 ∈ ℙ_{K} \S and L an intermediate field of K_{S}^{m𝔭}|K_{S}^{(m−1)𝔭} for some m ∈ ℕ. Then the genera of K and L are related as

(g_{L} − 1) deg S = [L : K](g_{K} − 1 + \frac{m}{2} deg 𝔭) − \frac{1}{2} ∑_{n=0}^{m−1}[K_{S}^{n𝔭} : K] deg 𝔭.

Thus computation of the discriminant (and thereby of the genus) of ray class field extensions amounts to determining their degrees. This has already been seen by Cohen et al. [CDO] in the number field case. As an illustration we give the following

5.10. Example (Ray Class Fields à la Hayes). Assume that S consists of exactly one place of degree d ∈ ℕ. Let 𝔪 be an S-cycle, h = h_{K} the (divisor) class number and g = g_{K} the genus of K. Then

(a) [K_{S}^{𝔬} : K] = hd and [K_{S}^{𝔪} : K] = \frac{hdϕ(𝔪)}{q−1} for 𝔪 ≠ 𝔬.

(b) The genera of the S-ray class fields are given by

g_{K_{S}^{𝔪}} = \begin{cases}1 + h(g − 1) if deg 𝔪 ≤ 1, \\ 1 + \frac{hq^{(m−1)n}(q^{n}−1)(g−1)}{q−1} + \frac{hn}{2} \frac{mq^{mn}−(m+1)q^{(m−1)n}−q+2}{q−1} \\ if 𝔪 = m𝔭 with 𝔭 ∈ ℙ_{K}^{n} and m, n ∈ ℕ, \\ 1 + \frac{h}{2} \frac{ϕ(𝔪)}{q−1}(2g − 2 + deg 𝔪 − ∑_{𝔭∈supp 𝔪} \frac{deg 𝔭}{ϕ(𝔭)}) if |supp 𝔪| > 1. \end{cases}

Note that the genus of K_{S}^{𝔪} does not depend on d but on g, h and 𝔪 only.

Proof. (a) By Proposition 2.1, |S| = 1 implies h_{S} = hd and O_{S}^{∗} = 𝔽_{q}^{∗}, which is mapped injectively into I_{S}^{𝔬}/I_{S}^{𝔪} when 𝔪 ≠ 𝔬. Hence the degree formulas follow from Proposition 5.2 and 5.1(e).

(b) Note that K_{S}^{𝔪} has the full constant field 𝔽_{q^{d}} according to Proposition 5.5. For 𝔪 ∈ ℙ_{K}^{1} we have ϕ(𝔪) = q − 1, hence K_{S}^{𝔪} = K_{S}^{𝔬} by (a). Since K_{S}^{𝔬}|K is unramified, its discriminant is 𝔬. Hence the assertion in case deg 𝔪 ≤ 1 follows from the Hurwitz Genus Formula 1.4.

For the other two cases, write 𝔪 = ∑_{𝔭} m_{𝔭}𝔭 and let 𝔭 ∈ supp 𝔪. Then

∑_{n=1}^{m_{𝔭}−1}[K_{S}^{𝔪\𝔭+n𝔭} : K] = \frac{hd}{q − 1} ∑_{n=1}^{m_{𝔭}−1} ϕ(𝔪)q^{(1−m_{𝔭}) deg 𝔭}q^{(n−1) deg 𝔭}

= \frac{hd}{q − 1} \frac{ϕ(𝔪)}{ϕ(𝔭)}(1 − q^{(1−m_{𝔭}) deg 𝔭}).

If 𝔪 = m𝔭 with 𝔭 ∈ ℙ_{K}^{n} and m, n ∈ ℕ, then 𝔪 \ 𝔭 = 𝔬, so (a) and the discriminant formula in Theorem 5.8 yield

𝔡(K_{S}^{𝔪}|K) = \frac{hdϕ(m𝔭)}{q − 1}m𝔭 − (hd + \frac{hd}{q − 1} \frac{ϕ(m𝔭)}{ϕ(𝔭)}(1 − q^{(1−m)n}))𝔭

= \frac{hd}{q − 1}(mq^{(m−1)n}(q^{n} − 1) − q + 1 − q^{(m−1)n} + 1)𝔭.

If 𝔭 is not the only place in supp 𝔪, then [K_{S}^{𝔪\𝔭} : K] = \frac{hd}{q−1} \frac{ϕ(𝔪)}{ϕ(𝔭)}q^{(1−m_{𝔭}) deg 𝔭}, whence

𝔡(K_{S}^{𝔪}|K) = \frac{hdϕ(𝔪)}{q − 1}𝔪 − ∑_{𝔭∈supp 𝔪} \frac{hd}{q − 1} \frac{ϕ(𝔪)}{ϕ(𝔭)}𝔭.

In both cases the genus of K_{S}^{𝔪} turns out as asserted by applying the Hurwitz Genus Formula 1.4.

As soon as S consists of more than one place, the determination of the degrees is much more difficult. We shall dedicate the next section to working out this problem.

6 Computation of Degrees

In view of the previous section (especially Theorem 5.8), our task is to determine the degrees [K_{S}^{𝔪} : K] for a fixed non-empty subset S of ℙ_{K}, which for practical reasons is now assumed to be finite, and for a varying S-cycle 𝔪. We divide the problem into three steps by considering two intermediate fields of K_{S}^{𝔪}|K, namely the Hilbert class field K_{S}^{𝔬} and the field K𝔪̃ ; S with

𝔪̃ := ∑_{𝔭∈supp 𝔪}𝔭.

By Proposition 5.2(a) the extension K_{S}^{𝔬}|K has group G(K_{S}^{𝔬}|K) ≃ Cℓ(O_{S}). Its order h_{S} can be determined via Corollary 2.2 after a computation of O_{S}^{∗} and the divisor class number h_{K} of K. In the next section we shall learn about yet another method to determine (or at least estimate) h_{S} when S contains “many” rational places, which may also help us with the S-units.

As for the second step, G(K𝔪̃ ; S|K_{S}^{𝔬}) is by 5.2(b) isomorphic to I_{S}^{𝔬}/O_{S}^{∗}I𝔪̃ ; S. Therefore the degree [K𝔪̃ ; S : K_{S}^{𝔬}] divides ϕ(𝔪̃)/(q − 1) (cf. also Example 5.10), and can be determined as follows (for technical reasons we exclude the trivial case of 𝔪 = 𝔬). Let s := |S| − 1 ∈ ℕ_{0}, and suppose we have computed S-units u_{1}, …, u_{s} such that O_{S}^{∗} = ⟨α, u_{1}, …, u_{s}⟩, where α is a generator of 𝔽_{q}^{∗}. Write R := supp 𝔪 = {𝔭_{1}, …, 𝔭_{r}} with r = |R| ∈ ℕ. For each j ∈ {1, …, r} we set d_{j} := q^{deg 𝔭_{j}} − 1 and c_{j} := d_{j}/(q − 1) and choose a generator ω_{j} of 𝔽_{𝔭_{j}}^{∗} such that ω_{j}^{c_{j}} = α. Let D = (d_{ij})_{1≤i,j≤r} ∈ ℤ^{r×r} with

d_{ij} := \begin{cases}d_{j} & for 1 ≤ i = j < r, \\ c_{j} & for i = r, \\ 0 & otherwise, \end{cases}

be the matrix obtained from the diagonal matrix with entries d_{1}, …, d_{r} by replacing the last row with (c_{1}, …, c_{r}). Note that the sacrificed row (0, …, 0, d_{r}) ∈ ℤ^{r} is a ℤ-linear combination of the rows of D. Since S ∩ R = ∅, there exists an A = (a_{ij})_{{1≤i≤s \atop 1≤j≤r}} ∈ ℤ^{s×r} such that ω_{j}^{a_{ij}} = u_{i} + 𝔭_{j} ∈ 𝔽_{𝔭_{j}} for all i ∈ {1, …, s} and j ∈ {1, …, r}. The (non-canonical) surjective group morphism

ℤ^{r} → I_{S}^{𝔬}/I𝔪̃ ; S ≃ ∏_{j=1}^{r} 𝔽_{𝔭_{j}}^{∗}

(a_{1}, …, a_{r}) ↦ (ω_{j}^{a_{j}})_{1≤j≤r}

gives rise to the exact sequence

0 → N → ℤ^{r} → I_{S}^{𝔬}/O_{S}^{∗}I𝔪̃ ; S → 1,

where N is the subgroup of ℤ^{r} generated by the rows of the matrix \binom{A}{D} ∈ ℤ^{(s+r)×r} of rank r. After some row transformations T = (t_{ij})_{1≤i,j≤s+r} ∈ GL_{s+r}(ℤ), we find a B ∈ ℤ^{r×r} with T \binom{A}{D} = \binom{0}{B}. Because G(K𝔪̃ ; S|K_{S}^{𝔬}) ≃ ℤ^{r}/N, it now follows that

[K𝔪̃ ; S : K_{S}^{𝔬}] = |det B| .

At the same time we have found a basis for the group K^{∗} ∩ I𝔪̃ ; S, which is needed in step three below.

6.1. Lemma. With notation as above, the S-units

ũ_{i} := α^{t_{i,s+r}} ∏_{k=1}^{s} u_{k}^{t_{ik}}, 1 ≤ i ≤ s,

form a basis of the free abelian group K^{∗} ∩ I𝔪̃ ; S = O_{S}^{∗} ∩ I𝔪̃ ; S.

Proof. For i ∈ {1, …, s} and j ∈ {1, …, r} we have ∑_{k=1}^{s} t_{ik}a_{kj} + t_{i,s+r}c_{j} ≡ 0 mod d_{j} by definition of T; hence,

ũ_{i} + 𝔭_{j} = ω_{j}^{t_{i,s+r}c_{j}} ∏_{k=1}^{s} ω_{j}^{t_{ik}a_{kj}} = 1.

This shows ũ_{1}, …, ũ_{s} ∈ I𝔪̃ ; S.

Conversely, suppose u = α^{t_{0}} ∏_{k=1}^{s} u_{k}^{t_{k}} ∈ O_{S}^{∗} with t_{0}, …, t_{s} ∈ ℤ is contained in I𝔪̃ ; S. Then there are t_{s+1}, …, t_{s+r} ∈ ℤ such that

t_{0}c_{j} + ∑_{k=1}^{s} t_{k}a_{kj} + t_{s+j}d_{j} = 0 ∀j ∈ {1, …, r}.

Thus t := (t_{1}, …, t_{s+r−1}, t_{0}) satisfies t \binom{A}{D} = 0. Putting t̃ = (t̃_{1}, …, t̃_{s+r}) := tT^{−1}, from

t̃\binom{0}{B} = t̃T \binom{A}{D} = t \binom{A}{D} = 0

we can conclude t̃_{s+1} = ⋯ = t̃_{s+r} = 0 since the rows of B are linearly independent. Therefore

∏_{i=1}^{s} ũt̃_{i} ; i = ∏_{i=1}^{s}(α^{t̃_{i}t_{i,s+r}} ∏_{k=1}^{s} u_{k}^{t̃_{i}t_{ik}}) = α^{t_{0}} ∏_{k=1}^{s} u_{k}^{t_{k}} = u.

It remains to show that K^{∗} ∩ I𝔪̃ ; S = O_{S}^{∗} ∩ I𝔪̃ ; S is free of rank s. The morphism K^{∗} ∩I𝔪̃ ; S → D_{K}, u ↦ (u) has kernel 𝔽_{q}^{∗} ∩I𝔪̃ ; S = 1, since we have assumed 𝔪̃ ≠ 𝔬. Therefore, by 2.2, K^{∗} ∩ I𝔪̃ ; S ≃ (K^{∗} ∩ I𝔪̃ ; S) ⊆ (O_{S}^{∗}) is free of rank ≤ s. On the other hand, for any common multiple d of d_{1}, …, d_{r}, we have O_{S}^{∗}^{d} ⊆ I𝔪̃ ; S, from which it follows that ((O_{S}^{∗}) : (K^{∗} ∩ I𝔪̃ ; S)) ≤ ((O_{S}^{∗}) : d(O_{S}^{∗})) = s^{d} < ∞.

The rest of this section is devoted to the treatment of step three, the extension K_{S}^{𝔪}|K𝔪̃ ; S. The Reciprocity Law tells us that

G(K_{S}^{𝔪}|K𝔪̃ ; S) ≃ K^{∗}I𝔪̃ ; S/K^{∗}I_{S}^{𝔪} ≃ I𝔪̃ ; S/(K^{∗} ∩ I𝔪̃ ; S)I_{S}^{𝔪}. (4)

Writing 𝔪 = ∑_{𝔭} m_{𝔭}𝔭, we see that I𝔪̃ ; S/I_{S}^{𝔪} ≃ ∏_{𝔭∈supp 𝔪} U_{𝔭}^{(1)}/U_{𝔭}^{(m_{𝔭})} is a p-group. We need to know what the structure of the groups U_{𝔭}^{(1)}/U_{𝔭}^{(n)} with 𝔭 ∈ ℙ_{K} and n ∈ ℕ is. In order to answer this question it is useful to agree upon some notation. Recall that p is the characteristic of K. For any real a > 0, let

⌈a⌉_{p} := min{p^{l} | a ≤ p^{l}, l ∈ ℕ_{0}}

be the least p-power integer ≥ a. Moreover denote by ℕ_{p}^{∗} := ℕ \ pℤ the set of positive integers prime to p and by n^{∗} := n/p^{v_{p}(n)} ∈ ℕ_{p}^{∗} the non-p-part of n ∈ ℕ. The following proposition could easily be derived from Hasse’s One Unit Theorem in [H, p. 227]. For reasons of constructiveness, however, we write down a direct proof.

6.2. Proposition. Let 𝔭 be a place of K, π ∈ K_{𝔭} a uniformizer at 𝔭, B an 𝔽_{p}-basis of 𝔽_{𝔭} and n ∈ ℕ.

(a) For each j ∈ ℕ and β ∈ 𝔽_{𝔭}^{∗}, the coset (1 + βπ^{j})U_{𝔭}^{(n)} ∈ U_{𝔭}^{(1)}/U_{𝔭}^{(n)} has order ⌈n/j⌉_{p}.

(b) There is a decomposition

U_{𝔭}^{(1)}/U_{𝔭}^{(n)} = ∏_{{j∈ℕ^{∗}_{p} \atop β∈B}} ⟨(1 + βπ^{j})U_{𝔭}^{(n)}⟩

as a direct product of cyclic subgroups.

Proof. (a) For l ∈ ℕ_{0}, one has (1 + βπ^{j})^{p^{l}} = 1 + β^{p^{l}}π^{jp^{l}} ∈ U_{𝔭}^{(n)} iff p^{l} ≥ n/j.

(b) Let H be the subgroup of U_{𝔭}^{(1)}/U_{𝔭}^{(n)} generated by all cosets (1 + βπ^{j})U_{𝔭}^{(n)} with j ∈ ℕ_{p}^{∗} and β ∈ B. We prove that

U_{𝔭}^{(m)}/U_{𝔭}^{(n)} ⊆ H

for m ∈ {1, …, n} by induction on n−m. For m = n there is nothing to show. Let 1 ≤ m < n and u ∈ U_{𝔭}^{(m)}. Set l := v_{p}(m), then u ≡ 1 + ω^{p^{l}}π^{m} mod U_{𝔭}^{(m+1)} for some ω ∈ 𝔽_{𝔭}. Writing ω = ∑_{β∈B} a_{β}β with a_{β} ∈ {0, …, p − 1}, we see that

∏_{β∈B}(1 + βπ^{j})^{a_{β}} ≡ 1 + ωπ^{j} mod U_{𝔭}^{(j+1)}

with j := m^{∗} ∈ ℕ_{p}^{∗}; hence,

∏_{β∈B}(1 + βπ^{j})^{a_{β}p^{l}} ≡ (1 + ωπ^{j})^{p^{l}} ≡ u mod U_{𝔭}^{(m+1)}.

By the induction hypothesis, U_{𝔭}^{(m+1)}/U_{𝔭}^{(n)} ⊆ H, from which we may conclude uU_{𝔭}^{(n)} ∈ H. So we have proved U_{𝔭}^{(1)}/U_{𝔭}^{(n)} = H. Since (U_{𝔭}^{(1)} : U_{𝔭}^{(n)}) = q^{(n−1) deg 𝔭}, the directness of the product follows from (a) and part (b) of the next lemma.

