Schanuel Property for Elliptic and Quasi--Elliptic Functions
Michel Waldschmidt
TL;DR
The paper investigates a strong, almost-everywhere version of Schanuel's Conjecture in the setting of elliptic and quasi--elliptic functions. It develops a general framework showing that, for almost all tuples, the values of Weierstrass functions $z$, $\wp(z)$, $\zeta(z)$, $\sigma(z)$, exponential functions, and Serre functions $f_u$ are algebraically independent, with explicit constructions of such tuples. It provides detailed independence results in both sigma-free and sigma-including contexts, including CM-extensions, and establishes transcendence properties of the sigma function over fields generated by the other functions. The results connect Schanuel-style transcendence with parametrizations of the exponential of commutative algebraic groups and motivate conjectures of Grothendieck-André generalized period type, highlighting geometric and motivic underpinnings.
Abstract
For almost all tuples $(x_1,\dots,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots,x_n, {\mathrm e}^{x_1},\dots, {\mathrm e}^{x_n}$ are algebraically independent. Similar statements hold when one replaces the exponential function ${\mathrm e}^z$ with algebraically independent functions. We give examples involving elliptic and quasi--elliptic functions, that we prove to be algebraically independent: $z$, $\wp(z)$, $ζ(z)$, $σ(z)$, exponential functions, and Serre functions related with integrals of the third kind.