# Colloquium: Ehud de Shalit (HUJI) — Difference equations and elliptic functions

A power series f is said to satisfy a p-Mahler equation (p>1 a natural number) if it satisfies a functional equation of the form
a_n(x).f(x^{p^n}) + ... + a_1(x).f(x^p) + a_0(x).f(x) = 0
where the coefficients a_i(x) are polynomials. These functional equations were studied by Kurt Mahler with relation to transcendence theory.

In 2017 Adamczewski and Bell proved a 30-year old conjecture of Loxton and van der Poorten saying that a power series satisfying simultaneously a p-Mahler equation and a q-Mahler equation for multiplicatively independent natural numbers p and q, is (the Laurent expansion of) a rational function.

I will discuss a similar theorem where the coefficients are elliptic functions, and the Mahler operator x^p is replaced by an isogeny of the elliptic curve. Issues of periodicity and of classification of vector bundles over elliptic curves make this case distinctly different, but the resulting theorem has a similar flavor.