6.3. Lemma. The positive integers ⌈\frac{m}{j}⌉_{p} with m, j ∈ ℕ satisfy the following properties.

(a) ⌈\frac{m+1}{j}⌉_{p}/⌈\frac{m}{j}⌉_{p} = \begin{cases}p & if there is l ∈ ℕ_{0} such that m = jp^{l}, \\ 1 & otherwise. \end{cases}

(b) ∏_{j∈ℕ_{p}^{∗}} ⌈\frac{m}{j}⌉_{p} = p^{m−1}.

Proof. (a) Clearly, ⌈\frac{m+1}{j}⌉_{p} = ⌈\frac{m}{j}⌉_{p} unless \frac{m}{j} = p^{l} for some l ∈ ℕ_{0}, in which case ⌈\frac{m+1}{j}⌉_{p} = p^{l+1} = p⌈\frac{m}{j}⌉_{p}.

(b) By induction on m. The assertion is trivial for m = 1. Write m = ip^{l} with i := m^{∗} ∈ ℕ_{p}^{∗} and l := v_{p}(m), then

∏_{j∈ℕ_{p}^{∗}} ⌈\frac{m+1}{j}⌉_{p} = ⌈\frac{m+1}{i}⌉_{p} ∏_{j∈ℕ_{p}^{∗}\\{i\}} ⌈\frac{m}{j}⌉_{p} = p ∏_{j∈ℕ_{p}^{∗}} ⌈\frac{m}{j}⌉_{p} = p^{m}

by (a) and the induction hypothesis.

For simplicity we restrict to the case of |supp 𝔪| = 1, i.e. 𝔪̃ =: 𝔭 ∈ ℙ_{K} \ S. In this case the result is as follows.

6.4. Theorem. Let 𝔭 ∈ ℙ_{K} \ S. There are s = |S| − 1 positive integers n_{1}, …, n_{s} depending only on S and 𝔭 such that

[K_{S}^{m𝔭} : K_{S}^{𝔭}] = q^{(m−1) deg 𝔭}/ ∏_{i=1}^{s} ⌈\frac{m}{n_{i}}⌉_{p}

for all m ∈ ℕ.^{4}

With n_{1}, …, n_{s} as in the theorem we define the polynomial

δ := \frac{q^{deg 𝔭} − 1}{(q − 1)[K_{S}^{𝔭} : K_{S}^{𝔬}]} + ∑_{i=1}^{s} t^{n_{i}} = ∑_{n∈ℕ_{0}} δ_{n}t^{n} ∈ ℤ[t]

4In general, all degrees [K_{S}^{𝔪} : K𝔪̃ ; S], for supp 𝔪 within a fixed set R ⊆ ℙ_{K} \ S of r places, can be given in terms of (a tree of at most) ∑_{i=1}^{s} r^{i} positive integers. A complete exposition of this result would make this section even more technical than it is already.

and set δ^{(m)} := ∑_{l=0}^{v_{p}(m)} δ_{m/p^{l}} = |{i ∈ {1, …, s} | n_{i} ≤ m, n_{i}^{∗} = m^{∗}}| for m ∈ ℕ and δ^{(0)} := δ_{0}. From 6.4 and 6.3(a) we derive the recursive degree formula

[K_{S}^{(m+1)𝔭} : K_{S}^{m𝔭}] = q^{deg 𝔭}/p^{δ^{(m)}} ∀m ∈ ℕ, (5)

which is more convenient for practical computations. Since the coefficients δ_{n}, n ∈ ℕ_{0}, of δ can be reconstructed out of the numbers δ^{(m)}, m ∈ ℕ_{0}, from (5) we see that δ is uniquely determined by S and 𝔭. We call δ_{S,𝔭} := δ the S-description at 𝔭. Note that δ_{S,𝔭}(1) − δ_{S,𝔭}(0) = s = |S| − 1. Together with 5.3(c) we obtain the following

6.5. Corollary. Let ∅ ≠ S′ ⊆ S. Then δ_{S′,𝔭}^{(m)} ≤ δ_{S,𝔭}^{(m)} for all m ∈ ℕ_{0}.^{5}

Before we enter the proof of Theorem 6.4, we want to take a closer look at the structure of U_{𝔭}^{(1)}. As usual we denote by ℤ_{p} = \underleftarrow{lim}_{l}ℤ/p^{l}ℤ the ring of p-adic integers, the completion of ℤ at v_{p}. Identifying u ∈ U_{𝔭}^{(1)} with the family (uU_{𝔭}^{(n)})_{n∈ℕ}, we can write

U_{𝔭}^{(1)} = \underleftarrow{lim}_{n}U_{𝔭}^{(1)}/U_{𝔭}^{(n)}.

Since U_{𝔭}^{(1)}/U_{𝔭}^{(n)} is naturally a module over ℤ/ ⌈n⌉_{p} ℤ, and thereby a fortiori a ℤ_{p}-module, the definition of u^{a} := ((uU_{𝔭}^{(n)})^{a})_{n∈ℕ} for u ∈ U_{𝔭}^{(1)} and a ∈ ℤ_{p} makes U_{𝔭}^{(1)} a ℤ_{p}-module (cf. [H, pp. 215ff]). For the proof of 6.4 it is important to know how linear independence in U_{𝔭}^{(1)} over ℤ and ℤ_{p} are related.

6.6. Theorem (Kisilevsky). Let I be a set. A family (u_{i})_{i∈I} of one-units u_{i} ∈ U_{𝔭}^{(1)}, i ∈ I, is independent over ℤ iff it is independent over ℤ_{p}.

Proof. See [Ki].

By Proposition 6.2 and with π and B as therein, for each n ∈ ℕ there is a (non-canonical) group isomorphism

ζ^{(n)} : \begin{matrix}U_{𝔭}^{(1)}/U_{𝔭}^{(n)} & → & Z^{(n)} := ∏_{j∈ℕ_{p}^{∗}} (ℤ/ℤ ⌈n/j⌉_{p})^{B} \\ ∏_{{j∈ℕ^{∗}_{p} \atop β∈B}} (1 + βπ^{j})^{c_{jβ}}U_{𝔭}^{(n)} & ↦ & (c_{jβ} + ℤ ⌈n/j⌉_{p})_{{j∈ℕ^{∗}_{p} \atop β∈B}} . \end{matrix}

5Moreover, one has deg δ_{S′,𝔭} ≤ deg δ_{S,𝔭}. But we shall not prove this here.

Passing to projective limits on both sides yields an isomorphism ζ : U_{𝔭}^{(1)} ≃ Z := \underleftarrow{lim}_{n}Z^{(n)} = ℤ_{p}^{ℕ_{p}^{∗}×B} of ℤ_{p}-modules (cf. Hasse’s One-Unit Theorem in [H, p. 227]), which makes the diagram

U_{𝔭}^{(1)}  ⟶^{ζ} Z
↓  ↓^{(n)}
U_{𝔭}^{(1)}/U_{𝔭}^{(n)} ⟶^{ζ^{(n)}} Z^{(n)}

commutative for all n ∈ ℕ. Here ^{(n)} means the canonical projection Z ↠ Z^{(n)}. A crucial role in the proof of Theorem 6.4 is played by the map

ν : \begin{matrix}Z & → & ℕ ∪ \{∞\} \\ (z_{jβ})_{{j∈ℕ^{∗}_{p} \atop β∈B}} & ↦ & min\{jp^{v_{p}(z_{jβ})} | j ∈ ℕ_{p}^{∗}, β ∈ B\}, \end{matrix}

where we agree that ν(0) = p^{∞} = ∞. It satisfies the following properties.

6.7. Lemma. Let z, z′ ∈ Z, a ∈ ℤ_{p} and m ∈ ℕ. Then

(a) ν(az) = p^{v_{p}(a)}ν(z).

(b) ν(z + z′) ≥ min{ν(z), ν(z′)}.

(c) The subgroup of Z^{(m)} generated by z^{(m)} has order |⟨z^{(m)}⟩| = ⌈m/ν(z)⌉_{p}.

(d) z^{(m)} = 0 if and only if m ≤ ν(z).

(e) If m > ν(z) then ν(z) can be read from z^{(m)}.

Proof. Write z = (z_{jβ})_{{j∈ℕ^{∗}_{p} \atop β∈B}} and z′ = (z′_{jβ})_{{j∈ℕ^{∗}_{p} \atop β∈B}}.

(a) ν(az) = min{jp^{v_{p}(a)}p^{v_{p}(z_{jβ})} | j ∈ ℕ_{p}^{∗}, β ∈ B} = p^{v_{p}(a)}ν(z).

(b) ν(z + z′) ≥ min{jp^{v_{p}(z_{jβ})}, jp^{v_{p}(z′_{jβ})} | j ∈ ℕ_{p}^{∗}, β ∈ B} = min{ν(z), ν(z′)}.

(c) Write z^{(m)} = (c_{jβ} + ℤ ⌈m/j⌉_{p})_{{j∈ℕ^{∗}_{p} \atop β∈B}} with c_{jβ} ∈ ℤ. For j ∈ ℕ_{p}^{∗} and β ∈ B,

we then have z_{jβ} ≡ c_{jβ} mod ⌈m/j⌉_{p}, and therefore c_{jβ} + ℤ ⌈m/j⌉_{p} ∈ ℤ/ℤ ⌈m/j⌉_{p} has order

|⟨c_{jβ} + ℤ ⌈m/j⌉_{p}⟩| = max{1, ⌈m/j⌉_{p} /p^{v_{p}(c_{jβ})}} = ⌈m/(jp^{v_{p}(z_{jβ})})⌉_{p} .

It follows that

|⟨z^{(m)}⟩| = max_{{j∈ℕ^{∗}_{p} \atop β∈B}} ⌈m/(jp^{v_{p}(z_{jβ})})⌉_{p} = ⌈m/ν(z)⌉_{p}

by definition of ν.

(d) This is deduced from (c) and entails (e).

After these preparations we can finally give the

Proof of Theorem 6.4. By Lemma 6.1, U_{S,𝔭} := O_{S}^{∗} ∩ U_{𝔭}^{(1)} = K^{∗} ∩ I_{S}^{𝔭} is free of rank s, say with basis^{6} (u_{1}, …, u_{s}). Set z_{i} = (z_{ijβ})_{{j∈ℕ^{∗}_{p} \atop β∈B}} := ζ(u_{i}) for i ∈

{1, …, s}. According to Kisilevsky’s Theorem 6.6, u_{1}, …, u_{s} and hence also z_{1}, …, z_{s} ∈ Z are ℤ_{p}-linearly independent. We can assume w.l.o.g. that

ν(z_{1}) = n_{1} := min_{1≤i≤s} ν(z_{i}).

Write n_{1} = j_{1}p^{l} with j_{1} := n_{1}^{∗} and l := v_{p}(n_{1}). By definition of ν, there is β_{1} ∈ B such that v_{p}(z_{1j_{1}β_{1}}) = l. For i ∈ {2, …, s} we have l ≤ v_{p}(z_{ij_{1}β_{1}}) and can therefore define

z̃_{i} := z_{i} − \frac{z_{ij_{1}β_{1}}}{z_{1j_{1}β_{1}}} z_{1} ∈ Z.

Replacing z_{2}, …, z_{s} by z̃_{2}, …, z̃_{s} yields a direct decomposition

⟨z_{1}^{(m)}, …, z_{s}^{(m)}⟩ = ⟨z_{1}^{(m)}⟩ ⊕ ⟨z̃_{2}^{(m)}, …, z̃_{s}^{(m)}⟩

for all m ∈ ℕ, while ⟨z_{1}^{(m)}⟩ ≃ ℤ/ℤ ⌈m/n_{1}⌉_{p} by Lemma 6.7(c). We note that ν(z̃_{i}) ≥ ν(z_{i}) ≥ n_{1} for i ∈ {2, …, s} and that z̃_{2}, …, z̃_{s} are again ℤ_{p}-linearly independent. Hence we can apply the same procedure to z̃_{2}, …, z̃_{s} once more and so on. In this way, we obtain positive integers n_{1} ≤ ⋯ ≤ n_{s} such that

U_{S,𝔭}U_{𝔭}^{(m)}/U_{𝔭}^{(m)} ≃ ⟨z_{1}^{(m)}, …, z_{s}^{(m)}⟩ ≃ ∏_{j=1}^{s} ℤ/ℤ⌈\frac{m}{n_{i}} ⌉_{p}

and, consequently,

[K_{S}^{m𝔭} : K_{S}^{𝔭}] = (U_{𝔭}^{(1)} : U_{S,𝔭}U_{𝔭}^{(m)}) = q^{(m−1) deg 𝔭}/ ∏_{i=1}^{s} ⌈\frac{m}{n_{i}}⌉_{p}

for all m ∈ ℕ by the isomorphism (4) on page 38.

6In fact it would suffice to assume ⟨u_{1}, …, u_{s}⟩ of finite index prime to p in U_{S,𝔭}.

The previous proof describes an algorithm for determining the positive integers n_{1} ≤ ⋯ ≤ n_{s}. In practice we shall perform it with some “precision” n ∈ ℕ, i.e. we apply it to z_{1}^{(n)}, …, z_{s}^{(n)} rather than to z_{1}, …, z_{s}. By Lemma 6.7(e) this works fine, if we chose n > n_{s} (which we cannot know beforehand). Otherwise, it is impossible to finish with the algorithm because of 6.7(d), and we have to redo the computations with larger precision. We shall illustrate this algorithm in an

6.8. Example. We continue Example 2.8 and retain the notation therein. Let k ∈ {0, …, 3}. We want to determine the S_{k}-description at 𝔭 := 𝔭_{4}. Note that U_{𝔭}^{(0)} = U_{𝔭}^{(1)} and δ_{S_{k},𝔭}^{(0)} = 1. It will suffice to work with “precision” n = 6. We abbreviate (a + 8ℤ, b + 2ℤ, c + 2ℤ) ∈ Z^{(6)} by (a b c) and let

ζ^{(6)} : \begin{matrix}U_{𝔭}^{(1)}/U_{𝔭}^{(6)} & → & Z^{(6)} \\ (1 + x)^{a}(1 + x^{3})^{b}(1 + x^{5})^{c}U_{𝔭}^{(6)} & ↦ & (a b c) \end{matrix}

be the group isomorphism defined above. The expansion of y at 𝔭 in x has been determined in Example 1.7(a). Using the constructive proof of Proposition 6.2, we compute the “matrix”

C := (ζ^{(6)}(u_{i}/x^{v_{𝔭}(u_{i})}U_{𝔭}^{(6)}))_{1≤i≤4} = ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 1 & 1 \\ 1 & 1 & 0 \end{matrix} ) ∈ (Z^{(6)})^{4}.

Let T_{k} ∈ ℤ^{k×4} consist of the first k rows of T. Then

(ζ^{(6)}(u′_{i}U_{𝔭}^{(6)}))_{1≤i≤3} = T_{3}C = ( \begin{matrix}6 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) ∈ (Z^{(6)})^{3}.

Applying the algorithm of the previous proof yields δ_{S_{1},𝔭} = 1 + t^{2}, δ_{S_{2},𝔭} = 1 + t^{2} + t^{5} and δ_{S_{3},𝔭} = 1 + t + t^{3} + t^{5}. By means of Theorem 6.4 or formula (5) we can now calculate the degrees [K_{S_{k}}^{m𝔭} : K] = [K_{S_{k}}^{m𝔭} : K_{S_{k}}^{𝔭}] for any m ∈ ℕ. The genera are obtained from Corollary 5.9. For m ∈ ℕ_{0} we put K^{(m)} := K_{S_{2}}^{m𝔭}. Note that 𝔭 = 𝔭_{4} ramifies totally and 𝔭_{0}, 𝔭_{1}, 𝔭_{2} split completely in K^{(m)}. From [K_{S_{3}}^{2𝔭} : K] = 1 < 2 = [K_{S_{2}}^{2𝔭} : K] we conclude by Proposition 5.5 that 𝔭_{3} ∉ S(K^{(m)}|K) and, consequently,

N_{K^{(m)}} := |ℙ_{K^{(m)}}^{1}| = 1 + 3[K^{(m)} : K]

for m ≥ 2. The genus and number of rational places of K^{(m)} with m ∈ {2, …, 17} are given in the following table.

q = 2

m  2-3 4-6 7 8-9 10-11 12 13 14  15 16-17
[K^{(m)} : K] 2  4 8 16  32  64 128 256 512 1024
g_{K^{(m)}}  4 10 28 68 164 388 868 1892 4068 8676
N_{K^{(m)}}  7 13 25 49  97 193 385 769 1537 3073

In the previous example, K^{(4)} has the maximal number of rational places possible for a function field L|𝔽_{2} of genus 10, and K^{(7)} and K^{(8)} have more rational places than any example L|𝔽_{2} of genus 28 and 68 found before. We shall further pursue this method of producing global function fields with many rational places in Part III.

