#### Vol. 15, No. 5, 2021

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Elliptic $(p,q)$-difference modules

### Ehud de Shalit

Vol. 15 (2021), No. 5, 1303–1342
##### Abstract

Let $p$ and $q$ be multiplicatively independent natural numbers, and $K$ the field $ℂ\left({x}^{1∕s}\mid s=1,2,3\dots \right)$. Let $p$ and $q$ act on $K$ as the Mahler operators $x↦{x}^{p}$ and $x↦{x}^{q}$. Schäfke and Singer (2019) showed that a finite-dimensional vector space over $K$, carrying commuting structures of a $p$-Mahler module and a $q$-Mahler module, is obtained via base change from a similar object over $ℂ$. As a corollary, they gave a new proof of a conjecture of Loxton and van der Poorten, which had been proved before by Adamczewski and Bell (2017). When $K=ℂ\left(x\right)$, and $p$ and $q$ are complex numbers of absolute value greater than 1, acting on $K$ via dilations $x↦px$ and $x↦qx$, a similar theorem has been obtained by Bézivin and Boutabaa (1992). Underlying these two examples are the algebraic groups ${\mathbb{𝔾}}_{m}$ and ${\mathbb{𝔾}}_{a}$, respectively, with $K$ the function field of their universal covering, and $p$, $q$ acting as endomorphisms.

Replacing the multiplicative or additive group by the elliptic curve $ℂ∕\Lambda$, and $K$ by the maximal unramified extension of the field of $\Lambda$-elliptic functions, we study similar objects, which we call elliptic $\left(p,q\right)$-difference modules. Here $p$ and $q$ act on $K$ via isogenies. When $p$ and $q$ are relatively prime, we give a structure theorem for elliptic $\left(p,q\right)$-difference modules. The proof is based on a periodicity theorem, which we prove in somewhat greater generality. A new feature of the elliptic modules is that their classification turns out to be fibered over Atiyah’s classification of vector bundles on elliptic curves (1957).

Only the modules whose associated vector bundle is trivial admit a $ℂ$-structure as in thc case of ${\mathbb{𝔾}}_{m}$ or ${\mathbb{𝔾}}_{a}$, but all of them can be described explicitly with the aid of (logarithmic derivatives of) theta functions. We conclude with a proof of an elliptic analogue of the conjecture of Loxton and van der Poorten.

##### Keywords
difference equations, elliptic functions
##### Mathematical Subject Classification
Primary: 12H10, 14H52, 39A10