Semi-abelian analogues of Schanuel Conjecture and applications

TL;DR

This paper extends the Schanuel Conjecture to semi-abelian varieties, establishing new conjectures and demonstrating their implications for the algebraic independence of iterated exponential and logarithm values in this broader context.

Contribution

It introduces the Semi-abelian analogue of Schanuel Conjecture as a generalization of existing conjectures, expanding the scope to 1-motives with semi-abelian structures.

Findings

Defines the Semi-abelian Schanuel Conjecture as a Generalized Period Conjecture for 1-motives.

Shows that the conjecture implies algebraic independence of iterated semi-abelian exponential and logarithm values.

Extends previous results from abelian varieties to semi-abelian varieties.

Abstract

In this article we study Semi-abelian analogues of Schanuel conjecture. As showed by the first author, Schanuel Conjecture is equivalent to the Generalized Period Conjecture applied to 1-motives without abelian part. Extending her methods, the second, the third and the fourth authors have introduced the Abelian analogue of Schanuel Conjecture as the Generalized Period Conjecture applied to 1-motives without toric part. As a first result of this paper, we define the Semi-abelian analogue of Schanuel Conjecture as the Generalized Period Conjecture applied to 1-motives. C. Cheng et al. proved that Schanuel conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm, answering a question of M. Waldschmidt. The second, the third and the fourth authors have investigated a similar question in the setup of abelian varieties: the Weak Abelian Schanuel conjecture implies the algebraic independence of the values of the iterated abelian exponential and the values of an iterated generalized abelian logarithm. The main result of this paper is that a Relative Semi-abelian conjecture implies the algebraic independence of the values of the iterated semi-abelian exponential and the values of an iterated generalized semi-abelian logarithm.