Part III

Many Rational Places

As announced in the title, we are now trying to find global function fields K for which the number

N_{K} := |ℙ_{K}^{1} |

is large. In view of our preceding investigations, we are searching for such fields among ray class fields. This is done systematically in Section 9.

Before, in the next section, we turn to the question of how large N_{K} can be. In connection with this problem we shall discuss an upper bound for S-class numbers as well. In Section 8 we shall treat certain ray class fields of the rational function field with many rational places.

7 Upper Bounds

Let K|𝔽_{q} be as before. In 1948 Weil proved the famous function field analogue of the Riemann Hypothesis, stating that the inverse roots ω_{1}, …, ω_{2g_{K}} of the numerator polynomial P_{K}(t) = ∏_{i=1}^{2g_{K}}(1−ω_{i}t) all have absolute value |ω_{i}| = \sqrt{q} (see [St2, Ch. V]). As an immediate consequence one obtains the so-called Hasse-Weil bound for the number of rational places

N_{K} ≤ q + 1 + 2g_{K} \sqrt{q}. (6)

In case equality holds, we call K maximal. Obviously, K can only be maximal if g_{K} = 0 or if q is a square. A consideration involving the constant field extension of degree 2 (see [St2, p. 182]) shows that in addition one must have g_{K} ≤ (q − \sqrt{q})/2. In fact (for each square q), the hermitian function field (see [St2, p. 198]) is a maximal function field of genus (q − \sqrt{q})/2. Stichtenoth and Xing [SX] conjectured a large gap below this genus and showed that there are indeed smaller values of g_{K}, for which K cannot be maximal. Finally Fuhrmann and Torres [FT] succeeded in completing the proof of this conjecture.

7.1. Theorem (Fuhrmann and Torres). Let K|𝔽_{q} be maximal. Then either g_{K} = (q − \sqrt{q})/2 or g_{K} ≤ (\sqrt{q} − 1)^{2}/4.

For non-square q, Serre [S2] could improve (6). We denote by ⌊a⌋ the integer part of a ∈ ℝ.

7.2. Theorem (Serre). For any K|𝔽_{q} we have:

(a) N_{K} ≤ q + 1 + g_{K}⌊2\sqrt{q}⌋.

(b) If N_{K} = q + g_{K}⌊2\sqrt{q}⌋ then g_{K} = 1 or g_{K} = 2.

Proof. (a) See [St2, pp. 180f].

(b) This is proven in [S5, pp. 5–11a] with the aid of a theorem by Siegel on totally positive algebraic integers.

The bound in 7.2(a) is called the Serre bound. Combining 7.2(b) with 7.1, we obtain:

7.3. Corollary. If q is a square ≥ 16 and (\sqrt{q} − 1)^{2}/4 < g_{K} ≠ (q − \sqrt{q})/2, then N_{K} ≤ q − 1 + 2g\sqrt{q}.

The maximum number of rational places a global function field of genus g ∈ ℕ_{0} with full constant field 𝔽_{q} can have is denoted N_{q}(g). We call K|𝔽_{q} optimal if N_{K} = N_{q}(g_{K}). Note that, trivially, N_{q}(0) = q + 1. Also N_{q}(1) and N_{q}(2) have been determined in general. (For details see [S3].) For larger g, the precise value of N_{q}(g) is still unknown in most cases. An overview of what is known about N_{q}(g) for g ≤ 50 and q a not too large power of 2 or 3 is given in [GV7].

For large genera one can improve the above bounds by means of the so-called explicit Weil formulas (see [St2, p. 183]): Each polynomial ψ(t) = c_{1}t + ⋯ + c_{m}t^{m} ∈ ℝ[t] with c_{1}, …, c_{m} ≥ 0 satisfying

1 + ψ(t) + ψ(t^{−1}) ≥ 0 ∀t ∈ ℂ with |t| = 1

(write ψ ≫ 0 for short) provides an estimate

g_{K} ≥ (N_{K} − 1)ψ(q^{−1/2}) − ψ(q^{1/2}). (7)

The maximization of the right hand side of (7) over all ψ ≫ 0 goes back to an unpublished manuscript of Oesterlé’s, given to Serre [S6, pp. 29–37a]. In the following we shall give an outline of this result, which is also presented in [Sf] and [Th].

First note that, for each m ∈ ℕ, the map

F_{m} : \begin{matrix}[\frac{π}{m+1}, \frac{π}{m}] & → & [0, 1] \\ φ & ↦ & −\frac{cos \frac{m+1}{2}φ}{cos \frac{m−1}{2}φ} \end{matrix}

is an isotonous homeomorphism because it has derivative F′_{m}(φ) = (sin mφ + m sin φ)/(2 cos^{2} \frac{m−1}{2}φ). Next we define a map

ϑ_{q} : [q + 1, ∞) → [0, 1)

piecewise as follows. Given a real number N ≥ q + 1, let m ≥ 2 be the unique integer such that N ∈ [q^{m/2} + 1, q^{(m+1)/2} + 1) and set

u := \frac{1}{\sqrt{q}} \frac{q^{(m+1)/2} − N + 1}{N − 1 − q^{(m−1)/2}} ∈ (0, 1].

Then we define ϑ_{q}(N) := cos(F_{m}^{−1}(u)). It is easily verified that ϑ_{q} is again an isotonous homeomorphism. Finally we claim that also

γ_{q} : \begin{matrix}[q + 1, ∞) & → & [0, ∞) \\ N & ↦ & 1 + \frac{(\sqrt{q}ϑ_{q}(N) − 1)N}{q − 2\sqrt{q}ϑ_{q}(N) + 1}. \end{matrix}

is strictly increasing. The isotony of γ_{q} on [q^{2} + 1, ∞) is evident since ϑ_{q}(q^{2} + 1) = 1/\sqrt{2} ≥ 1/\sqrt{q}. A direct computation shows that

γ_{q}(N) = (N − q − 1)/(2\sqrt{q}) for N ∈ [q + 1, q^{3/2} + 1] (8)

and

\begin{aligned}γ_{q}(N) & = \frac{1}{4q}(N − q^{2} − 1 \\ & + \sqrt{(8q + 1)N^{2} − (18q^{2} + 16q + 2)N + q^{4} + 8q^{3} + 18q^{2} + 8q + 1}) \end{aligned} (9)

for N ∈ [q^{3/2} + 1, q^{2} + 1],

whence we see that γ_{q} is strictly increasing on [q + 1, q^{2} + 1] as well.

7.4. Theorem (Oesterlé). Suppose that N_{K} ≥ q + 1. Then g_{K} ≥ γ_{q}(N_{K}). For q ≥ 3 (but not always for q = 2), γ_{q}(N_{K}) is the maximum of the right hand side of (7) over all ψ ≫ 0.

Proof. See [Th, pp. 45-63].

Let N̄_{q} := γ_{q}^{−1} : [0, ∞) → [q + 1, ∞) be the inverse map. Then clearly

N_{q}(g) ≤ N̄_{q}(g) ∀g ∈ ℕ_{0}. (10)

We call this estimate the Oesterlé bound. Equation (8) implies N̄_{q}(g) = q + 1 + 2g\sqrt{q}, i.e. the Oesterlé bound coincides with the Hasse-Weil bound, for g ∈ [0, (q − \sqrt{q})/2]. Furthermore, from (9) we deduce

N̄_{q}(g) = q + 1 + \frac{1}{2}(\sqrt{(8q + 1)g^{2} + 4(q^{2} − q)g} − g) (11)

for g ∈ [\frac{1}{2}(q − \sqrt{q}), \sqrt{\frac{q}{2}}(q − 1)].

The right hand side of (11) is also known as the Ihara bound for N_{q}(g) and has been directly and more elegantly established for all g ∈ ℕ_{0} in [I].

7.5. Example. (a) We want to find a good upper bound for N_{7}(3). The Hasse-Weil bound and the Serre bound both say N_{7}(3) ≤ 23. From (11) we see that

22 < N̄_{7}(3) = (3\sqrt{113} + 13)/2 < 23.

Since N_{7}(3) ≠ 22 by 7.2(b), we can conclude N_{7}(3) ≤ 21. The exact value of N_{7}(3), as determined by Serre [S4], is 20.

(b) By (11) we have N̄_{4}(3) = 5 + \frac{1}{2}(\sqrt{441} − 3) = 14. This shows that the global function field in Example 2.7 is optimal and that N_{4}(3) = 14.

For non-empty S ⊆ ℙ_{K}, the S-class number h_{S} can be estimated by applying Oesterlé’s Theorem to the Hilbert class field of O_{S}.

7.6. Proposition. Assume that K|𝔽_{q} has genus g ≥ 1 and that S ⊆ ℙ_{K} contains at least N > (\sqrt{q} − 1)(g − 1) rational places. Then

h_{S} ≤ ℏ_{q}(g, N) := ϑ_{q}^{−1}(t)/N with t := \frac{N + (q + 1)(g − 1)}{\sqrt{q}(N + 2g − 2)}.

Proof. The restrictions on g and N ensure that 1/\sqrt{q} ≤ t < 1, whereby ℏ_{q} is well-defined. Let K′ := K_{S}^{𝔬} be the Hilbert class field of O_{S}. By Proposition 5.5, its field of constants is again 𝔽_{q}, and the extension K′|K is unramified of degree [K′ : K] = h_{S}. Hence g′ := g_{K′} = 1 + h_{S}(g − 1) by 1.4. Moreover, N′ := N_{K′} ≥ h_{S}N since every place of S splits completely in K′|K. For N′ ≤ q + 1, the claim of the proposition is obvious. Assume N′ ≥ q + 1. Then, by Oesterlé’s Theorem,

\frac{g − 1}{N} ≥ \frac{g′ − 1}{N′} ≥ \frac{γ_{q}(N′) − 1}{N′} = \frac{\sqrt{q}ϑ_{q}(N′) − 1}{q − 2\sqrt{q}ϑ_{q}(N′) + 1}.

Isolating ϑ_{q}(N′) in this inequality results in ϑ_{q}(N′) ≤ t, from which, by the isotony of ϑ_{q}, we conclude the assertion.

From the following lemma, which also helps us compute the bounds ℏ_{q}(g, N), we see that ϑ_{q}^{−1} is piecewise a ℚ(\sqrt{q})-rational function.

7.7. Lemma. For each n ∈ ℕ, define a polynomial

f_{n}(t) := ∑_{i=0}^{⌊n/2⌋} \frac{n}{n−i} \binom{n−i}{i}(−2t − 2)^{⌊n/2⌋−i} ∈ ℤ[t].

(a) The f_{n} satisfy f_{n−1} + f_{n} = f_{n+1} if n is even, and f_{n−1}(t) − 2(t + 1)f_{n}(t) = f_{n+1}(t) if n is odd.

(b) Let t ∈ [cos \frac{π}{m}, cos \frac{π}{m+1}] with 2 ≤ m ∈ ℕ. Then

ϑ_{q}^{−1}(t) = 1 + \frac{q^{(m+1)/2} + q^{m/2}u}{1 + \sqrt{q}u} with u := \frac{f_{m+1}(t)}{f_{m−1}(t)}.

Proof. (a) This is verified using \frac{n−1}{n−i} \binom{n−i}{i−1} + \frac{n}{n−i} \binom{n−i}{i} = \frac{n+1}{n+1−i} \binom{n+1−i}{i}.

(b) Using (a) and the formula cos \frac{n−1}{2}φ + cos \frac{n+1}{2}φ = 2 cos \frac{φ}{2} cos \frac{n}{2}φ, one can show by induction that

(−1)^{⌊n/2⌋} cos \frac{n}{2}φ = \begin{cases}\frac{1}{2}f_{n}(cos φ) & if n is even, \\ (cos \frac{φ}{2})f_{n}(cos φ) & if n is odd, \end{cases}

for all n ∈ ℕ and φ ∈ ℝ. Writing t = cos φ with φ ∈ [\frac{π}{m+1}, \frac{π}{m}], it follows that F_{m}(φ) = f_{m+1}(t)/f_{m−1}(t). Hence the assertion is clear from the definition of ϑ_{q}.

7.8. Example. By means of 7.7(a) we obtain

f_{1}(t) = 1, f_{2}(t) = −2t, f_{3}(t) = −2t + 1, f_{4}(t) = 4t^{2} − 2, f_{5}(t) = 4t^{2} − 2t − 1, f_{6}(t) = −8t^{3} + 6t, f_{7}(t) = −8t^{3} + 4t^{2} + 4t − 1, f_{8}(t) = 16t^{4} − 16t^{2} + 2, f_{9}(t) = 16t^{4} − 8t^{3} − 12t^{2} + 4t + 1, f_{10}(t) = −32t^{5} + 40t^{3} − 10t, f_{11}(t) = −32t^{5} + 16t^{4} + 32t^{3} − 12t^{2} − 6t + 1, f_{12}(t) = 64t^{6} − 96t^{4} + 36t^{2} − 2, … (a) Let K and S be as in Example 2.7. By Lemma 7.7 we have

14ℏ_{4}(3, 14) = ϑ_{4}^{−1}(\frac{2}{3}) = 1 + \frac{16 + 8u}{1 + 2u} with u = \frac{f_{4}(2/3)}{f_{2}(2/3)} = \frac{1}{6}

since \frac{2}{3} ∈ [cos \frac{π}{3}, cos \frac{π}{4}]. Hence h_{S} ≤ ℏ_{4}(3, 14) = 1 according to Proposition 7.6. By Corollary 2.2, we can therefore conclude that K has divisor class number h_{K} = reg_{S} = 512.

(b) Let K = 𝔽_{3}(x, y) where x is an indeterminate over 𝔽_{3} and y satisfies the Artin-Schreier equation y^{3} − y = x^{7} − x^{5}. The numbers of places of K up to degree g_{K} = 6 are given by (|ℙ_{K}^{d} |)_{1≤d≤6} = (10, 9, 0, 18, 36, 108). Hence Z_{K}(t) = 1 + 10t + 64t^{2} + 310t^{3} + 1273t^{4} + 4648t^{5} + 15700t^{6} + ⋯,

P_{K}(t) = (1 − 4t + 3t^{2})Z_{K}(t) = 1 + 6t + 27t^{2} + 84t^{3} + 225t^{4} + 486t^{5}

+ 927t^{6} + 1458t^{7} + 2025t^{8} + 2268t^{9} + 2187t^{10} + 1458t^{11} + 729t^{12}

and h_{K} = P_{K}(1) = 11881 according to Theorem 1.2. Let S ⊆ ℙ_{K} contain at least 5 rational places. We want to show that then h_{S} = 1. We have t := \frac{5}{9} \sqrt{3} ∈ [cos \frac{π}{11}, cos \frac{π}{12}] and u := f_{12}(t)/f_{10}(t) = \frac{1633}{6525} \sqrt{3}, consequently,

ϑ_{3}^{−1}(t) = 1 + \frac{3^{6} + 3^{5}\sqrt{3}u}{1 + \sqrt{3}u} = \frac{993101}{1904}

by Lemma 7.7. From Proposition 7.6 we obtain h_{S} ≤ ℏ_{3}(6, 5) = ϑ_{3}^{−1}(t)/5 = 993101/9520 < 105. Since, on the other hand, h_{S} divides h_{K} = 109^{2}, we can conclude that h_{S} = 1.

8 Rational Function Field

Let K = 𝔽_{q}(x), where x is an indeterminate over 𝔽_{q}. For convenience we again fix an algebraic closure K̄, in which all algebraic extensions of K considered here, especially the ray class fields, are assumed to lie.

Throughout this section we assume that S ⊆ ℙ_{K}^{1} is non-empty and that 𝔭 ∈ ℙ_{K}^{1} \ S is another rational place. Note that under these assumptions we have h_{S} = h_{K} = 1 and I_{S}^{𝔬}/O_{S}^{∗}I_{S}^{𝔭} ≃ U_{𝔭}/𝔽_{q}^{∗}U_{𝔭}^{(1)} = 1, hence K_{S}^{𝔭} = K by Proposition 5.2. For α ∈ 𝔽_{q} denote by 𝔭_{α} the zero of x − α. After a transformation of the variable x we can assume that 𝔭 = ∞ is the pole of x and that 𝔭_{0} ∈ S.

We shall consider two special situations in this section. First we restrict to the case q = p, and in the end we allow q to be an arbitrary p-power again but require |S| = q. In both cases we shall explicitly give the S-description at 𝔭 and provide defining equations for the first few ray class fields.

From now on we assume q = p and set A := {α ∈ 𝔽_{p}^{∗} | 𝔭_{α} ∈ S}. Then |A| = |S| − 1 =: s ∈ {0, …, p − 1}.

