no. 4
Algebraic independence and difference equations over elliptic function fields
de Shalit, Ehud1
1Einstein Institute of Mathematics, The Hebrew University of Jerusalem (Israel)
Annales de l'Institut Fourier, Volume 75 (2025) no. 4, pp. 1509-1554

For a lattice $\Lambda $ in the complex plane, let $K_{\Lambda }$ be the field of $\Lambda $-elliptic functions. For two relatively prime integers $p$ (respectively $q$) greater than 1, consider the endomorphisms $\psi $ (resp. $\phi )$ of $K_{\Lambda }$ given by multiplication by $p$ (resp. $q$) on the elliptic curve $\mathbb{C}/\Lambda $. We prove that if $f$ (resp. $g$) are complex Laurent power series that satisfy linear difference equations over $K_{\Lambda }$ with respect to $\phi $ (resp. $\psi $) then there is a dichotomy. Either, for some sublattice $\Lambda ^{\prime }$ of $\Lambda ,$ one of $f$ or $g$ belongs to the ring $K_{\Lambda ^{\prime }}[z,z^{-1},\zeta (z,\Lambda ^{\prime })]$, where $\zeta (z,\Lambda ^{\prime })$ is the Weierstrass zeta function, or $f$ and $g$ are algebraically independent over $K_{\Lambda }.$ This is an elliptic analogue of a recent theorem of Adamczewski, Dreyfus, Hardouin and Wibmer (over the field of rational functions).

Pour un reseau $\Lambda $ dans le plan complexe, soit $K_{\Lambda }$ le corps des fonctions $\Lambda $-elliptiques. Pour deux entiers $p$ (respectivement $q$), premiers entre eux, considérons les endomorphismes $\psi $ (resp. $\phi $) de $K_{\Lambda }$ donnés par multiplication par $p$ (resp. $q$) sur la courbe elliptique $\mathbb{C}/\Lambda $. Nous prouvons que si $f$ (resp. $g$) sont des séries de Laurent complexes qui satisfont les équations aux différences linéaires sur $K_{\Lambda }$ par rapport à $\phi $ (resp. $\psi $), alors il y a une dichotomie. Soit, pour un sous-réseau $\Lambda ^{\prime }$ de $\Lambda ,$ l’un de $f$ ou $g$ appartient à l’anneau $K_{\Lambda ^{\prime }}[z,z^{-1},\zeta (z,\Lambda ^{\prime })],$ où $\zeta (z,\Lambda ^{\prime }$) est la fonction zeta de Weierstrass, ou $f$ et $g$ sont algébriquement indépendents sur $K_{\Lambda }.$ C’est un analogue elliptique d’un théorème récent d’Adamczewski, Dreyfus, Hardouin et Wibmer (sur le corps des fonctions rationelles).

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DOI: 10.5802/aif.3694
Classification: 12H10, 14H52, 39A10
Keywords: Difference equations, elliptic functions, algebraic independence
Mots-clés : équations aux différences, fonctions elliptiques, indépendence algébrique

de Shalit, Ehud  1

1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem (Israel)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
de Shalit, Ehud. Algebraic independence and difference equations over elliptic function fields. Annales de l'Institut Fourier, Volume 75 (2025) no. 4, pp. 1509-1554. doi: 10.5802/aif.3694
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