Difference equations over fields of elliptic functions

Abstract: Adamczewski and Bell proved in 2017 a 30-year old conjecture of Loxton and van der Poorten, asserting that a Laurent power series, which simultaneously satisfies a p-Mahler equation and a q-Mahler equation for multiplicatively independent integers p and q, is a rational function. Similar looking theorems have been proved by Bezivin-Boutabaa and Ramis for pairs of difference, or difference-differential equations. Recently, Schafke and Singer gave a unified treatment of all these theorems.

In this talk we shall discuss a similar theorem for (p,q)-difference equations over fields of elliptic functions. Despite having the same flavor, there are substantial differences, having to do with issues of periodicity, and with the existence of non-trivial (p,q)-invariant vector bundles on the elliptic curve.

dynamical systems

Audience: researchers in the topic

BIRS workshop: Algebraic Dynamics and its Connections to Difference and Differential Equations

Series comments: The field of algebraic dynamics has emerged over the past two decades at the confluence of algebraic geometry, discrete dynamical systems, and diophantine geometry. In recent work, striking connections have been observed between algebraic dynamics and much older theories of difference and differential equations. This meeting brings together mathematicians with expertise in such diverse fields as ring theory, complex dynamics, differential and difference algebra, combinatorics and algebraic geometry. New work towards the dynamical Mordell-Lang and dense orbit conjectures as well as theorems on hypertranscendence and functional independence proven by connecting difference Galois theory, algebraic dynamics and other algebraic approaches to the study offunctional equations will be presented at this meeting.