8.1. Theorem. Under the above conditions, the S-description at 𝔭 equals

δ_{S,𝔭} = ∑_{n=0}^{s} t^{n}.

The proof depends on the following

8.2. Lemma. Let U_{S,𝔭} := O_{S}^{∗} ∩ U_{𝔭}^{(1)}. Then U_{𝔭}^{(1)} = U_{S,𝔭}U_{𝔭}^{(s+1)}.

Proof of the lemma. Write A = {α_{1}, …, α_{s}} and take π := 1/x as a uniformizer at 𝔭 = ∞. Then U_{S,𝔭} = ⟨1 − α_{1}π, …, 1 − α_{s}π⟩. By Proposition 6.2, it suffices to show that the product

∏_{i=1}^{s} ⟨(1 − α_{i}π)U_{𝔭}^{(s+1)}⟩ ⊆ U_{𝔭}^{(1)}/U_{𝔭}^{(s+1)}

is direct. To this end, let c_{1}, …, c_{s} ∈ {0, …, p − 1} and write

y := ∏_{i=1}^{s}(1 − α_{i}π)^{c_{i}} = ∑_{n∈ℕ_{0}} σ_{n}π^{n}

with 1 = σ_{0}, σ_{1}, σ_{2}, … ∈ 𝔽_{p}. For j ∈ ℕ, set β_{j} := ∑_{i=1}^{s} c_{i}α_{i}^{j} and recall Newton’s Formulas

nσ_{n} + ∑_{j=1}^{n} β_{j}σ_{n−j} = 0 ∀n ∈ ℕ. (12)

Now we assume that y ∈ U_{𝔭}^{(s+1)}, i.e. σ_{1} = ⋯ = σ_{s} = 0. By (12) it follows inductively that β_{1} = ⋯ = β_{s} = 0. In other words, (c_{1}, …, c_{s}), viewed as a vector in 𝔽_{p}^{s}, lies in the kernel of the matrix (α_{i}^{j})_{1≤i,j≤s} ∈ GL_{s}(𝔽_{p}), i.e. c_{1} = ⋯ = c_{s} = 0.

Proof of the theorem. Write δ_{S,𝔭} = 1 + ∑_{n∈ℕ} δ_{n}t^{n} with δ_{1}, δ_{2}, … ∈ {0, 1}. By the lemma we know that K_{S}^{(s+1)𝔭} = K_{S}^{𝔭}. According to the recursive degree formula (5) in Section 6, this implies δ_{S,𝔭}^{(m)} = 1 for m ∈ {1, …, s}, which in turn yields δ_{1} = ⋯ = δ_{s} = 1. Since δ_{S,𝔭}(1) = s+1, we obtain the assertion.

Note that 𝔭 is totally ramified in K_{S}^{m𝔭} for all m ∈ ℕ_{0} since K_{S}^{𝔬} = K is the inertia field of 𝔭 in K_{S}^{m𝔭}. Furthermore, if S′ ⊆ ℙ_{K}^{1} satisfies |S′| − 1 > s, then K_{S′}^{(s+2)𝔭} = K ⊊ K_{S}^{(s+2)𝔭} according to Theorem 8.1. Consequently for all m ≥ s + 2, by Proposition 5.5, S = S(K_{S}^{m𝔭}|K) ∩ ℙ_{K}^{1} and therefore K_{S}^{m𝔭} has exactly

N_{K_{S}^{m𝔭}} = 1 + [K_{S}^{m𝔭} : K](s + 1)

rational places. The genus can be calculated by means of 5.9.

8.3. Example. For p = q ∈ {5, 7} and different values of s = |S| − 1 and m we compute genus and number of rational places of the ray class fields K_{S}^{m𝔭}. As an orientation we include (the integer part of) the Oesterlé bound in the following tables.

q = 5
s  1 2 3 4 1 2  3  4  1  2  3  4
m  3 4 5-6 7 4 5-6 7  8 5-6 7  8  9
[K_{S}^{m𝔭} : K] 5 5 5 5 25 25 25 25 125 125 125 125
g_{K_{S}^{m𝔭}}  2 4 6 10 22 34 56 70 172 284 356 420
N_{K_{S}^{m𝔭}}  11 16 21 26 51 76 101 126 251 376 501 626
⌊N̄_{5}(g_{K_{S}^{m𝔭}})⌋ 14 19 25 34 60 83 125 150 324 506 622 723
q = 7
s  1 2 3 4 5 6 1  2  3  4  5  6
m  3 4 5 6 7-8 9 4  5  6 7-8 9 10
[K_{S}^{m𝔭} : K] 7 7 7 7 7 7 49 49 49 49 49 49
g_{K_{S}^{m𝔭}}  3 6 9 12 15 21 45 69 93 117 162 189
N_{K_{S}^{m𝔭}}  15 22 29 36 43 50 99 148 197 246 295 344
⌊N̄_{7}(g_{K_{S}^{m𝔭}})⌋ 22 32 42 52 60 77 140 198 254 311 413 472

We want to write the first few K_{S}^{m𝔭} as Artin-Schreier extensions in the sense of Example 3.3. Recall the definition of ℕ_{p}^{∗} = ℕ \ pℤ from Section 6.

8.4. Proposition. For s < n ∈ ℕ we set

u_{n} := x^{n−s} ∏_{α∈A}(x − α).

Let m ∈ {s̃+ 1, …, p(s̃+ 1)} with s̃:= s if s < p − 1 and s̃:= p if s = p − 1.^{7} Then

K_{S}^{m∞} = K(℘^{−1}u_{n} | n ∈ ℕ_{p}^{∗}, s̃ < n < m).

7It is not hard to derive from the proof of 6.4 that if m is larger than this, then G(K_{S}^{m∞}|K)

is no longer of exponent p, i.e. K_{S}^{m∞}|K can no longer be written as an Artin-Schreier extension in the sense of 3.3.

Proof. Set J := {n ∈ ℕ_{p}^{∗} | s̃ < n < m} and L := K(℘^{−1}u_{n} | n ∈ J). By 3.3(b) and (c), the extension L|K has conductor 𝔣(L|K) ≤ m∞ and splitting set S(L|K) ⊇ S; hence, L ⊆ K_{S}^{m∞} according to Proposition 5.5. We show equality by comparing degrees. Set

V := ∑_{n∈J} 𝔽_{p}u_{n} ⊆ 𝔽_{p}[x];

then we have deg u ∈ J ⊆ ℕ_{p}^{∗} for all u ∈ V \ {0}. Therefore V ∩ ℘K = V ∩ ℘𝔽_{p}[x] = 0, thus

log_{p}[L : K] = dim_{𝔽_{p}} V = |J|

by 3.3(d). For n ∈ ℕ, Theorem 8.1 and the recursive degree formula (5) yield [K_{S}^{(n+1)∞} : K_{S}^{n∞}] = p if s < n^{∗} and K_{S}^{(n+1)∞} = K_{S}^{n∞} otherwise. Hence

log_{p}[K_{S}^{m∞} : K] = |{n ∈ ℕ | s < n^{∗}, n < m}| .

The inclusion {n ∈ ℕ | s < n^{∗}, n < m} ⊆ J holds due to the restrictions on m.

Now we turn to a different situation. We let q = p^{e} with e ∈ ℕ arbitrary and take S as large as possible under the general assumptions of this section, i.e. S = ℙ_{K}^{1} \ {𝔭}. In this case, the S-description at 𝔭 has been explicitly determined by Lauter [La3]. We present her result in the following.

Take Q := {1, …, q − 1} ⊆ ℤ as a set of representatives for the cyclic group ℤ/(q − 1)ℤ ≃ 𝔽_{q}^{∗}. Via this latter isomorphism (and independent of its specific form), the group G := G(𝔽_{q}|𝔽_{p}) acts on Q. Clearly, two elements n and n′ of Q lie in the same G-orbit Gn = Gn′ iff n′ ≡ p^{l}n mod q − 1 for some l ∈ ℕ_{0}. For n ∈ ℕ we define

e_{n} := \begin{cases}|Gn| & if n ∈ Q and n = min Gn, \\ 0 & otherwise. \end{cases}

Note that if e_{n} > 0, then n ∈ ℕ_{p}^{∗} ∩ Q and e_{n} divides e = |G|. Some other facts concerning the numbers e_{n} are collected in Lemma 8.7 below.

8.5. Theorem (Lauter). Under the above assumptions with |S| = q, the Sdescription at 𝔭 is

δ_{S,𝔭} = 1 + ∑_{n∈ℕ}e_{n}t^{n}.

Proof. See [La3].

Thus the recursive degree formula (5) simplifies to

[K_{S}^{(m+1)𝔭} : K_{S}^{m𝔭}] = q/p^{e_{m^{∗}}} ∀m ∈ ℕ. (13)

By Galois theory, for each l ∈ ℕ_{0} we choose an extension L_{l}|K of degree p^{l} such that L_{l} is an intermediate field of K_{S}^{m𝔭}|K_{S}^{(m−1)𝔭} for some m ∈ ℕ. The genus g_{L_{l}} of L_{l} can be computed by means of 5.9. According to 5.8, the inertia field of 𝔭 in L_{l} is L_{l}^{0}(𝔭) = L_{l} ∩ K_{S}^{𝔬} = K; hence, 𝔭 is totally ramified in L_{l}. Therefore

N_{L_{l}} = 1 + p^{l}q ∀l ∈ ℕ_{0}. (14)

In fact, for small l, the L_{l} tend to have many rational places compared to their genera.

8.6. Example. Let q = 16. Then the G-orbits in Q are

{1, 2, 4, 8}, {3, 6, 9, 12}, {5, 10}, {7, 11, 13, 14} and {15}.

Hence δ_{S,𝔭} = 1 + 4t + 4t^{3} + 2t^{5} + 4t^{7} + t^{15} by Lauter’s Theorem. The degrees of the extensions K_{S}^{m𝔭}|K, as computed by means of (13) for the first few values of m, are as follows

\begin{matrix}m & 0–5 & 6–9 & 10 & 11 & 12–13 & 14–15 & 16–17 & 18 & 19 & 20 & 21 & 22 \\ [K_{S}^{m𝔭} : K] & 1 & 4 & 64 & 2^{8} & 2^{12} & 2^{16} & 2^{19} & 2^{23} & 2^{27} & 2^{31} & 2^{33} & 2^{37} \end{matrix}

Using (14) and the genus formula of Corollary 5.9, we obtain the genera and numbers of rational places of the fields L_{l} for 1 ≤ l ≤ 11 in the table below, the last row of which gives (the integer part of) the Oesterlé bound.

q = 16

l 1 2 3 4 5 6 7 8 9 10 11
g_{L_{l}}  2 6 22 54 118 246 534 1110 2390 4950 10070
N_{L_{l}}  33 65 129 257 513 1025 2049 4097 8193 16385 32769
⌊N̄_{16}(g_{L_{l}})⌋ 33 65 150 309 590 1135 2271 4496 9211 18489 36503

8.7. Lemma. Write q = p^{e} with e ∈ ℕ.

(a) We have e_{q−1} = 1, e_{n} = 0 for q − q/p ≤ n < q − 1, and e_{q−1−q/p} = e.

(b) Let e be even and r := \sqrt{q}. Then e_{n} = e for n ∈ \{1, …, 2r\}\(pℤ∪\{r+1\}). Moreover, e_{r+1} = e/2 and e_{2r+1} = 0.

(c) Let e be odd and r := \sqrt{pq}. Then e_{n} = e for n ∈ \{1, …, r\} \ pℤ and e_{n} = 0 for n ∈ {r, …, r + p}.

Proof. (a) Clearly, the G-orbit of q − 1 is G(q − 1) = {q − 1}; hence e_{q−1} = 1. Let q−q/p ≤ n < q−1. Then q ≠ p and p−1 ≤ n′ := pn−(p−1)(q−1) < q−1. Hence n′ ∈ Gn, and from n − n′ = (p − 1)(q − 1 − n) > 0 we conclude that e_{n} = 0. Finally let n := q − 1 − q/p. Then

p^{l}n = p^{l}(q − 1 − q/p) ≡ q − 1 − p^{l−1} mod q − 1

for all l ∈ {1, …, e}. Thus Gn = {q − 1 − p^{l−1} | 1 ≤ l ≤ e} has e elements and minimum n.

(b) Let n ∈ {1, …, 2r} \ pℤ. Then

n ≤ p^{l}n ∈ {p^{l}, …, 2q/p − p^{l}} ⊆ Q for 0 ≤ l < e/2.

In particular, Gn has at least e/2 elements. Now let e/2 ≤ l < e and write n = a + p^{e−l}b with a ∈ {1, …, p^{e−l} − 1} and b ∈ {0, …, 2p^{l−e/2} − 1}. Then

n′ := p^{l}n − (q − 1)b = p^{l}a + b ∈ {p^{l}, …, q − 1 − (r − 2)p^{l−e/2}} ⊆ Q.

Thus again n < 2r ≤ pr < p^{l} ≤ n′ for e/2 < l < e. If l = e/2, then n = a + rb and n′ = ra + b with a ∈ {1, …, r − 1} and b ∈ {0, 1}; hence n′ − n = (r − 1)(a − b) ≥ 0, and n′ = n only for a = b = 1, i.e. for n = r + 1. This shows n = min Gn and |Gn| > e/2 ⟺ n ≠ r + 1.

Finally we have p^{e/2}(2r + 1) = 2q + r ≡ r + 2 mod q − 1; hence e_{2r+1} = 0.

(c) Let n ∈ {1, …, r} \ pℤ. Then

n ≤ p^{l}n ∈ {p^{l}, …, q − p^{l}} ⊆ Q for 0 ≤ l < (e − 1)/2.

In particular, |Gn| = e. Now let (e + 1)/2 ≤ l < e and write n = a + p^{e−l}b with a ∈ {1, …, p^{e−l} − 1} and b ∈ {0, …, 2p^{l−(e−1)/2} − 1}. Then

n < p^{l} ≤ p^{l}a + b = p^{l}n − (q − 1)b ≤ q − 1 − (r − p)p^{l−(e−1)/2};

hence n = min Gn. As for the second assertion, let n ∈ {r + 1, …, r + p − 1}. Then p^{(e−1)/2}n − q + 1 = r(n − r)/p + 1 < r + 1 ≤ n, hence e_{n} = 0.

As in the lemma, we set r := \sqrt{q} or \sqrt{pq} according to whether q is a square or not. From (13) and 8.7(b) and (c) we conclude K_{S}^{(r+1)𝔭} = K. Furthermore we obtain two canonical fields, namely K_{S}^{(r+2)𝔭} and K_{S}^{(2r+2)𝔭} of degree r and rq over K if q is a square, and p − 1 canonical fields K_{S}^{(r+i+1)𝔭} with 1 ≤ i < p of degree q^{i} over K in case q is non-square. Below we shall give defining equations for these canonical fields. Their genera and those of the intermediate fields L_{l} introduced above are easily computed by means of 5.9.

8.8. Proposition. Write q = p^{e} with e ∈ ℕ.

(a) If e is even, then g_{L_{l}} = \begin{cases}\frac{r}{2}(p^{l} − 1) & for 0 ≤ l ≤ e/2, \\ \frac{r}{2}(2p^{l} − r − 1) & for e/2 ≤ l ≤ 3e/2. \end{cases}

(b) If e is odd, then

g_{L_{l}} = \frac{1}{2} (p^{l}(r + i − 1) − r − \frac{q^{i} − q}{q − 1} )

for (i − 1)e ≤ l ≤ ie with 1 ≤ i < p.

Proof. (a) If 0 ≤ l ≤ e/2, then L_{l} is an intermediate field of K_{S}^{(r+2)𝔭}|K_{S}^{(r+1)𝔭}, whence g_{L_{l}} = 1 + p^{l} \frac{r}{2} − \frac{r+2}{2} = \frac{r}{2}(p^{l} − 1) by Corollary 5.9. For e/2 ≤ l ≤ e/3, L_{l} is an intermediate field of K_{S}^{(2r+2)𝔭}|K_{S}^{(2r+1)𝔭}, thus g_{L_{l}} = 1 + p^{l}r − \frac{1}{2}(r + 2 + q) = \frac{r}{2}(2p^{l} − r − 1) by 5.9.

(b) If (i − 1)e ≤ l ≤ ie with i ∈ {1, …, p − 1}, then L_{l} is an intermediate field of K_{S}^{(r+i)𝔭}|K_{S}^{(r+i+1)𝔭}, thus g_{L_{l}} = 1 + p^{l} \frac{r+i−1}{2} − \frac{1}{2}(r + 1 + ∑_{j=1}^{i−1} q^{j}) = \frac{1}{2}(p^{l}(r + i − 1) − r − \frac{q^{i}−q}{q−1}) by 5.9.

