Hypertranscendence and $q$-difference equations over elliptic functionfields

TL;DR

This paper explores the differential properties of solutions to linear difference equations over elliptic functions, establishing conditions under which these solutions satisfy polynomial differential equations and extending previous results to the elliptic setting.

Contribution

It introduces the first elliptic extension of recent theorems on differential transcendence of difference equation solutions, utilizing parametrized Picard-Vessiot theory and elliptic function analysis.

Findings

Solutions satisfy polynomial differential equations iff they belong to a specific elliptic function ring.

Established an elliptic analogue of integrability results for difference-differential systems.

Extended the understanding of differential transcendence in the context of elliptic functions.

Abstract

The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $\phi f(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},\zeta(z,\Lambda)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $\zeta$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\''afke and Singer over the projective line.