Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.14048 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915250167611392 |
|---|---|
| author | Bertolin, Cristiana Waldschmidt, Michel |
| author_facet | Bertolin, Cristiana Waldschmidt, Michel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variations on Schanuel's Conjecture for elliptic and quasi-elliptic functions I: the split case Bertolin, Cristiana Waldschmidt, Michel Number Theory 11J81 11J89 14K25 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. |
| title | Variations on Schanuel's Conjecture for elliptic and quasi-elliptic functions I: the split case |
| topic | Number Theory 11J81 11J89 14K25 |
| url | https://arxiv.org/abs/2504.14048 |