A conjecture in Schanuel style for 1-motives

Abstract

Schanuel Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function. In particular it implies the Lindemann-Weierstrass Theorem. In my Ph.D. I showed that Schanuel Conjecture has a geometrical origin: it is equivalent to the Grothendieck-Andr\'e periods Conjecture applied to a 1-motive without abelian part. In this paper, we state a conjecture in Schanuel style, which will imply conjectures in Lindemann-Weierstrass style, for the semi-elliptic exponential function, that is for the exponential map of an extension G of an elliptic curve E by a multiplicative group. We propose the semi-elliptic Conjecture, which concerns the exponential function, the Weierstrass $\wp,$ $\zeta$ functions and Serre functions. The case of a trivial extension has been treated in \cite{BW}, where we introduced the split semi-elliptic Conjecture. As in Schanuel's case, we expect that the semi-elliptic Conjecture contains all ``reasonable" statements that can be made on the values of the exponential function, of the Weierstrass $\wp$, $\zeta$ functions and of Serre functions. We show that the semi-elliptic Conjecture has a geometrical origin (as Schanuel Conjecture): it is equivalent to the Grothendieck-Andr\'e periods Conjecture applied to a 1-motive whose underlying abelian part is an elliptic curve. We prove the Grothendieck-Andr\'e periods Conjecture for 1-motives defined by an elliptic curve with algebraic invariants and complex multiplication and by torsion points. We introduce the $\sigma$-Conjecture which involves the Weierstrass $\wp$, $\zeta$ and $\sigma$ functions and we show that this conjecture is a consequence of the Grothendieck-Andr\'e periods Conjecture applied to an adequate 1-motive.