In particular, if e is even, then L_{0}, …, L_{e/2} are maximal, and moreover, according to [RS], L_{e/2} = K_{S}^{(r+2)𝔭} is a hermitian function field. If e is odd and p = 2 or p = 3, then K_{S}^{(r+p)𝔭} = L_{(p−1)e,S} is optimal (cf. [S3]). There is also a connection with Deligne-Lusztig curves which has been elucidated by [La2]. The following equations have already been found by Pedersen [Ped] and Hansen and Stichtenoth [HS] in special cases (cf. also [GS1, GS2]).

8.9. Proposition. (a) Assume that r := \sqrt{q} ∈ ℕ and let y, z ∈ K̄ satisfy y^{r} + y = x^{r+1} and z^{q} − z = x^{2r}(x^{q} − x). Then

K_{S}^{(r+2)∞} = K(y) and K_{S}^{(2r+2)∞} = K(y, z).

(b) Assume that r := \sqrt{pq} ∈ ℕ and let y_{1}, …, y_{p−1} ∈ K̄ satisfy y_{i}^{q} − y_{i} = x^{ir/p}(x^{q} − x). Then

K_{S}^{(r+i+1)∞} = K(y_{1}, …, y_{i}) for i ∈ {1, …, p − 1}.

Proof. Write q = p^{e} with e ∈ ℕ and let ℘ : K̄ → K̄ be the Artin-Schreier operator as defined in Example 3.3. For z ∈ K̄ and β ∈ 𝔽_{q}, define z^{β} := ∑_{j=0}^{e−1}(βz)^{p^{j}}. Then, clearly,

℘z^{β} = β(z^{q} − z). (15)

(a) By [St2, p. 203], the extension K(y)|K has degree r, and S(K(y)|K) contains S. Define the 𝔽_{p}-space Γ := {γ ∈ 𝔽_{q} | γ^{r} + γ = 0} = {β^{r} − β | β ∈ 𝔽_{q}} ≃ 𝔽_{q}/𝔽_{r} ≃ 𝔽_{r} and set

y_{γ} := − ∑_{j=0}^{e/2−1}(γy)^{p^{j}}

for γ ∈ Γ; then ℘y_{γ} = −(γy)^{r} + γy = γx^{r+1}. Since Γx^{r+1} ∩ ℘K = 0, from 3.3 we can conclude K(y) = K(y_{γ} | γ ∈ Γ) and 𝔣(K(y)|K) = (r + 2)∞. Hence K_{S}^{(r+2)∞} = K(y) by the characterization of the ray class fields in Proposition 5.5. For β ∈ 𝔽_{q} we have ℘z^{β} = βx^{2r}(x^{q} − x) =: u_{β} by (15), and

t_{β} := ∑_{j=0}^{e/2−1} (β^{r}x^{r+2})^{p^{j}}

satisfies ℘t_{β} = βx^{q+2r} − β^{r}x^{r+2}, implying ũ_{β} := u_{β} − ℘t_{β} = β^{r}x^{r+2} − βx^{2r+1} and thus v_{∞}^{∗}(u_{β}) = v_{∞}(ũ_{β}) = −(2r + 1) by 3.3(a). Because the 𝔽_{p}-space

Ṽ := Γx^{r+1} ⊕ {ũ_{β} | β ∈ 𝔽_{q}}

has rq elements and Ṽ ∩ ℘K = 0, from 3.3(b)–(d) we obtain K(y, z) = K(y_{γ}, z^{β} | γ ∈ Γ, β ∈ 𝔽_{q}), 𝔣(K(y, z)|K) = (2r + 2)∞ and S ⊆ S(K(y, z)|K). Therefore, again by 5.5, K_{S}^{(2r+2)∞} = K(y, z).

(b) Let i ∈ {1, …, p − 1}. Then K_{i} := K(y_{1}, …, y_{i}) has degree [K_{i} : K] ≤ q^{i} over K. For β ∈ 𝔽_{q} we have ℘y_{i}^{β} = βx^{ir/p}(x^{q} − x) =: u_{iβ} by (15), and

t_{iβ} := ∑_{j=0}^{(e−3)/2} (β^{r}x^{r+i})^{p^{j}}

satisfies ℘t_{iβ} = βx^{q+ir/p}−β^{r}x^{r+i}, implying ũ_{iβ} := u_{iβ} −℘t_{iβ} = β^{r}x^{r+i}−βx^{ir/p+1} and thus v_{∞}^{∗}(u_{iβ}) = v_{∞}(ũ_{iβ}) = −(r + i) by 3.3(a). Since the 𝔽_{p}-space

Ṽ_{i} := {ũ_{jβ} | 1 ≤ j ≤ i, β ∈ 𝔽_{q}}

has q^{i} elements and Ṽ_{i} ∩ ℘K = 0, applying 3.3(b)–(d) shows that K_{i} = K(y_{j}^{β} | 1 ≤ j ≤ i, β ∈ 𝔽_{q}), [K_{i} : K] = q^{i}, 𝔣(K_{i}|K) = (r + i + 1)∞ and S ⊆ S(K_{i}|K). Therefore K_{S}^{(r+i+1)∞} = K_{i} according to Proposition 5.5.

9 Tables of Examples

Global function fields with many rational places have been constructed before by Serre [S2–S6], Hansen and Stichtenoth [Han, HS, St1], Wirtz [Wi], Schoof [Sf], van der Geer and van der Vlugt [GV1–GV7], Lauter [La1, La3], Niederreiter and Xing [XN,NX1–NX9], Shabat [Sh], Garcia, Stichtenoth and Xing [GSX, GS3], Doumen [Do], Özbudak and Stichtenoth [OS], and certainly many others. They use various methods, some of which are quite similar to the ones introduced in this work.

In this section we want to reproduce or, where possible, improve some of the known results by means of ray class field extensions. As a reference point we take the tables in [GV7] and consider function fields of genus up to 50 over 𝔽_{q} for q ∈ {2, 3, 4, 8, 9}. We refrain from also listing the results for larger values of q obtained in [Au] by employing a special method for determining the degrees of ray class fields over a rational function field, that we did not discuss here.

As before, we start with a global function field K|𝔽_{q} of “small” genus g_{K} having “sufficiently” many rational places, which is given by an explicit equation. We choose a non-empty subset S ⊆ ℙ_{K}^{1} and a place 𝔭 ∈ ℙ_{K} \ S and consider the intermediate fields L of K_{S}^{m𝔭}|K_{S}^{(m−1)𝔭} for m ∈ ℕ. From Section 6 and Corollaries 2.2 and 5.9 we recall the formulas

h_{S} reg_{S} = h_{K}, (16)

[K_{S}^{𝔬} : K] = h_{S}, [K_{S}^{𝔭} : K_{S}^{𝔬}] = \frac{q^{deg 𝔭} − 1}{(q − 1)δ_{S,𝔭}^{(0)}}, (17)

[K_{S}^{(m+1)𝔭} : K_{S}^{m𝔭}] = q^{deg 𝔭}/p^{δ_{S,𝔭}^{(m)}} ∀m ∈ ℕ (18)

and

g_{L} = 1 + [L : K](g_{K} − 1 + \frac{m}{2} deg 𝔭) − \frac{1}{2} ∑_{n=0}^{m−1}[K_{S}^{n𝔭} : K] deg 𝔭. (19)

By Theorem 5.8, the inertia degree of 𝔭 in L is [K_{S}^{𝔬} : K_{S∪{𝔭}}^{𝔬}] = h_{S}/h_{S∪{𝔭}}. Therefore L has

(20) N_{L} ≥ [L : K] |S| + \begin{cases}h_{S} & if h_{S} = h_{S∪\{𝔭\}} and deg 𝔭 = 1 \\ 0 & otherwise \end{cases}

rational places, and equality holds iff S = S(L|K) ∩ ℙ_{K}^{1}.

Now the computations are arranged as follows. First we calculate h_{K} via P_{K}(t) by counting all places of K up to degree g_{K}. Next we “guess” S_{0}- units u_{1}, …, u_{s} with s = |S_{0}| − 1, where S_{0} is either equal to or (due to the specific equation for K) slightly larger than ℙ_{K}^{1}. We verify that the upper bound ℏ_{q}(g_{K}, N_{K}) for h_{S_{0}} established in Proposition 7.6 is less than 2 and that reg⟨u_{1}, …, u_{s}⟩ = h_{K} and thereby prove O_{S_{0}}^{∗} = 𝔽_{q}^{∗} × ⟨u_{1}, …, u_{s}⟩. By means of Remark 2.4, Lemma 2.3 and equation (16) we then obtain h_{S} and a ℤ-basis of O_{S}^{∗}/𝔽_{q}^{∗} for any non-empty subset S ⊆ ℙ_{K}^{1}.

These calculations are performed automatically by a program written in KANT/KASH. Also the algorithms for computing S-descriptions discussed in Section 6 have been implemented in KASH. By (17), (18) and (19) we can then calculate the genera of the intermediate fields L. The program is in fact fast enough to run through all non-empty subsets S of ℙ_{K}^{1} \ {𝔭}, so that we can always take S = S(L|K) ∩ ℙ_{K}^{1} and achieve equality in (20).

The whole procedure is applied to several examples of ground fields K|𝔽_{q}, the results for each q being summarized in a corresponding N_{q}(g)-table. In each example, K|𝔽_{q} is defined via an equation, in which x means an indeterminate over 𝔽_{q} and y is algebraic over 𝔽_{q}(x). Furthermore the relevant data of the ground field K|𝔽_{q} are provided. These are the field invariants g_{K}, N_{K} and h_{K}, the set S_{0} containing ℙ_{K}^{1}, a small table of S_{0}-units together with their valuations at all 𝔭 ∈ S_{0}, the S_{0}-regulator and the bound ℏ_{q}(g_{K}, N_{K}). While the rational places are specified by the table of S_{0}-units, other places 𝔭, which are possibly needed, are given in the form 𝔭 = (z_{0}, …), meaning that 𝔭 is the common zero of z_{0}, … in ℙ_{K}. Finally the S-class numbers and S-descriptions leading to an entry in the corresponding N_{q}(g)-table are listed, each S being labelled with its cardinality |S| and a small letter in parantheses.

Every N_{q}(g)-table is superscribed by the corresponding value of q. In its second column it provides a lower and an upper bound, connected by a hyphen, for the number N_{q}(g) with g given in the first column, or a single value if these bounds coincide. Except for some cases in which explicit improvements have been achieved by Serre [S2–S5] or Lauter [La4], the upper bound is (the integer part of) the Oesterlé bound N̄_{q}(g); it is thereby the same as in the tables of [GV7]. The lower bound given here is always greater than or equal to the one in those tables, and it is realized by a certain field L of genus g_{L} = g as outlined above. The last three columns give reference to how this field L is obtained. The column entitled “K,S” gives the number of the example, from which the ground field K is taken, and the label of a set S in this example satisfying S = S(L|K) ∩ ℙ_{K}^{1}. The last column tells us the conductor of L|K in the form 𝔣(L|K) = m𝔭 with 𝔭 ∈ ℙ_{K} and m ∈ ℕ_{0}. Thus L is any intermediate field of K_{S}^{m𝔭}|K_{S}^{(m−1)𝔭} other than K_{S}^{(m−1)𝔭} (or of K_{S}^{𝔬}|K if m = 0) having degree [L : K] as given in the table’s fourth column. The lower bound for N_{g}(g) is set in boldface if our construction by ray class fields improves its previously known value. (Note that many of these improvements have already been added to the tables in [GV7]).

Examples for q = 2

9.1. Example. Let K = 𝔽_{2}(x) and consider the places 𝔭_{0} = (x), 𝔭_{1} = (x+1), 𝔭_{2} = (1/x), 𝔭_{3} = (x^{2} + x + 1), 𝔭_{4} = (x^{4} + x + 1) and 𝔭_{5} = (x^{6} + x^{3} + 1) of K. The relevant S-class numbers and S-descriptions are given below. (1a) S = {𝔭_{0}}, h_{S} = 1, δ_{S,𝔭_{2}} = 1, δ_{S,𝔭_{4}} = 1.

(2a) S = {𝔭_{0}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{5}} = 9 + t.

(2b) S = {𝔭_{0}, 𝔭_{1}}, h_{S} = 1, δ_{S,𝔭_{2}} = 1 + t.

(3a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{3}} = 3 + 2t.

9.2. Example. Let K = 𝔽_{2}(x, y) with y^{2} + y = (x^{2} + x)/(x^{2} + x + 1). Then g_{K} = 1 and N_{K} = 4. The set S_{0} := {𝔭_{0}, …, 𝔭_{5}} and S_{0}-units u_{1}, …, u_{5} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5}
u_{1} = x  1 1 0 0 −1 0
u_{2} = x + 1  0 0 1 1 −1 0
u_{3} = x^{2} + x + 1 0 0 0 0 −2 2
u_{4} = y  1 0 1 0 0 −1
u_{5} = y + x  2 0 0 2 −1 −1

We have ℏ_{2}(1, 4) = 5/4 < 2 and reg⟨u_{1}, …, u_{5}⟩ = 4 = h_{K}. Apart from the rational places also 𝔭_{4} = (1/x) ∈ ℙ_{K}^{2}, 𝔭_{5} = (x^{2} + x + 1) ∈ ℙ_{K}^{2} are used. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{0}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{4}} = 1 + t.

9.3. Example. Let K = 𝔽_{2}(x, y) with y^{2} + y = x^{3} + x. Then g_{K} = 1 and N_{K} = 5. The set S_{0} := {𝔭_{0}, …, 𝔭_{4}} and S_{0}-units u_{1}, …, u_{4} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4}
u_{1} = x  1 1 0 0 −2
u_{2} = x + 1 0 0 1 1 −2
u_{3} = y  1 0 2 0 −3
u_{4} = y + x 2 0 0 1 −3

We have ℏ_{2}(1, 5) = 1 and reg⟨u_{1}, …, u_{4}⟩ = 5 = h_{K}. Apart from the rational places also 𝔭_{5} = (x^{3} + x + 1) ∈ ℙ_{K}^{6}, 𝔭_{6} = (x^{7} + x + 1, y + x^{6} + x^{5} + x^{2} + x) ∈ ℙ_{K}^{7} are used. The relevant S-class numbers and S-descriptions are given below.

(1a) S = {𝔭_{0}}, h_{S} = 5, δ_{S,𝔭_{4}} = 1.

(5a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 1, δ_{S,𝔭_{5}} = 63 + 4t, δ_{S,𝔭_{6}} = 127 + 4t.

9.4. Example. Let K|𝔽_{2} and 𝔭_{0}, …, 𝔭_{4} be as in Examples 1.7(a), 2.8 and 6.8. We recall that g_{K} = 2, N_{K} = 5 and h_{K} = 13. The relevant S-class numbers and S-descriptions are given below. (3a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{4}} = 1 + t^{2} + t^{5}.

9.5. Example. Let K = 𝔽_{2}(x, y) with y^{2} + y = (x^{2} + x)/(x^{3} + x + 1). Then g_{K} = 2 and N_{K} = 6. The set S_{0} := {𝔭_{0}, …, 𝔭_{6}} and S_{0}-units u_{1}, …, u_{6} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6}
u_{1} = x  1 1 0 0 −1 −1 0
u_{2} = x + 1  0 0 1 1 −1 −1 0
u_{3} = x^{3} + x + 1  0 0 0 0 −3 −3 2
u_{4} = y  1 0 1 0 1 0 −1
u_{5} = y + x  2 0 0 3 −1 −1 −1
u_{6} = (x^{3} + x + 1)y + 1 0 4 0 1 −2 −3 0

We have ℏ_{2}(2, 6) = (893 + 306\sqrt{2})/1262 < 2 and reg⟨u_{1}, …, u_{6}⟩ = 19 = h_{K}. Apart from the rational places also 𝔭_{6} = (x^{3} + x + 1) ∈ ℙ_{K}^{3} is used. The relevant S-class numbers and S-descriptions are given below. (1a) S = {𝔭_{1}}, h_{S} = 19.

(4a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{4}, 𝔭_{5}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{3} + t^{7}.

(5a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{3} + t^{5} + t^{7}.

