A conjecture in Schanuel style for 1-motives
Cristiana Bertolin
TL;DR
This paper extends Schanuel-style transcendence to a semi-elliptic setting by formulating the semi-elliptic Conjecture for the exponential of a 1-motive $G$ (an extension of an elliptic curve by a torus) and showing its equivalence to the Grothendieck-André periods Conjecture for the corresponding 1-motive with elliptic part. It provides a detailed algebraic-analytic apparatus: explicit addition/multiplication relations for Serre functions, endomorphism actions on $\sigma$ and Serre functions, and precise dimension counts for motivic Galois groups, enabling reductions to linearly independent data and torsion/torsion-transfer arguments. The paper proves GA conjecture in key CM and torsion cases and derives the $\sigma$-Conjecture as a corollary of GA for auto-dual 1-motives, linking deep period- and transcendence phenomena to explicit analytic functions on elliptic curves. The results thus connect transcendence theory with motivic Galois groups for 1-motives, providing a robust framework to derive Lindemann–Weierstrass-type statements for semi-elliptic exponential maps and their accompanying Weierstrass and Serre function values.
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-André 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 $\wp,$ $ζ$ 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 $\wp$, $ζ$ 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-André periods Conjecture applied to a 1-motive whose underlying abelian part is an elliptic curve. We prove the Grothendieck-André periods Conjecture for 1-motives defined by an elliptic curve with algebraic invariants and complex multiplication and by torsion points. We introduce the $σ$-Conjecture which involves the Weierstrass $\wp$, $ζ$ and $σ$ functions and we show that this conjecture is a consequence of the Grothendieck-André periods Conjecture applied to an adequate 1-motive.