Semi-abelian analogues of Schanuel conjecture and applications. (English) Zbl 1495.11085
In this paper, the authors study semi-abelian analogues of Schanuel’s conjecture. The first author
[J. Number Theory 97, No. 2, 204–221 (2002; Zbl 1067.11041)] showed that 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 [Ramanujan J. 52, No. 2, 381–392 (2020; Zbl 1496.11099)] have introduced in 2020 the abelian analogue of Schanuel’s conjecture as the generalized period conjecture applied to 1-motives without toric part. In the first result of the paper, the authors define the semi-abelian analogue of Schanuel’s conjecture as the Generalized Period conjecture applied to 1-motives. Answering a question of M. Waldschmidt, C. Cheng et al. [J. Number Theory 129, No. 6, 1464–1467 (2009; Zbl 1242.11052)] proved in 2009 that Schanuel’s conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm. In the setting of abelian varieties, P. Philippon, B. Saha and E. Saha have showed that the weak abelian Schanuel’s 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 the 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.
Reviewer: Tanguy Rivoal (Grenoble)
MSC:
| 11J81 | Transcendence (general theory) |
| 11J89 | Transcendence theory of elliptic and abelian functions |
| 14K20 | Analytic theory of abelian varieties; abelian integrals and differentials |
| 14K25 | Theta functions and abelian varieties |
Keywords:
semi-abelian analogue of Schanuel’s conjecture; linear disjointness; algebraic independence; semi-abelian exponential; generalized semi-abelian logarithmReferences:
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