9.6. Example. Let K = 𝔽_{2}(x, y) with y^{2}+y = (x^{3}+x^{2})/(x^{6}+x^{5}+x^{3}+x+1). Then g_{K} = 3 and N_{K} = 6. The set S_{0} := {𝔭_{0}, …, 𝔭_{6}} and S_{0}-units u_{1}, …, u_{6} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6}
u_{1} = x  1 1 0 0 −1 −1 0
u_{2} = x + 1  0 0 1 1 −1 −1 0
u_{3} = x^{2} + x + 1  0 0 0 0 −2 −2 2
u_{4} = y  2 0 1 0 3 0 −3
u_{5} = (x^{2} + x + 1)y + 1  0 1 0 3 0 −2 −1
u_{6} = (x^{2} + x + 1)y + x^{2}  3 0 0 2 −2 −1 −1

We have ℏ_{2}(3, 6) = 195/146 < 2 and reg⟨u_{1}, …, u_{6}⟩ = 52 = h_{K}. Apart from the rational places also 𝔭_{6} = (x^{2} + x + 1) ∈ ℙ_{K}^{2}, 𝔭_{7} = (x^{3} + x^{2} + 1) ∈ ℙ_{K}^{6} are used. The relevant S-class numbers and S-descriptions are given below. (3a) S = {𝔭_{0}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 4, δ_{S,𝔭_{7}} = 21 + 2t.

(4a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{5} + t^{9}.

9.7. Example. Let K = 𝔽_{2}(x, y) with y^{3}+(x^{2}+x+1)y^{2}+(x^{3}+x^{2})y = x^{2}+x. Then g_{K} = 3 and N_{K} = 7. The set S_{0} := {𝔭_{0}, …, 𝔭_{6}} and S_{0}-units u_{1}, …, u_{6} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6}
u_{1} = x  2 1 0 0 −1 −1 −1
u_{2} = x + 1  0 0 2 1 −1 −1 −1
u_{3} = y  1 0 1 0 1 −2 −1
u_{4} = y + 1  0 2 0 1 0 −2 −1
u_{5} = y + x  1 0 0 2 −1 −2 0
u_{6} = y + x^{2}  1 0 0 1 −2 2 −2

We have ℏ_{2}(3, 7) = 911/853 < 2 and reg⟨u_{1}, …, u_{6}⟩ = 71 = h_{K}. The relevant S-class numbers and S-descriptions are given below. (6a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{3} + t^{5} + t^{7} + t^{9}. 9.8. Example. Let K = 𝔽_{2}(x, y) with y^{4}+xy^{2}+(x^{3}+x^{2})y = x^{7}+x^{6}+x^{5}+x^{4}. Then g_{K} = 4 and N_{K} = 8. The set S_{0} := {𝔭_{0}, …, 𝔭_{7}} and S_{0}-units u_{1}, …, u_{7} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7}
u_{1} = x  1 1 2 0 0 0 0 −4
u_{2} = x + 1  0 0 0 1 1 1 1 −4
u_{3} = y  2 1 1 2 0 1 0 −7
u_{4} = y + 1  0 0 0 0 1 0 6 −7
u_{5} = y + x^{2}  3 1 1 0 1 0 2 −8
u_{6} = y^{2} + xy  3 3 2 2 2 1 1 −14
u_{7} = y^{2} + xy + x^{3} + x^{2}  2 2 2 1 1 3 3 −14

We have ℏ_{2}(4, 8) = 1571/1472 < 2 and reg⟨u_{1}, …, u_{7}⟩ = 260 = h_{K}. The relevant S-class numbers and S-descriptions are given below. (4a) S = {𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{7}} = 1 + t^{3} + t^{4} + t^{9}. (6a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{3}} = 1 + t + t^{3} + t^{5} + t^{7} + t^{13}. (7a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{3}} = 1 + t + t^{3} + t^{5} + t^{7} + t^{9} + t^{13}. 9.9. Example. Let K = 𝔽_{2}(x, y) with y^{4} + (x^{2} + x + 1)y^{2} + (x^{2} + x)y = x^{7} + x^{6} + x^{5} + x^{4}. Then g_{K} = 5 and N_{K} = 9. The set S_{0} := {𝔭_{0}, …, 𝔭_{8}} and S_{0}-units u_{1}, …, u_{8} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8}
u_{1} = x  1 1 1 1 0 0 0 0 −4
u_{2} = x + 1  0 0 0 0 1 1 1 1 −4
u_{3} = y  3 0 1 0 2 0 1 0 −7
u_{4} = y + x  1 0 3 0 0 1 0 2 −7
u_{5} = y^{2} + y  3 3 1 1 2 2 1 1 −14
u_{6} = y^{2} + y + x^{3} + x^{2}  2 2 1 1 1 1 3 3 −14
u_{7} = y^{2} + (x^{2} + x + 1)y  3 1 1 2 2 1 1 4 −15
u_{8} = y^{2} + (x^{2} + x + 1)y + x^{3} + x^{2}  2 1 1 3 1 4 2 1 −15

We have ℏ_{2}(5, 9) = \frac{24230319+1494360\sqrt{2}}{24972319} < 2 and reg⟨u_{1}, …, u_{8}⟩ = 975 = h_{K}. The relevant S-class numbers and S-descriptions are given below. (7a) S = {𝔭_{1}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{3} + t^{5} + t^{7} + t^{9} + t^{15}. (8a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{3} + t^{5} + t^{7} + t^{9} +

t^{11} + t^{15}.

The outgrowth of Examples 9.1–9.9 is gathered in the table below.

q = 2

g N_{2}(g) K, S  [L : K] 𝔣(L|K)
6  10  9.3(1a)  10  2𝔭_{4}
7  10  9.3(5a)  2  2𝔭_{5}
8  11  9.5(5a)  2 10𝔭_{0}
9  12  9.1(3a)  4  4𝔭_{3}
10  13  9.4(3a)  4  4𝔭_{4}
12 14–15 9.1(2a)  7  𝔭_{5}
14 15–16 9.1(1a)  15  𝔭_{4}
15  17  9.1(2b)  8  7𝔭_{2}
16 17–18 9.9(8a)  2 14𝔭_{0}
17 17–18 9.1(1a)  16  5𝔭_{2}
19  20  9.3(5a)  4  2𝔭_{5}
20 19–21 9.5(1a)  19  𝔬

g N_{2}(g) K, S  [L : K] 𝔣(L|K)
22 21–22 9.5(5a)  4 12𝔭_{0}
27 24–25 9.2(2a)  12  3𝔭_{4}
28 25–26 9.4(3a)  8  7𝔭_{4}
29 25–27 9.7(6a)  4 14𝔭_{0}
30 25–27 9.8(6a)  4 12𝔭_{3}
35 29–31 9.8(7a)  4 16𝔭_{3}
37 29–32 9.9(7a)  4 14𝔭_{0}
39  33  9.1(2b)  16  8𝔭_{2}
41 33–35 9.8(4a)  8  6𝔭_{7}
42 33–35 9.6(4a)  8  8𝔭_{0}
44 33–37 9.5(4a)  8 11𝔭_{0}
49 36–40 9.6(3a)  12  𝔭_{7}
50  40  9.3(5a)  8  2𝔭_{6}

Examples for q = 3

9.10. Example. Let K = 𝔽_{3}(x) and consider the places 𝔭_{0} = (x), 𝔭_{1} = (x − 1), 𝔭_{2} = (x + 1), 𝔭_{3} = (1/x), 𝔭_{4} = (x^{2} + 1) and 𝔭_{5} = (x^{4} − x^{3} + x^{2} + 1) of K. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{0}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{5}} = 5 + t.

(2b) S = {𝔭_{0}, 𝔭_{1}}, h_{S} = 1, δ_{S,𝔭_{3}} = 1 + t.

(3a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{3}} = 1 + t + t^{2}, δ_{S,𝔭_{4}} = 4 + 2t.

9.11. Example. Let K = 𝔽_{3}(x, y) with y^{2} = x^{3} + x^{2} − x. Then g_{K} = 1 and N_{K} = 6. The set S_{0} := {𝔭_{0}, …, 𝔭_{5}} and S_{0}-units u_{1}, …, u_{5} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5}
u_{1} = x  2 0 0 0 0 −2
u_{2} = x − 1 0 1 1 0 0 −2
u_{3} = x + 1 0 0 0 1 1 −2
u_{4} = y − 1  0 0 1 0 2 −3
u_{5} = y − x 1 0 1 1 0 −3

We have ℏ_{3}(1, 6) = 7/6 < 2 and reg⟨u_{1}, …, u_{5}⟩ = 6 = h_{K}. Apart from the rational places also 𝔭_{6} = (x^{2} − x − 1, y + x − 1) ∈ ℙ_{K}^{2} is used. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{0}, 𝔭_{4}}, h_{S} = 2, δ_{S,𝔭_{6}} = 1 + t.

(3a) S = {𝔭_{0}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 2, δ_{S,𝔭_{6}} = 4 + t + t^{2}.

(3b) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{5}}, h_{S} = 2, δ_{S,𝔭_{0}} = 1 + t + t^{2}.

(4a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 1, δ_{S,𝔭_{6}} = 4 + t + 2t^{2}.

(4b) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{2} + t^{5}. (5a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{2} + t^{4} + t^{5}.

9.12. Example. Let K = 𝔽_{3}(x, y) with y^{2} = x^{3} − x + 1. Then g_{K} = 1 and N_{K} = 7. The set S_{0} := {𝔭_{0}, …, 𝔭_{6}} and S_{0}-units u_{1}, …, u_{6} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6}
u_{1} = x  1 1 0 0 0 0 −2
u_{2} = x − 1  0 0 1 1 0 0 −2
u_{3} = x + 1  0 0 0 0 1 1 −2
u_{4} = y − 1  0 1 0 1 0 1 −3
u_{5} = y − x  0 0 0 2 1 0 −3
u_{6} = y − x − 1 0 2 1 0 0 0 −3

We have ℏ_{3}(1, 7) = 1 and reg⟨u_{1}, …, u_{6}⟩ = 7 = h_{K}. Apart from the rational places also 𝔭_{7} = (x^{3} − x + 1) ∈ ℙ_{K}^{3}, 𝔭_{8} = (x^{2} + 1) ∈ ℙ_{K}^{4} are used. The relevant S-class numbers and S-descriptions are given below. (5a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{7}} = 13 + 2t + 2t^{2}. (6a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{2} + t^{4} + t^{5} + t^{7}. (7a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{8}} = 40 + 4t + 2t^{2}.

9.13. Example. Let K = 𝔽_{3}(x, y) with y^{2} = x^{6} − x^{2} + 1. Then g_{K} = 2 and N_{K} = 8. The set S_{0} := {𝔭_{0}, …, 𝔭_{7}} and S_{0}-units u_{1}, …, u_{7} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7}
u_{1} = x  1 1 0 0 0 0 −1 −1
u_{2} = x − 1  0 0 1 1 0 0 −1 −1
u_{3} = x + 1  0 0 0 0 1 1 −1 −1
u_{4} = y − x^{3}  0 0 0 1 1 0 −3 1
u_{5} = y − x^{3} − 1  0 2 1 0 0 0 −3 0
u_{6} = y − x^{3} + 1  2 0 0 0 0 1 −3 0
u_{7} = y + x^{3} − x + 1 1 0 2 0 1 0 −1 −3

We have ℏ_{3}(2, 8) = 13/11 < 2 and reg⟨u_{1}, …, u_{7}⟩ = 35 = h_{K}. Apart from the rational places also 𝔭_{8} = (x^{2} + 1, y + 1) ∈ ℙ_{K}^{2} is used. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{2}, 𝔭_{4}}, h_{S} = 7, δ_{S,𝔭_{8}} = 1 + t.

9.14. Example. Let K = 𝔽_{3}(x, y) with y^{3} − y = (x^{2} − 1)/x. Then g_{K} = 2 and N_{K} = 8. The set S_{0} := {𝔭_{0}, …, 𝔭_{7}} and S_{0}-units u_{1}, …, u_{7} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} u_{1} = x  3 0 0 0 0 0 0 −3 u_{2} = x − 1  0 1 1 1 0 0 0 −3 u_{3} = x + 1  0 0 0 0 1 1 1 −3 u_{4} = y  −1 1 0 0 1 0 0 −1 u_{5} = y − 1  −1 0 0 1 0 1 0 −1 u_{6} = xy + 1  0 0 2 0 0 2 0 −4 u_{7} = xy + x + 1  0 0 0 2 2 0 0 −4

We have ℏ_{3}(2, 8) = 13/11 < 2 and reg⟨u_{1}, …, u_{7}⟩ = 36 = h_{K}. Apart from the rational places also 𝔭_{8} = (x^{2} + 1, xy − 1) ∈ ℙ_{K}^{2} is used. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{1}, 𝔭_{4}}, h_{S} = 12, δ_{S,𝔭_{8}} = 2 + t.

(4a) S = {𝔭_{1}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}}, h_{S} = 2, δ_{S,𝔭_{7}} = 1 + t + t^{2} + t^{7}.

(6a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{7}} = 1 + t + t^{2} + t^{5} + t^{7} + t^{8}.

(8a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{8}} = 4 + 2t + 2t^{2} + t^{4} + 2t^{5}.

9.15. Example. Let K = 𝔽_{3}(x, y) with y^{3} − y = x^{4} − x^{2}. Then g_{K} = 3 and N_{K} = 10. The set S_{0} := {𝔭_{0}, …, 𝔭_{9}} and S_{0}-units u_{1}, …, u_{9} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8} 𝔭_{9}
u_{1} = x  1 1 1 0 0 0 0 0 0 −3
u_{2} = x − 1  0 0 0 1 1 1 0 0 0 −3
u_{3} = x + 1  0 0 0 0 0 0 1 1 1 −3
u_{4} = y  2 0 0 1 0 0 1 0 0 −4
u_{5} = y − 1  0 0 2 0 0 1 0 0 1 −4
u_{6} = y − x  1 0 0 0 0 2 0 1 0 −4
u_{7} = y − x − 1 0 0 1 0 2 0 1 0 0 −4
u_{8} = y + x  1 0 0 0 1 0 0 0 2 −4
u_{9} = y + x − 1 0 0 1 1 0 0 0 2 0 −4

We have ℏ_{3}(3, 10) = (2414+270\sqrt{3})/2531 < 2 and reg⟨u_{1}, …, u_{9}⟩ = 196 = h_{K}. Apart from the rational places also 𝔭_{10} = (x^{5} + x^{4} − 1, y + x^{4} + x^{3} + x) ∈ ℙ_{K}^{5} is used. The relevant S-class numbers and S-descriptions are given below. (5a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{5}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t^{2} + t^{4} + t^{5} + t^{10}.

(9a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{2} + t^{4} + t^{5} + t^{7} + t^{8} + t^{10} + t^{14}.

(10a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 1, δ_{S,𝔭_{10}} = 121 + 5t + 4t^{2}.

9.16. Example. Let K = 𝔽_{3}(x, y) with y^{3} −y = (x^{3} −x)/(x^{4} −x^{2} +1). Then g_{K} = 4 and N_{K} = 12. The set S_{0} := {𝔭_{0}, …, 𝔭_{12}} and S_{0}-units u_{1}, …, u_{12} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8} 𝔭_{9} 𝔭_{10} 𝔭_{11} 𝔭_{12}
u_{1} = x  1 1 1 0 0 0 0 0 0 −1 −1 −1 0
u_{2} = x − 1  0 0 0 1 1 1 0 0 0 −1 −1 −1 0
u_{3} = x + 1  0 0 0 0 0 0 1 1 1 −1 −1 −1 0
u_{4} = x^{2} + 1  0 0 0 0 0 0 0 0 0 −2 −2 −2 3
u_{5} = y  1 0 0 1 0 0 1 0 0 1 0 0 −2
u_{6} = y − 1  0 0 1 0 1 0 0 1 0 0 0 1 −2
u_{7} = y − x  3 0 0 0 2 0 0 0 2 −1 −1 −1 −2
u_{8} = y − x − 1  0 0 3 0 0 2 2 0 0 −1 −1 −1 −2
u_{9} = (x^{2} + 1)y + 1  0 1 0 0 1 0 0 3 0 −1 −2 −2 0
u_{10} = (x^{2} + 1)y − x^{2}  1 0 0 0 0 1 0 0 3 −2 −2 −1 0
u_{11} = (x^{2} + 1)y − x  3 0 0 0 0 1 0 1 0 −1 −2 −2 0
u_{12} = (x^{2} + 1)y − x^{2} − x − 1 0 0 3 1 0 0 0 0 1 −2 −2 −1 0

We have ℏ_{3}(4, 12) = (610+120\sqrt{3})/759 < 2 and reg⟨u_{1}, …, u_{12}⟩ = 1225 = h_{K}. Apart from the rational places also 𝔭_{12} = (x^{2} + 1) ∈ ℙ_{K}^{2} is used. The relevant S-class numbers and S-descriptions are given below. (10a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{11}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + t + t^{2} + t^{4} +

t^{5} + t^{7} + t^{8} + t^{10} + t^{11} + t^{14}.

