2025/04/18 by Bertolin, Cristiana, Waldschmidt, Michel · 1 citation
#11J81 11J89 14K25 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2504.14048
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 Gm. In the present paper we assume that the extension is split, that is G=\mathbb Gm× \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:ℤ → \mathbb Gms × \mathcal En ], 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.