Global Function Fields Research Articles

Stickelberger series and Main Conjecture for function fields

Let F be a global function field of characteristic p with ring of integers A and let Φ be a Hayes module on the Hilbert class field HA of F. We prove an Iwasawa Main Conjecture for the Z∞p extension F/F generated by the p-power torsion of Φ (p a prime of A). The main tool is a Stickelberger series whose specialization provides a generator for the Fitting ideal of the class group of F. Moreover we prove that the same series, evaluated at complex or p-adic characters, interpolates the Goss Zeta-function or some p-adic L-function, thus providing the link between the algebraic structure (class groups) and the analytic functions, which is the crucial part of Iwasawa Main Conjecture.

On the μ-invariants of abelian varieties over function fields of positive characteristic

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $\mu$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $\mu$-invariant) in terms of other quantities including the Faltings height of $A$ and Frobenius slopes of the numerator of the Hasse-Weil $L$-function of $A / K$ assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this $\mu$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$\mu=0$" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

Analogues of Alladi's formula over global function fields

In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field K, if a set S of prime divisors has a natural density δ(S) within prime divisors, then−limn→∞⁡∑1≤deg⁡D≤nD∈D(K,S)μ(D)|D|=δ(S), where μ(D) is the Möbius function on divisors and D(K,S) is the set of all effective distinguishable divisors whose smallest prime factors are in S. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the Möbius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article.

Computing Singular Elements Modulo Squares

The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtwängler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and S-singular elements (modulo squares) in a global function field.

Annihilators of the ideal class group of a cyclic extension of a global function field

Let $K$ be a global function field and fix a place $\infty $ of $K$. Let $L/K$ be a finite real abelian extension, i.e. a finite, abelian extension such that $\infty $ splits completely in $L$. Then we define a group $C_L$ of elliptic units in $\mathcal O

Universally defining finitely generated subrings of global fields

It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in number fields and rings of S -integers in global function fields of odd characteristic. In this article a proof is presented which is uniform in all global fields, including the characteristic two case, where the result is entirely novel. Furthermore, the proposed method results in universal formulae requiring significantly fewer quantifiers than the formulae that can be derived through the previous approaches.

Modular invariant of rank 1 Drinfeld modules and class field generation

The modular invariant of rank 1 Drinfeld modules is introduced and used to prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is the global function field version of Shimura's Main Theorem of Complex Multiplication.

Even points on an algebraic curve

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.

Rational points of rationally simply connected varieties over global function fields

A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing elementary obstruction has a rational point if it deforms to a rationally simply connected variety in characteristic 0. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre's Conjecture II, and Lang's $C_2$ property.

The discriminant problem for Artin-Schreier extensions: Explicit results

Given a global function field K of characteristic p, for all effective divisors D in the divisor group of K we count the number of cyclic extensions F⊃K of degree p where the relative discriminant DiscK(F)=(p−1)D.

Hall algebras and graphs of Hecke operators for elliptic curves

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.

Category of C-motives over finite fields

In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure L of a finite field with coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category generalizes the previous construction due to Anderson and is more relevant for applications to the theory of G-Shtukas, such as formulating the analog of the Langlands-Rapoport conjecture over function fields. We further develop the analogy with the category of motives over L with coefficients in Q for which the existence of the expected fiber functors depends on famous unproven conjectures.

On Galois extensions with prescribed decomposition groups

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of Q all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realizations with only cyclic decomposition groups and with nonsolvable Galois group. We also investigate the analogous problem over global function fields.

A prime analogue Erdős-Pomerance result for Drinfeld modules with arbitrary endomorphism rings

Let k \mathbf {k} be a global function field, and let A \mathbf {A} be the elements of k \mathbf {k} regular outside a fixed place ∞ \infty . Let ϕ : A → K { τ } \phi :\mathbf {A}\to K\{\tau \} be a Drinfeld module of generic characteristic and rank n n . For a prime ℘ \wp of K K of good reduction, let F ℘ \mathbb {F}_\wp be the residue field at ℘ \wp , and let χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) be the Euler-Poincaré characteristic of F ℘ \mathbb {F}_\wp viewed as an A \mathbf {A} -module. We determine the normal order of the number of distinct prime ideals of A \mathbf {A} dividing χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) , denoted by ω A ( χ A ( ϕ ( F ℘ ) ) ) \omega _\mathbf {A}\Big (\chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big )\Big ) , as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Furthermore, let a ∈ K a\in K be non-torsion for ϕ \phi , and let f a ( ℘ ) f_a(\wp ) be the Euler-Poincaré characteristic of the submodule of ϕ ( F ℘ ) \phi (\mathbb {F}_\wp ) generated by a a modulo ℘ \wp . We also consider the problem of determining the distribution of ω A ( f a ( ℘ ) ) \omega _\mathbf {A}\big (f_a(\wp ) \big ) as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Note that we do not make restrictions on A \mathbf {A} , ϕ \phi , or its endomorphism ring E n d K {sep} ( ϕ ) \rm {End}_{K^\textrm {{sep}}}(\phi ) .

Translation-invariant line bundles on linear algebraic groups

We study the Picard groups of connected linear algebraic groups and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.

Wild sets in global function fields

Abstract Given a self-equivalence of a global function field, its wild set is the set of points where the self-equivalence fails to preserve parity of valuation. In this paper we describe structure of finite wild sets.

On Kronecker terms over global function fields

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical $\Delta$. This leads to analytic means of deriving a Colmez-type formula for "stable Taguchi height" of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for "totally real" function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-Selberg $L$-functions and the Godement-Jacquet $L$-functions associated to automorphic cuspidal representations over global function fields.

Witt equivalence of function fields of conics

Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely function fields over global fields and function fields of curves over local fields, were investigated by the authors in their earlier works [5] and [6]. In the present work, which can be viewed as a sequel to the earlier papers, we discuss the previously obtained results in the specific case of function fields of conic sections, and apply them to provide a few theorems of a somewhat quantitive flavour shedding some light on the question of numbers of Witt non-equivalent classes of such fields.

Unit equations and Fermat surfaces in positive characteristic

In this article we study the three-variable unit equation $x + y + z = 1$ to be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is the $S$-unit group of some global function field. We give upper bounds for the height of solutions and the number of solutions. We also apply these techniques to study the Fermat surface $x^N + y^N + z^N = 1$.

Incidence relation for primes of a global function field

We show that primes of any global function field (of characteristic $\not =2$ and containing a square root of $-1$) form a connected graph of diameter $2$. This fact generalizes our earlier result concerning the graph of even points.