The outgrowth of Examples 9.10–9.16 is gathered in the table below.

q = 3

g N_{3}(g) K, S  [L : K] 𝔣(L|K)
5 12–13 9.11(4a)  3  2𝔭_{6}
7  16  9.10(2a)  8  𝔭_{5}
9  19  9.14(6a)  3  5𝔭_{7}
10 19–21 9.10(2b)  9  5𝔭_{3}
13 24–25 9.14(2a)  12  𝔬
14 24–26 9.14(8a)  3  5𝔭_{8}
15  28  9.10(3a)  9  6𝔭_{3}
16 27–29 9.10(3a)  9  3𝔭_{4}
17 24–30 9.14(4a)  6  5𝔭_{7}
19 28–32 9.15(9a)  3 12𝔭_{0}
22 30–36 9.15(10a)  3 3𝔭_{10}
24 31–38 9.16(10a)  3 14𝔭_{0}

g N_{3}(g) K, S  [L : K] 𝔣(L|K)
30 37–46 9.11(4b)  9  8𝔭_{0}
33 46–49 9.15(5a)  9  4𝔭_{0}
34 45–50 9.12(5a)  9  3𝔭_{7}
36 46–52 9.11(5a)  9  9𝔭_{0}
37 48–54 9.14(2a)  24  𝔭_{8}
39 48–56 9.11(2a)  24  2𝔭_{6}
43 55–60 9.12(6a)  9 11𝔭_{0}
45 54–62 9.11(3a)  18  3𝔭_{6}
46 55–63 9.10(2b)  27  6𝔭_{3}
47 54–65 9.11(3b)  18  6𝔭_{0}
48 55–66 9.14(6a)  9 11𝔭_{7}
49 63–67 9.12(7a)  9  3𝔭_{8}
50 56–68 9.13(2a)  28  𝔭_{8}

Examples for q = 4

Let α be the multiplicative generator of 𝔽_{4}^{∗} satisfying α^{2} = α + 1.

9.17. Example. Let K = 𝔽_{4}(x) and consider the places 𝔭_{0} = (x), 𝔭_{1} = (x + 1), 𝔭_{2} = (x + α), 𝔭_{3} = (x + α^{2}), 𝔭_{4} = (1/x), 𝔭_{5} = (x^{2} + x + α) and 𝔭_{6} = (x^{3} +α) of K. The relevant S-class numbers and S-descriptions are given below. (3a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{4}} = 1 + 2t.

(3b) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}}, h_{S} = 1, δ_{S,𝔭_{6}} = 7 + 2t.

(4a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}}, h_{S} = 1, δ_{S,𝔭_{4}} = 1 + 2t + t^{3}, δ_{S,𝔭_{6}} = 21 + 3t.

(5a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}}, h_{S} = 1, δ_{S,𝔭_{5}} = 5 + 4t, δ_{S,𝔭_{6}} = 21 + 4t.

9.18. Example. Let K = 𝔽_{4}(x, y) with y^{2} + y = x^{2}/(x + 1). Then g_{K} = 1 and N_{K} = 8. The set S_{0} := {𝔭_{0}, …, 𝔭_{7}} and S_{0}-units u_{1}, …, u_{7} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7}
u_{1} = x  1 1 0 0 0 0 0 −2
u_{2} = x + 1  0 0 2 0 0 0 0 −2
u_{3} = x + α  0 0 0 1 1 0 0 −2
u_{4} = x + α^{2}  0 0 0 0 0 1 1 −2
u_{5} = y  2 0 −1 0 0 0 0 −1
u_{6} = y + α  0 0 −1 1 0 0 1 −1
u_{7} = y + x  1 0 −1 1 0 1 0 −2

We have ℏ_{4}(1, 8) = 9/8 < 2 and reg⟨u_{1}, …, u_{7}⟩ = 8 = h_{K}. The relevant S-class numbers and S-descriptions are given below. (3a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{7}}, h_{S} = 2, δ_{S,𝔭_{0}} = 1 + t + t^{3}.

(7a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + 2t + 2t^{3} + t^{5} + t^{7}.

9.19. Example. Let K = 𝔽_{4}(x, y) with y^{2} + y = x^{3}. Then g_{K} = 1 and N_{K} = 9. The set S_{0} := {𝔭_{0}, …, 𝔭_{8}} and S_{0}-units u_{1}, …, u_{8} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8}
u_{1} = x  1 1 0 0 0 0 0 0 −2
u_{2} = x + 1  0 0 1 1 0 0 0 0 −2
u_{3} = x + α  0 0 0 0 1 1 0 0 −2
u_{4} = x + α^{2}  0 0 0 0 0 0 1 1 −2
u_{5} = y  3 0 0 0 0 0 0 0 −3
u_{6} = y + α  0 0 1 0 1 0 1 0 −3
u_{7} = y + x  1 0 0 0 1 0 0 1 −3
u_{8} = y + αx 1 0 1 0 0 1 0 0 −3

We have ℏ_{4}(1, 9) = 1 and reg⟨u_{1}, …, u_{8}⟩ = 9 = h_{K}. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{0}, 𝔭_{1}}, h_{S} = 3, δ_{S,𝔭_{8}} = 1 + t^{3}.

9.20. Example. Let K = 𝔽_{4}(x, y) with y^{2} + y = x^{5} + x^{4}. Then g_{K} = 2 and N_{K} = 9. The set S_{0} := {𝔭_{0}, …, 𝔭_{8}} and S_{0}-units u_{1}, …, u_{8} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8}
u_{1} = x  1 1 0 0 0 0 0 0 −2
u_{2} = x + 1  0 0 1 1 0 0 0 0 −2
u_{3} = x + α  0 0 0 0 1 1 0 0 −2
u_{4} = x + α^{2}  0 0 0 0 0 0 1 1 −2
u_{5} = y  4 0 1 0 0 0 0 0 −5
u_{6} = y + x  1 0 0 2 1 0 0 1 −5
u_{7} = y + x^{2}  2 0 0 1 0 1 1 0 −5
u_{8} = y + αx^{3}  3 0 0 0 1 0 2 0 −6

We have ℏ_{4}(2, 9) = 57/41 < 2 and reg⟨u_{1}, …, u_{8}⟩ = 45 = h_{K}. The relevant S-class numbers and S-descriptions are given below. (4a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{8}}, h_{S} = 3, δ_{S,𝔭_{0}} = 1 + t + t^{3} + t^{5}.

9.21. Example. Let K = 𝔽_{4}(x, y) with y^{2} +y = x/(x^{3} +x+1). Then g_{K} = 2 and N_{K} = 10. The set S_{0} := {𝔭_{0}, …, 𝔭_{10}} and S_{0}-units u_{1}, …, u_{10} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8} 𝔭_{9} 𝔭_{10}
u_{1} = x  1 1 0 0 0 0 0 0 −1 −1 0
u_{2} = x + 1  0 0 1 1 0 0 0 0 −1 −1 0
u_{3} = x + α  0 0 0 0 1 1 0 0 −1 −1 0
u_{4} = x + α^{2}  0 0 0 0 0 0 1 1 −1 −1 0
u_{5} = x^{3} + x + 1  0 0 0 0 0 0 0 0 −3 −3 2
u_{6} = y  1 0 0 0 0 0 0 0 2 0 −1
u_{7} = y + α  0 0 1 0 0 1 1 0 0 0 −1
u_{8} = y + x  3 0 0 0 0 1 0 1 −1 −1 −1
u_{9} = (x^{3} + x + 1)y + α  0 0 1 0 0 0 0 3 −1 −3 0
u_{10} = (x^{3} + x + 1)y + x^{2}  1 0 0 0 0 2 0 2 −2 −3 0

We have ℏ_{4}(2, 10) = 41/34 < 2 and reg⟨u_{1}, …, u_{10}⟩ = 55 = h_{K}. Apart from the rational places also 𝔭_{10} = (x^{3} + x + 1) ∈ ℙ_{K}^{3} is used. The relevant S-class numbers and S-descriptions are given below. (3a) S = {𝔭_{1}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 5, δ_{S,𝔭_{0}} = 1 + t + t^{5}.

(4a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 5, δ_{S,𝔭_{10}} = 7 + 2t + t^{2}.

(7a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{2}} = 1 + 2t + t^{3} + 2t^{5} + t^{7}.

(8a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 1, δ_{S,𝔭_{10}} = 21 + 4t + t^{2} + 2t^{3}.

(9a) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + 2t + 2t^{3} + 2t^{5} + t^{7} + t^{9}.

9.22. Example. Let K = 𝔽_{4}(x, y) with y^{4} + y = x^{3} + x^{2} + x. Then g_{K} = 3 and N_{K} = 13. The set S_{0} := {𝔭_{0}, …, 𝔭_{12}} and S_{0}-units u_{1}, …, u_{12} are given by the following table:

𝔭_{0} 𝔭_{1} 𝔭_{2} 𝔭_{3} 𝔭_{4} 𝔭_{5} 𝔭_{6} 𝔭_{7} 𝔭_{8} 𝔭_{9} 𝔭_{10} 𝔭_{11} 𝔭_{12}
u_{1} = x  1 1 1 1 0 0 0 0 0 0 0 0 −4
u_{2} = x + α  0 0 0 0 1 1 1 1 0 0 0 0 −4
u_{3} = x + α^{2}  0 0 0 0 0 0 0 0 1 1 1 1 −4
u_{4} = y  1 0 0 0 1 0 0 0 1 0 0 0 −3
u_{5} = y + 1  0 1 0 0 0 1 0 0 0 1 0 0 −3
u_{6} = y + α  0 0 1 0 0 0 1 0 0 0 1 0 −3
u_{7} = y + x  2 0 0 0 0 0 1 0 0 0 0 1 −4
u_{8} = y + x + 1  0 2 0 0 0 0 0 1 0 0 1 0 −4
u_{9} = y + x + α  0 0 2 0 1 0 0 0 0 1 0 0 −4
u_{10} = y + αx  1 0 0 0 0 0 0 2 0 1 0 0 −4
u_{11} = y + αx + 1  0 1 0 0 0 0 2 0 1 0 0 0 −4
u_{12} = y + αx + α 0 0 1 0 0 2 0 0 0 0 0 1 −4

We have ℏ_{4}(3, 13) = 183/163 < 2 and reg⟨u_{1}, …, u_{12}⟩ = 441 = h_{K}. Apart from the rational places also 𝔭_{13} = (x^{3} + x^{2} + 1, y + x^{2} + x) ∈ ℙ_{K}^{3} is used. The relevant S-class numbers and S-descriptions are given below. (10a) S = {𝔭_{1}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{11}, 𝔭_{12}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + 2t + 2t^{3} +

t^{5} + 2t^{7} + t^{9} + t^{11}.

(10b) S = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{10}, 𝔭_{11}, 𝔭_{12}}, h_{S} = 1, δ_{S,𝔭_{0}} = 1 + 2t + 2t^{3} + 2t^{5} + 2t^{9} + t^{11}.

(13a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{11}, 𝔭_{12}}, h_{S} = 1, δ_{S,𝔭_{13}} = 21 + 6t + 5t^{3} + t^{5}.

9.23. Example. Let K|𝔽_{4} and 𝔭_{0}, …, 𝔭_{13} be as in Example 2.7. We recall that g_{K} = 3, N_{K} = 14 and h_{K} = 512. The relevant S-class numbers and S-descriptions are given below. (10a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{4}, 𝔭_{5}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{13}}, h_{S} = 1, δ_{S,𝔭_{12}} = 1 + 2t + 2t^{3} +

2t^{5} + t^{9} + 2t^{11}.

(11a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{13}}, h_{S} = 1, δ_{S,𝔭_{12}} = 1 + 2t + 2t^{3} + 2t^{5} + t^{7} + t^{9} + 2t^{11}.

(12a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{11}}, h_{S} = 1, δ_{S,𝔭_{12}} = 1 + 2t + 2t^{3} + 2t^{5} + 2t^{7} + 2t^{11} + t^{15}.

(13a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}, 𝔭_{9}, 𝔭_{10}, 𝔭_{11}, 𝔭_{13}}, h_{S} = 1, δ_{S,𝔭_{12}} = 1 + 2t + 2t^{3} + 2t^{5} + 2t^{7} + t^{9} + 2t^{11} + t^{15}.

The outgrowth of Examples 9.17–9.23 is gathered in the table below.

q = 4

g N_{4}(g) K, S  [L : K] 𝔣(L|K)
4  15  9.18(7a)  2  6𝔭_{0}
5 17–18 9.17(4a)  4  6𝔭_{4}
6  20  9.17(5a)  4  2𝔭_{6}
8 21–24 9.22(10a)  2  6𝔭_{0}
9  26  9.18(3a)  8  3𝔭_{0}
10 27–28 9.19(2a)  12  2𝔭_{8}
11 26–30 9.22(13a)  2 4𝔭_{13}
12 29–31 9.18(7a)  4  8𝔭_{0}
13  33  9.17(4a)  8  6𝔭_{4}
14 32–35 9.17(4a)  8  2𝔭_{6}
19 37–43 9.21(9a)  4 10𝔭_{0}
20 40–45 9.17(5a)  8  3𝔭_{6}
21 41–47 9.22(10b)  4  8𝔭_{0}
22 41–48 9.22(10a)  4 10𝔭_{0}
23 45–50 9.23(11a)  4 10𝔭_{12}

g N_{4}(g) K, S  [L : K] 𝔣(L|K)
24 49–52 9.23(12a)  4 10𝔭_{12}
25 51–53 9.20(4a)  12  3𝔭_{0}
27 49–56 9.17(3a)  16  6𝔭_{4}
28 53–58 9.23(13a)  4 14𝔭_{12}
31 60–63 9.21(4a) 15 𝔭_{10}
32 57–65 9.18(7a)  8 10𝔭_{0}
33 65–66 9.17(4a)  16  7𝔭_{4}
34 57–68 9.21(7a)  8  8𝔭_{2}
36 64–71 9.21(8a)  8 3𝔭_{10}
41 65–78 9.21(3a)  20  3𝔭_{0}
43 72–81 9.17(3b)  24  2𝔭_{6}
45 80–84 9.17(5a)  16  4𝔭_{5}
47 73–87 9.21(9a)  8 12𝔭_{0}
48 80–89 9.17(5a) 16 3𝔭_{6}
49 81–90 9.23(10a)  8 10𝔭_{12}

Examples for q = 8

Let α be the multiplicative generator of 𝔽_{8}^{∗} satisfying α^{3} = α + 1.

9.24. Example. Let K = 𝔽_{8}(x) and consider the places 𝔭_{0} = (x), 𝔭_{1} = (x+1), 𝔭_{2} = (x+α), 𝔭_{3} = (x+α^{2}), 𝔭_{4} = (x+α^{3}), 𝔭_{5} = (x+α^{4}), 𝔭_{6} = (x+α^{5}), 𝔭_{7} = (x + α^{6}), 𝔭_{8} = (1/x), 𝔭_{9} = (x^{2} + x + 1) and 𝔭_{10} = (x^{3} + x + α) of K. The relevant S-class numbers and S-descriptions are given below.

