A conjecture in Schanuel style for 1-motives
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914171693563904 |
|---|---|
| author | Bertolin, Cristiana |
| author_facet | Bertolin, Cristiana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A conjecture in Schanuel style for 1-motives Bertolin, Cristiana Number Theory Algebraic Geometry 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. |
| title | A conjecture in Schanuel style for 1-motives |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2509.08700 |