Cyclotomic units in function fields

In this paper, we construct a new maximal independent system of units in cyclotomic function fields and their subfields. We also calculate its index in the full units group and show that it is smaller than the index of Feng-Yin's system.

Let K=𝔽 q (C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field 𝔽 q . The ring of regular functions on C-S where S≠∅ is any finite set of closed points on C is a Dedekind domain 𝒪 S of K. For a semisimple 𝒪 S -group G ̲ with a smooth fundamental group F ̲, we aim to describe both the set of genera of G ̲ and its principal genus (the latter if G ̲⊗ 𝒪 S K is isotropic at S) in terms of abelian groups depending on 𝒪 S and F ̲ only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain G ̲. We also use it to express the Tamagawa number τ(G) of a semisimple K-group G by the Euler–Poincaré invariant. This facilitates the computation of τ(G) for twisted K-groups.

This paper shows that the group is boundedly elementary generated for and the ring of algebraic integers in a global function field. Contrary to previous statements for number fields and earlier statements for global function fields, the bounds proven in this paper for elementary bounded generation are independent of the underlying global function field and only depend on the integer . Combining our main result with earlier results, we further establish that elementary bounded generation always has bounds independent from the global field in question, only depending on .

Double complex and gamma-monomials in global function fields

On ring class eigenspaces of Mordell–Weil groups of elliptic curves over global function fields

Groups of Elliptic Units in Global Function Fields

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in [16] for PGL2 and generalized to GLn in [1]. After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann [7] and Fratila [8] to develop an algorithm which explicitly calculates these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section.

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.

Polynomial factorization over finite fields by computing Euler–Poincaré characteristics of Drinfeld modules

We present algorithms for computing j j -invariants, modular polynomials and explicit isogenies for ordinary rank 2 Drinfeld modules over finite fields and describe how Drinfeld modular polynomials can be used to compute isogeny graphs and endomorphism rings of ordinary rank 2 Drinfeld modules. Our technique for computing Drinfeld modular polynomials is based on the traditional analytic approach for obtaining classical modular polynomials. Our ideas for generating isogeny graphs and finding endomorphism rings for rank 2 Drinfeld modules closely follows the work of Kohel and Fouquet. All our algorithms were implemented in SAGE and numerical examples are included.

The purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field. In particular, we explore, among other things, their connection with the theory of Drinfeld modules of rank one (which is function field cyclotomic theory), their interpolations, functional equations, the nature of their special values (transcendence, algebraicity, connection with periods) and various analogies. Most of the facts about the classical gamma function are recalled when necessary. Readers may refer to [G-K] or Artin's book on the classical gamma function. Now we describe more precisely the contents of this paper. Let Fq be a finite field of characteristic p. Section 0 covers the background material on Drinfeld modules. It also describes the gamma function F for Fq[T] (with

Let F F be a finite field, A = F [ T ] A=F[T] , and k = F ( T ) k=F(T) . Let K m = k ( Λ m ) K_{m}=k(\Lambda _{m}) be the field extension of k k obtained by adjoining the m m -torsion on the Carlitz module. The class number h m h_{m} of K m K_{m} can be written as a product h m = h m + h m − h_{m}=h_{m}^{+}h_{m}^{-} . The number h m − h_{m}^{-} is called the relative class number. In this paper a formula for h m − h_{m}^{-} is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of p p -th roots of unity. Some consequences of this formula are also derived.

In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.

Given an elliptic curve $E$ over a global function field $K$, the Galois action on the $n$-torsion points of $E$ gives rise to a mod-$n$ Galois representation $\rho _{E,n}$. For $K$ satisfying some mild conditions, we show that the set of $E$ for which th