Workshop on the Arithmetic of Function Fields

Abstract: Goss has defined for an integer ring A of a function field a p-adic Zeta-function with values in entire functions on C_infty. For each p-adic integer n one can thus ask for the distribution of the roots of the associated entire power series f_n.

For A=F_q[t] work of Wan, Diaz-Vargas, Poonen and Sheats on the Newton polygons of the power series f_n yields that all roots of all f_n are simple and have pairwise distinct valuations. In the talk we shall describe the Newton polygons of the f_n for A=F_2[x,y]/(y^2+y+x^3+x+1) explicitly. It follows that, with the exception of the two smallest roots (in absolute value), all roots of the f_n are simple and have pairwise distinct absolute values. For general A no conjecture seems known. In some cases, we present numerical evidence.

Speaker: Dale Brownawell

Title: The algebraic independence of the divided derivatives of the Carlitz period

Abstract: L. Denis ingeniously noted that the logarithmic derivatives of a certain function whose value at T equals the Carlitz period satisfy functional equations appropriate for Mahler's method. In that way he could show that the derivatives of order < p of the Carlitz period with respect to T are algebraically independent over the rational functions in T. In joint work with the late A.J. van der Poorten, we extend Denis' result to all divided derivatives of the Carlitz period. As with Denis, after the insight to consider logarithmic derivatives, the crux is to establish the algebraic independence of the functions underlying the logarithmic derivatives, whose functional equations are now appreciably more complicated.

Speaker: Florian Breuer

Title: Drinfeld modular forms in higher rank

Abstract: Drinfeld modular forms of rank 2 are an interesting analogue of classical modular forms, and have been much studied in the literature. However, very little works seems to have been done on Drinfeld modular forms of higher rank. In this talk I will report on joint work in progress with Richard Pink, where we define general Drinfeld modular forms analytically, and classify all Drinfeld modular forms over F_q[T] with certain simple level structures.

Speaker: David Burns

Title: On main conjectures of geometric Iwasawa theory and related conjectures

Speaker: Chieh-Yu Chang

Title: Algebraic independence of Drinfeld quasi-logarithms

Abstract: In this talk, I will present the joint work with Matt Papanikolas on the algebraic independence of Drinfeld quasi-logarithms under the hypothesis that the fraction field of the endomorphism ring of the given Drinfeld module is separable over the base field. If time is permitted, I will discuss the approaches of B. Conrad how to remove the separability hypothesis.

Speaker: Alina Cojocaru

Title: Almost all reductions of a generic Drinfeld module of arbitrary rank have a large exponent

Abstract: Let A = F_q[T], K a finite field extension of F_q(T), and psi a Drinfeld A-module over K of rank r > 1. For a prime P of K, of good reduction for psi, let F_P be the residue field of K at P and psi_P the reduction of psi modulo P. As an A-module via psi, F_P is isomorphic to A/d(1, P) A × . . . × A/d(r, P) A for uniquely determined monic polynomials d(1, P), . . . , d(r, P) in A with d(1, P) | . . . | d(r, P). We will discuss a result stating that for a density 1 of primes P of K, the infinity norm of the exponent d(r, P) grows as fast as possible (which is almost as fast as the infinity norm of P itself). This is joint work with Drew Shulman (University of Illinois at Chicago).

Speaker: David Goss

Title: The ongoing binomial revolution

Abstract: We will discuss the Binomial Theorem and how it has been essential for function field arithmetic in many ways. In return, function field arithmetic is contributing to a deeper understanding of this fundamental result.

Speaker: Urs Hartl

Title: A-motives with complex multiplication

Speaker: Patrik Hubschmid

Title: Modular subvarieties of Drinfeld modular varieties

Abstract: We consider Drinfeld modular varieties over global function fields as a natural analogue of Shimura varieties and give a definition of modular subvarieties in analogy to Shimura subvarieties. We show that, under mild assumptions, each modular subvariety can be realized as the image of a lower rank Drinfeld modular variety under a closed immersion. Furthermore, we give a classification of all possible modular subvarieties of a given Drinfeld modular variety. This allows us to prove that the degree is unbounded in any infinite set of modular subvarieties and is being applied in the proof of the analogue of the André-Oort conjecture for Drinfeld modular varieties in the separable case.

Speaker: Satoshi Kondo

Title: On the values of Drinfeld modular Beilinson-Kato type elements at supersingular points

Abstract: We consider an analogue of Beilinson-Kato type elements in the K-theory of the moduli of Drinfeld modular varieties. We compute the value at supersingular points via boundary maps. This is a Drinfeld modular analogue of one of first steps in the proof by Beilinson (as presented by Schappacher-Scholl) of the integrality of Beilinson elements. By restricting to Beilinson-Kato type elements (not Beilinson type elements), we prove a similar statement by explicit computation, which then generalizes to higher dimensions.