(3a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}}, h_{S} = 1, δ_{S,𝔭_{8}} = 1 + 2t. (4a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}}, h_{S} = 1, δ_{S,𝔭_{8}} = 1 + 3t. (4b) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{4}}, h_{S} = 1, δ_{S,𝔭_{8}} = 1 + 2t + t^{2}. (6a) S = {𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{9}} = 9 + 4t + t^{3}. (6b) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}}, h_{S} = 1, δ_{S,𝔭_{10}} = 73 + 5t. (7a) S = {𝔭_{0}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{9}} = 9 + 5t + t^{3}. (8a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}}, h_{S} = 1, δ_{S,𝔭_{8}} = 1+3t+3t^{3} +t^{7}, δ_{S,𝔭_{10}} =

73 + 7t. (9a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}}, h_{S} = 1, δ_{S,𝔭_{9}} = 9 + 6t + 2t^{3}, δ_{S,𝔭_{10}} =

73 + 8t. The outgrowth of Example 9.24 is gathered in the table below.

\begin{aligned}& & q = 8 \\ g & N_{8}(g) & K, S & [L : K] & 𝔣(L|K) \\ 1 & 14 & 9.24(7a) & 2 & 2𝔭_{9} \\ 2 & 18 & 9.24(9a) & 2 & 2𝔭_{10} \\ 3 & 24 & 9.24(6a) & 4 & 2𝔭_{9} \\ 6 & 33–36 & 9.24(8a) & 4 & 6𝔭_{8} \\ 7 & 33–39 & 9.24(4a) & 8 & 4𝔭_{8} \\ 11 & 48–54 & 9.24(6a) & 8 & 3𝔭_{9} \\ 14 & 65 & 9.24(8a) & 8 & 6𝔭_{8} \end{aligned} g N_{8}(g) K, S  [L : K] 𝔣(L|K)

21 72–86 9.24(9a)  8  4𝔭_{9}
27 96–103 9.24(6a)  16  3𝔭_{9}
29 97–109 9.24(3a)  32  4𝔭_{8}
30 96–112 9.24(6b)  16 2𝔭_{10}
38 129–135 9.24(8a)  16  8𝔭_{8}
45 144–156 9.24(9a)  16  4𝔭_{9}
46 129–158 9.24(4b)  32  6𝔭_{8}
48 128–164 9.24(8a)  16 3𝔭_{10}

Examples for q = 9

Let α be the multiplicative generator of 𝔽_{9}^{∗} satisfying α^{2} = α + 1. 9.25. Example. Let K = 𝔽_{9}(x) and consider the places 𝔭_{0} = (x), 𝔭_{1} = (x − 1), 𝔭_{2} = (x + 1), 𝔭_{3} = (x − α), 𝔭_{4} = (x − α^{2}), 𝔭_{5} = (x − α^{3}), 𝔭_{6} = (x − α^{5}), 𝔭_{7} = (x − α^{6}), 𝔭_{8} = (x − α^{7}) and 𝔭_{9} = (1/x) of K. The relevant S-class numbers and S-descriptions are given below. (2a) S = {𝔭_{0}, 𝔭_{1}}, h_{S} = 1, δ_{S,𝔭_{9}} = 1 + t. (5a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{5}}, h_{S} = 1, δ_{S,𝔭_{9}} = 1 + 2t + t^{2} + t^{4}. (6a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{9}} = 1 + 2t + 2t^{2} + t^{5}. (7a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}}, h_{S} = 1, δ_{S,𝔭_{9}} = 1 + 2t + 2t^{2} + t^{4} + t^{5}. (9a) S = {𝔭_{0}, 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}, 𝔭_{5}, 𝔭_{6}, 𝔭_{7}, 𝔭_{8}}, h_{S} = 1, δ_{S,𝔭_{9}} = 1 + 2t + 2t^{2} + t^{4} +

2t^{5} + t^{8}.

72 Part III – Many Rational Places

The outgrowth of Example 9.25 is gathered in the table below.

q = 9

g N_{9}(g) K, S  [L : K] 𝔣(L|K)
1  16  9.25(5a)  3  3𝔭_{9}
3  28  9.25(9a)  3  5𝔭_{9}
12 55–63 9.25(2a)  27  3𝔭_{9}
15 64–74 9.25(7a)  9  6𝔭_{9}
48 163–180 9.25(6a)  27  6𝔭_{9}

List of Symbols

Below we list most of the notation and symbols used in this work. The numbers on the right refer to the page of the definition or of the first occurrence.

α a generator of 𝔽_{q}^{∗} . . . . . . . . . . . . . . 13 α an idèle of K . . . . . . . . . . . . . . . . . 27 α_{𝔭} 𝔭-th component of α ∈ I_{K} … 26 B a basis of 𝔽_{𝔭} over 𝔽_{p} . . . . . . . . . . 41 ℂ the complex numbers ℂ((t)) the field of formal Laurent series over ℂ . . . . . . . . . . . . . . . 9

C_{K} idèle class group of K . . . . . . . 27 C_{S}^{𝔪} S-congruence subgroup mod 𝔪 . . . . . . . . . . . . . . . . . . . . . . 30

Cℓ(O_{S}) S-class group . . . . . . . . . . . . 14 Cℓ^{𝔪}(O_{S}) S-class group mod 𝔪 … 31 D_{K} divisor group of K . . . . . . . . . . . 8 D_{K}^{0} divisors of degree 0 . . . . . . . . . . . 8 D_{K}^{0}/(K^{∗}) divisor class group of K 9 D_{S} S-divisors . . . . . . . . . . . . . . . . . . . 14 D_{S}^{0} S-divisors of degree 0 . . . . . . . . 14

deg degree map on 𝔽_{q}[x] . . . . . . . . . . 9

deg 𝔪 degree of 𝔪 . . . . . . . . . . . . . . . . 8 deg 𝔭 degree of 𝔭 . . . . . . . . . . . . . . . . . 8 deg S degree of S . . . . . . . . . . . . . . . . 14 dim 𝔪 dimension of L(𝔪) . . . . . . . . . 9 𝔡(L|K) discriminant of L|K . . . . . 11 d(𝔮|𝔭) different exponent of 𝔮|𝔭 . . 10 δ a description . . . . . . . . . . . . . . . . . . 40 δ^{(m)} number derived from δ . . . . . 41 δ_{n} n-th coefficient of δ . . . . . . . . . . . 40 δ_{S,𝔭} S-description at 𝔭 . . . . . . . . . . . 41 e = log_{p} q . . . . . . . . . . . . . . . . . . . . . . . 54 e_{n} 0 or length of a G-orbit in Q . 54 e(𝔮|𝔭) ramification index of 𝔮|𝔭 . . 10 F_{m} homeomorphism from [\frac{π}{m+1}, \frac{π}{m}] onto [0, 1] . . . . . . . . . . . . . . . . . . . 47

𝔽_{𝔭} residue field at 𝔭 . . . . . . . . . . . . . . 8 𝔽_{𝔭}((π)) field of expansions in π at 𝔭 . . . . . . . . . . . . . . . . . . . . . 12

𝔽_{q} finite field with q elements . . . . . 8 𝔽_{q^{d}}K constant field extension of degree d over K . . . . . . . . . . . 11

𝔽_{q}(x) rational function field . . . . . . 9 [𝔽_{𝔮} : 𝔽_{𝔭}] inertia degree of 𝔮|𝔭 …. 10 f(L, 𝔭) conductor exponent of 𝔭 in L . . . . . . . . . . . . . . . . . . . . . 20

f_{n}(t) auxiliary polynomials over ℤ . . . . . . . . . . . . . . . . . . . . . 50

𝔣(L|K) conductor of L|K . . . . . . . . 20 G = G(𝔽_{q}|𝔽_{p}) . . . . . . . . . . . . . . . . . . . 54 G(L|K) Galois group of L|K . . . . . 8

GL_{n} invertible n × n-matrices

G^{s}(L, 𝔭) s-th upper ramification group of 𝔭 in L . . . . . . . . . 23

G_{s}(𝔮|𝔭) s-th lower ramification group of 𝔮|𝔭 . . . . . . . . . . . . . . . . . . . 19

G^{s}(𝔮|𝔭) s-th upper ramification group of 𝔮|𝔭 . . . . . . . . . . . . . 20

G^{0}(𝔮|𝔭) inertia group of 𝔮|𝔭 . . . . . . 20 G^{−1}(𝔮|𝔭) decomposition group of 𝔮|𝔭 . . . . . . . . . . . . . . . . . . 20

g_{K} genus of K . . . . . . . . . . . . . . . . . . . 9 γ_{q} lower bound for the genus …. 48 h_{K} divisor class number of K …. 9 h_{S} S-class number . . . . . . . . . . . . . . 14 ℏ_{q} upper bound for S-class numbers . . . . . . . . . . . . . . . . . . . . . 49

η continuous bijection on [−1, ∞) 20 I_{K} idèle group of K . . . . . . . . . . . . . 26 I^{𝔫} large idèle congruence subgroup mod 𝔫 . . . . . . . . . . . . . . . . . . . . . . . 31

I_{S}^{𝔪} S-idèle congruence subgroup mod 𝔪 . . . . . . . . . . . . . . . . . . . . . . 30

K global function field . . . . . . . . . . . 8 K_{𝔭} completion of K at 𝔭 . . . . . . . . 11 K_{S}^{𝔪} S-ray class field mod 𝔪 . . . . . 32 K_{S}^{𝔬} Hilbert class field of O_{S} . . . . . 32 K|𝔽_{q} global function field with full constant field 𝔽_{q} . . . . . . . . . . . . 8

K̄ a fixed algebraic closure of K . 19

K^{∗} group of invertible elements of K

(K^{∗}) group of principal divisors of K . . . . . . . . . . . . . . . . . . . . . . . . 9

L global function field . . . . . . . . . . . 10 L_{l} field of degree p^{l} over 𝔽_{q}(x) in between K_{S}^{m𝔭}|K_{S}^{(m−1)𝔭} . . . . . . . 55

L_{𝔭} completion of L at some 𝔮|𝔭 . . 30 L_{𝔮} completion of L at 𝔮 . . . . . . . . . 11 L_{𝔮}|K_{𝔭} local extension . . . . . . . . . . . 11 L^{s}(𝔭) s-th upper ramification field of 𝔭 in L . . . . . . . . . . . . . . . . . . 23

L^{0}(𝔭) inertia field of 𝔭 in L . . . . . . 24 L^{−1}(𝔭) decomposition field of 𝔭 in L . . . . . . . . . . . . . . . . . . . . . 24

L|𝔽_{q^{d}} extension of K|𝔽_{q} . . . . . . . . . 10 [L : K] degree of L|K L_{1}L_{2} compositum of L_{1} and L_{2} L(𝔪) Riemann-Roch space of 𝔪 . . 9 max{𝔪, 𝔫} coefficientwise maximum of 𝔪 and 𝔫 . . . . . . . . . . . 21

min{𝔪, 𝔫} coefficientwise minimum of 𝔪 and 𝔫 . . . . . . . . . . . 32

𝔪 a divisor . . . . . . . . . . . . . . . . . . . . . . . 8

𝔪̃ tame part ∑_{𝔭∈supp 𝔪} 𝔭 of 𝔪 … 36

𝔪 \ 𝔭 non-𝔭-part of 𝔪 . . . . . . . . . . . 34 N_{K} number of rational places of K . . . . . . . . . . . . . . . . . . . . . . . . 46

N_{L|K} norm map . . . . . . . . . . . . . . . . 10 N_{L|K}(β) norm of β ∈ I_{L} . . . . . . . . 27 N_{q}(g) maximum number of rational places for global function fields K|𝔽_{q} of genus g . . . . . 47

N̄_{q} Oesterlé bound . . . . . . . . . . . . . . 48

ℕ the positive integers ℕ_{0} the non-negative integers ℕ_{p}^{∗} the positive integers prime

to p . . . . . . . . . . . . . . . . . . . . . . . . . 39 N_{L} norm group of L in C_{K} . . . . . . 29 n^{∗} non-p-part of n ∈ ℕ . . . . . . . . . . 39 𝔫 a divisor . . . . . . . . . . . . . . . . . . . . . . . 8 ν valuation-like map on Z . . . . . . . 42 O_{S} ring of S-integral functions . . 14 O_{S}^{∗} group of S-units . . . . . . . . . . . . . 14 (O_{S}^{∗}) group of principal S-divisors . . . . . . . . . . . . . . . . . 14

𝔬 zero element in D_{K} . . . . . . . . . . . . . 9 ω_{j} a generator of 𝔽_{𝔭_{j}}^{∗} . . . . . . . . . . . . 37 ℘ Artin-Schreier operator . . . . . . . 21 φ_{q|𝔭} Frobenius automorphism of 𝔮|𝔭 . . . . . . . . . . . . . . . . . . . . . . . 20

φ_{L,𝔭} Frobenius automorphism of 𝔭 in L . . . . . . . . . . . . . . . . . . . . . . . . 23

ϕ(𝔪) Euler function of 𝔪 . . . . . . . . 31 ℙ_{K} set of all places of K . . . . . . . . . 8 ℙ_{K}^{d} set of places of K of degree d . 8 P_{K}(t) numerator of Z_{K}(t) . . . . . . . . 9 p the characteristic . . . . . . . . . . . . . . . 8 𝔭 a place . . . . . . . . . . . . . . . . . . . . . . . . . 8 𝔭_{α} the zero of x − α . . . . . . . . . . . . . 51 𝔭_{j} place in the support of 𝔪 . . . . . 37

𝔭̄ topological closure of 𝔭 in K_{𝔭} . . 11

π a uniformizer . . . . . . . . . . . . . . . . . 12 ψ(t) polynomial over ℝ . . . . . . . . . . 47 Q representatives {1, …, q − 1} of ℤ/(q − 1)ℤ . . . . . . . . . . . . . . . . . 54

ℚ the rational numbers q a power of p . . . . . . . . . . . . . . . . . . . . 8 𝔮 a place . . . . . . . . . . . . . . . . . . . . . . . . 10 𝔮|𝔭 𝔮 lies over 𝔭 . . . . . . . . . . . . . . . . . 10 R support of 𝔪 . . . . . . . . . . . . . . . . . . 37 R_{𝔭} discrete valuation ring at 𝔭 …. 8 R𝔭̄ topological closure of R_{𝔭} in K_{𝔭} . . . . . . . . . . . . . . . . . . . . . . . . 11

ℝ the real numbers ℝ[t] polynomial ring over ℝ . . . . . 47

reg_{S} S-regulator . . . . . . . . . . . . . . . . 14

reg U regulator of U . . . . . . . . . . . . . 14 S subset of ℙ_{K} . . . . . . . . . . . . . . . . . . 14 S(L|K) splitting set of L|K . . . . . 21 s = |S| − 1, the rank of O_{S}^{∗} . . . . . . 15 supp 𝔪 support of 𝔪 . . . . . . . . . . . . . . 8 ∑_{n} α_{n}π^{n} an expansion in π . . . . . 12 ϑ_{q} homeomorphism from [q + 1, ∞) onto [0, 1) . . . . . . . . . . . . . . . . . . . . 48

U subgroup of O_{S}^{∗} . . . . . . . . . . . . . . . 14 U_{𝔭} unit group at 𝔭 . . . . . . . . . . . . . . 13 U_{𝔭}^{(n)} n-th one-unit group at 𝔭 … 13 U_{𝔭}^{(1)} the one-unit group at 𝔭 . . . . . 13 𝔲 a map with values in (O_{S}^{∗}) …. 16 v_{l} l-adic valuation . . . . . . . . . . . . . . . 16 v_{p} p-adic valuation . . . . . . . . . . . . . . . 8 v_{𝔭} valuation at 𝔭 . . . . . . . . . . . . . . . . . 8 v_{𝔭}^{∗} Artin-Schreier reduced valuation at 𝔭 . . . . . . . . . . . . . . . . . . . . . . . . . . 21

v_{∞} degree valuation on 𝔽_{q}(x) . . . . . 9 x indeterminate over 𝔽_{q} . . . . . . . . . 51 y algebraic element over 𝔽_{q}(x) … 60 ℤ[t] polynomial ring over ℤ ℤ[[t]] formal power series ring over ℤ . . . . . . . . . . . . . . . . . . . . . . 9

Z_{K}(t) zeta function of K . . . . . . . . . 9 Z the ℤ_{p}-module ℤ_{p}^{ℕ_{p}^{∗}×B} . . . . . . . . 42 Z^{(n)} additive group isomorphic to U_{𝔭}^{(1)}/U_{𝔭}^{(n)} . . . . . . . . . . . . . . . 41

ℤ the integers

ℤ_{p} the p-adic integers . . . . . . . . . . . 41 ζ isomorphism U_{𝔭}^{(1)} ≃ Z of ℤ_{p}-modules . . . . . . . . . . . . . . . . . 42

ζ^{(n)} group isomorphism U_{𝔭}^{(1)}/U_{𝔭}^{(n)} ≃ Z^{(n)} . . . . . . . . . . . 41

( ) principal divisor map . . . . . . . . . 9

(z) principal divisor . . . . . . . . . . . . . . 9 (z)_{0} divisor of zeros . . . . . . . . . . . . . . 9 (z)_{∞} divisor of poles . . . . . . . . . . . . . 9

(n) projection of Z onto Z^{(n)} … 42
( , L|K) Artin map of L|K . . . . . . 27
( , L_{𝔭}|K_{𝔭}) Artin map of L_{𝔭}|K_{𝔭} . . 28
[ ]_{𝔭} embedding of K_{𝔭}^{∗} into I_{K} … 27

⟨u_{1}, …, u_{s}⟩ group generated by u_{1}, …, u_{s}

m|n m divides n

f|_{A} restriction of f to A

|a| absolute value of a

|S| cardinality of S

| |_{𝔭} absolute value at 𝔭 . . . . . . . . . . 11

⌊a⌋ integer part of a . . . . . . . . . . . . . 46

⌈a⌉_{p} least p-power integer ≥ a … 39

f ∘ g composite of maps f and g, f after g

n^{∗} non-p-part of n . . . . . . . . . . . . . . 39 ∞ pole of x in 𝔽_{q}(x) . . . . . . . . . . . . . 9

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