Variations on Schanuel's Conjecture for elliptic and quasi-elliptic functions I: the split case
Cristiana Bertolin, Michel Waldschmidt
TL;DR
The paper proposes a Split Semi-Elliptic Conjecture for the split extension $G=\mathbb{G}_m\times\mathcal{E}$, linking exponential/logarithmic values with Weierstrass $\wp$ and $\zeta$ data via a generalized-period framework. It proves that this conjecture is equivalent to the Elliptico–Toric Conjecture, i.e., the Grothendieck–André generalized period conjecture applied to a specific 1-motive, thereby embedding the problem into arithmetic geometry. The authors show that, assuming the conjecture, elliptic analogues of Hermite–Lindemann and Gel'fond–Schneider follow, and a Weierstrass $\zeta$ conjecture emerges as a corollary, with a rich set of consequences for algebraic independence. A thorough survey of elliptic and quasi-elliptic function theory—particularly Weierstrass $\wp$, $\zeta$, and $\sigma$—provides the technical backbone, including addition/multiplication formulae and CM-specific identities. The work sets the stage for a subsequent treatment of non-split extensions and the full generality of the conjectural framework.
Abstract
It is expected that Schanuel's Conjecture contains all ``reasonable" statements that can be made on the values of {\em the exponential function}. In particular it implies the Lindemann-Weierstrass Theorem and the Conjecture on algebraic independence of logarithms of algebraic numbers. Our goal is to state conjectures {\em à la Schanuel}, which imply conjectures {\em à la Lindemann-Weierstrass}, for the exponential map of an extension $G$ of an elliptic curve ${\mathcal E}$ by the multiplicative group ${\mathbb G}_m$. In the present paper we assume that the extension is split, that is $G={\mathbb G}_m\times {\mathcal E}$. In a second paper in preparation we will deal with the non-split case, namely when the extension is not a product. Here we propose the {\em split semi-elliptic Conjecture}, which involves the exponential function and the Weierstrass $\wp$ and $ζ$ functions, related with integrals of the first and second kind. In the second paper, our {\em non-split semi-elliptic Conjecture} will also involve Serre's functions, related with integrals of the third kind. We expect that our conjectures contain all ``reasonable" statements that can be made on the values of these functions. In the present paper we highlight the geometric origin of the split semi-elliptic Conjecture: it is {\em equivalent to} the Grothendieck-André generalized period Conjecture applied to the 1-motive $M=[u:\mathbb{Z} \rightarrow {\mathbb G}_m^s \times {\mathcal E}^n ]$, which is the Elliptico-Toric Conjecture of the first author. We show that our split semi-elliptic Conjecture implies three theorems of Schneider on elliptic analogs of the Hermite-Lindemann and Gel'fond-Schneider's theorems, as well as a conjecture on the Weierstrass zeta function.