Algebraic independence and difference equations over elliptic function fields
[Indépendence algébrique et équations aux différences sur les corps de fonctions elliptiques]
de Shalit, Ehud1
1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem (Israel)
Annales de l'Institut Fourier, Online first, 46 p.

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

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

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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) 
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de Shalit, Ehud. Algebraic independence and difference equations over elliptic function fields. Annales de l'Institut Fourier, Online first, 46 p.

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