[Indépendence algébrique et équations aux différences sur les corps de fonctions elliptiques]
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).
Révisé le :
Accepté le :
Première publication :
Publié le :
Keywords: Difference equations, elliptic functions, algebraic independence
Mots-clés : équations aux différences, fonctions elliptiques, indépendence algébrique
de Shalit, Ehud 1
CC-BY-ND 4.0
@article{AIF_2025__75_4_1509_0,
author = {de Shalit, Ehud},
title = {Algebraic independence and difference equations over elliptic function fields},
journal = {Annales de l'Institut Fourier},
pages = {1509--1554},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {75},
number = {4},
year = {2025},
doi = {10.5802/aif.3694},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3694/}
}
TY - JOUR AU - de Shalit, Ehud TI - Algebraic independence and difference equations over elliptic function fields JO - Annales de l'Institut Fourier PY - 2025 SP - 1509 EP - 1554 VL - 75 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3694/ DO - 10.5802/aif.3694 LA - en ID - AIF_2025__75_4_1509_0 ER -
%0 Journal Article %A de Shalit, Ehud %T Algebraic independence and difference equations over elliptic function fields %J Annales de l'Institut Fourier %D 2025 %P 1509-1554 %V 75 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3694/ %R 10.5802/aif.3694 %G en %F AIF_2025__75_4_1509_0
de Shalit, Ehud. Algebraic independence and difference equations over elliptic function fields. Annales de l'Institut Fourier, Tome 75 (2025) no. 4, pp. 1509-1554. doi : 10.5802/aif.3694. https://aif.centre-mersenne.org/articles/10.5802/aif.3694/
[1] A problem about Mahler functions, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 17 (2017), pp. 1301-1355 | DOI | MR | Zbl
[2] Hypertranscendence and linear difference equations, J. Am. Math. Soc., Volume 34 (2021), pp. 475-503 | DOI | MR | Zbl
[3] Algebraic independence and linear difference equations, J. Eur. Math. Soc. (2022) | DOI | MR | Zbl
[4] Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3), Volume 7 (1957), pp. 414-452 | DOI | MR | Zbl
[5] Sur les équations fonctionelles -adiques aux -différences, Collect. Math., Volume 43 (1992), pp. 125-140 | MR | Zbl
[6] Difference Galois theory of linear differential equations, Adv. Math., Volume 260 (2014), pp. 1-58 | DOI | MR | Zbl
[7] Difference algebraic relations among solutions of linear differential equations, J. Inst. Math. Jussieu, Volume 16 (2017) no. 1, pp. 59-119 | DOI | MR | Zbl
[8] Functional relations for solutions of -differential equations, Math. Z., Volume 298 (2021) no. 3-4, pp. 1751-1791 | DOI | MR | Zbl
[9] Galois groups of difference equations of order two on elliptic curves, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 11 (2015), 003, 23 pages | DOI | MR | Zbl
[10] Differential Galois theory of linear difference equations, Math. Ann., Volume 342 (2008) no. 2, pp. 333-377 | DOI | Zbl
[11] -Galois theory of linear difference equations, Int. Math. Res. Not., Volume 12 (2015), pp. 3962-4018 | DOI | MR | Zbl
[12] Remarks on automata, functional equations and transcendence, Séminaire de Théorie des Nombres de Bordeaux, Volume 1986-1987 (1986), 27, 11 pages
[13] Galois theory of difference equations, Lecture Notes in Mathematics, 1666, Springer, 1997 | DOI | MR | Zbl
[14] Consistent systems of linear differential and difference equations, J. Eur. Math. Soc., Volume 21 (2019), pp. 2751-2792 | DOI | MR | Zbl
[15] Cohomologie Galoisienne. Cours au College de France, 1962-1963, Lecture Notes in Mathematics, 5, Springer, 1973 | MR | Zbl
[16] Criteria for periodicity and an application to elliptic functions, Can. Math. Bull., Volume 64 (2021), pp. 530-540 | DOI | MR | Zbl
[17] Elliptic -difference modules, Algebra Number Theory, Volume 15 (2021), pp. 1303-1342 | DOI | MR | Zbl
[18] On the structure of certain -difference modules, Enseign. Math. (2), Volume 68 (2022) no. 3-4, pp. 341-377 | DOI | MR | Zbl
Cité par Sources :
