Generalized theta functions, Drinfeld modules and some arithmetic consequences (original) (raw)

Explicit formulas for Drinfeld modules and their periods

Journal of Number Theory, 2013

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure and an analytic expression for computing the periods of rank 2 Drinfeld modules and also a criterion for supersingularity.

Log-algebraic identities on Drinfeld modules and special L-values

Journal of the London Mathematical Society, 2018

We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules defined over the polynomial ring F q [θ]. This generalizes results of Anderson for the rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.

Identities for Anderson generating functions for Drinfeld modules

Monatshefte für Mathematik, 2014

Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic L-series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.

On a reduction map for Drinfeld modules

Acta Arithmetica, 2020

In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers O K of t-modules that are products of the Drinfeld modules ϕ = φ e1 1 × • • • × φ et t. Here K is a finite extension of the field of fractions of A = F q [t]. We assume that the rank(φ i) = d i and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. End K sep (φ i) = A for i = 1,. .. , t. Our main result is the following numeric criterion. Let N = N e1 1 ו • •×N et t be a finitely generated A submodule of the Mordell-Weil group ϕ(O K) = φ 1 (O K) e1 ו • •×φ t (O K) et , and let Λ ⊂ N be an A-submodule. If we assume d i ≥ e i and P ∈ N such that red W (P) ∈ red W (Λ) for almost all primes W of O K , then P ∈ Λ + N tor. We also build on the recent results of S.Barańczuk [B17] concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned t-modules. This theorem is in fact the detecting linear dependence problem for number fields and of the kind considered in Question 1.1. The reduction map is the usual reduction modulo a non Archimedean prime in a number field. It is well known that some questions concerning number fields can be translated to the context of abelian varieties. An analogous question

Differential characters of Drinfeld modules and de Rham cohomology

Algebra & Number Theory

We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical F-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This F-crystal is of a differential-algebraic nature, and the relation to the classical cohomological realizations is presently not clear.

On the Deuring polynomial for Drinfeld modules in Legendre form

Acta Arithmetica

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Linear Independence and Divided Derivatives of a Drinfeld Module II

In this note we extend our previous results on the linear indepen- dence of values of the divided derivatives of exponential and quasi-periodic functions related to a Drinfeld module to divided derivatives of values of iden- tity and quasi-periodic functions evaluated at the logarithm of an algebraic value. The change in point of view enables us to deal smoothly with divided derivatives of arbitrary order. Moreover we treat a full complement of quasi- periodic functions corresponding to a basis of de Rham cohomology.