What the study found
The authors propose a split semi-elliptic conjecture for the split case of an extension of an elliptic curve by the multiplicative group, written as G = Gm × E. They also state that this conjecture is equivalent to the Grothendieck-André generalized period conjecture for a specific 1-motive.
Why the authors say this matters
The authors say their conjectures are meant to capture "reasonable" statements about the values of the exponential function and related elliptic functions. They also conclude that the split semi-elliptic conjecture implies several known results and another conjecture about the Weierstrass zeta function.
What the researchers tested
The paper studies conjectures in the setting of an extension of an elliptic curve by the multiplicative group, focusing here on the split case where the extension is a direct product. It involves the exponential function and the Weierstrass wp and zeta functions, and relates the conjecture to a 1-motive of the form M = [u: Z -> Gm^s × E^n].
What worked and what didn't
The authors report that the split semi-elliptic conjecture is equivalent to the Grothendieck-André generalized period conjecture for the stated 1-motive. They also show that it implies three theorems of Schneider on elliptic analogs of the Hermite-Lindemann and Gel'fond-Schneider theorems, as well as a conjecture on the Weierstrass zeta function. The abstract does not report counterexamples or failed tests.
What to keep in mind
This paper treats only the split case; the non-split case is said to be addressed in a second paper in preparation. The abstract does not provide proofs, numerical evidence, or experimental limitations, and it does not describe any empirical study.
Key points
- The paper proposes a split semi-elliptic conjecture for G = Gm × E.
- The conjecture is stated to be equivalent to the Grothendieck-André generalized period conjecture for a specific 1-motive.
- The authors say the conjecture involves the exponential function, Weierstrass wp, and Weierstrass zeta functions.
- The abstract says the split semi-elliptic conjecture implies three theorems of Schneider and a conjecture on the Weierstrass zeta function.
- The non-split case is reserved for a second paper in preparation.
Disclosure
- Research title:
- Split semi-elliptic conjecture links elliptic functions and period conjectures
- Authors:
- Cristiana Bertolin, Michel Waldschmidt
- Institutions:
- University of Padua, Sorbonne Université
- Publication date:
- 2026-04-27
- OpenAlex record:
- View
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