Speaker: Dmitry Logachev

Title: Duality of Anderson T-motives and related results

Abstract: Let M be an Anderson T-motive. Using the notion of duality for M and a theorem of Anderson, we shall discuss the following results. Let M be uniformizable, over F_q[T], of rank r, dimension n, and let the nilpotent operator N=N(M) be 0.

1. A Siegel matrix of the dual of M is the transposed of a Siegel matrix of M.

2. Let n=r-1. There is a 1--1 correspondence between pure T-motives (all they are uniformizable), and lattices of rank r in C^n having dual (not all such matrices have dual).

3. Let M have good ordinary reduction at P. Then the dual of M also has good ordinary reduction at P, and the kernels of the reduction maps at the groups of P-torsion points of M and of its dual are in perfect duality. We show that in some cases this result holds even if N(M) is not 0.

Speaker: Ignazio Longhi

Title: Stark-Heegner points in the function field setting

Abstract: Let F be a global function field and E/F a non-isotrivial elliptic curve over F. It is possible to mimic Darmon's construction of Stark-Heegner points in this setting, replacing classical modular curves with Drinfeld modular curves. Even more, one can prove that in the function field case Stark-Heegner points are indeed algebraic.

Speaker: Amilcar Pacheco

Title: Torsion points of abelian varieties over function fields

Speaker: Ambrus Pal

Title: The Brauer-Manin obstruction to the local-global principle for the embedding problem

Abstract: We study an analogue of the Brauer-Manin obstruction to the local-global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the Brauer-Manin obstruction is the only one to strong approximation when the embedding problem has abelian kernel and show that the analogue of the algebraic Brauer-Manin obstruction is equivalent to the analogue of the abelian descent obstruction. In the course of our investigations we give a new, elegant description of the Tate duality pairing and prove a new theorem on the cup product in group cohomology. (Joint work with Tomer Schlank.)

Speaker: Mihran Papikian

Title: Modular curves over function fields and odd jacobians

Abstract: In this talk we explain how the jacobians of modular curves arising from quaternion algebras over function fields can be used to construct examples of Tate-Shafarevich groups having non-square order. We will also discuss an explicit relationship of these jacobians with the
jacobians of Drinfeld modular curves.

Speaker: Mohamed Saidi

Title: The anabelian geometry of hyperbolic curves over finite fields

Abstract: I will discuss my joint result with Akio Tamagawa where we prove that the isomorphy type of a hyperbolic curve over a finite field can be recovered from the isomorphy type of its geometrically pro-Sigma arithmetic fundamental group where Sigma is a "large" set of prime integers. I will also discuss a Hom-form of this result.

Speaker: Lenny Taelman

Title: Cohomology of the Carlitz module

Abstract: I will discuss several constructions of invariants of the Carlitz module and more generally of Drinfeld modules. These invariants are A-modules of finite type, analogues to the group of units and the class group of a number field (or the Mordell-Weil and Tate-Shafarevich groups of an elliptic curve). I will give some applications of these constructions, as well as state some open questions.

Speaker: Yuichiro Taguchi

Title: On congruences of Galois representations of function fields

Abstract: We give a simple criterion for two v-adic Galois representations of a global function field K to be locally isomorphic at a place u in terms of their reductions mod v. As an application, we prove that there exist no t-motives over K which have very special properties (in particular, very special types of v-torsion points) if v is "too large".

Speaker: Tom Tucker

Title: Proportions of periodic points on varieties over finite fields

Abstract: Let f: X --> X be a self-map of a variety over a finite field. Every point on X must be preperiodic under f (since we are working over a finite field). Heuristically, one expects that a typical point z is not periodic, though, only preperiodic (that is, some iterate of z is periodic, though z is itself not periodic). By ordering the points in some manner, either by varying the degree or the characteristic, this can be made into a reasonably precise conjecture about proportions of periodic points. We will describe one approach to this problem, using Galois theory of function fields, following ideas of Odoni.

Speaker: Dinesh Thakur

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Speaker: Fabien Trihan

Title: On the Iwasawa main conjecture for abelian varieties over function fields of characteristic p>0

Abstract: We prove an analogue of the Iwasawa main conjecture for abelian varieties over function fields of characteristic p>0 in two cases: CM-abelian varieties over Z_p^d-extensions ramifying at a finite set of places; Without CM abelian varieties over the everywhere unramified Z_p-extension.

Speaker: Douglas Ulmer

Title: Explicit points on the Legendre curve

Abstract: I will explain an elementary and explicit construction of elliptic curves over function fields with Mordell-Weil group of arbitrarily large rank. More advanced methods then lead to precise information on Tate-Shafarevich groups and to a number of open questions.