Abstract
In this article we study an abelian analogue of Schanuel’s conjecture. This conjecture falls in the realm of the generalised period conjecture of André. As shown by Bertolin, the generalised period conjecture includes Schanuel’s conjecture as a special case. Extending methods of Bertolin, it can be shown that the abelian analogue of Schanuel’s conjecture we consider also follows from André’s conjecture. Cheng et al. showed that the classical Schanuel’s conjecture implies the algebraic independence of the values of the iterated exponential function and the values of the iterated logarithmic function, answering a question of Waldschmidt. We then investigate a similar question in the setup of abelian varieties.
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Notes
As previously for the field \({{\mathbb {Q}}}\), we denote by \(\overline{K}\) the algebraic closure in \({{\mathbb {C}}}\) of a subfield \(K\subset {{\mathbb {C}}}\).
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Acknowledgements
The authors would like to thank D. Bertrand for helpful discussions and useful comments on an earlier version of this article. The second and the third authors would like to thank the Institut de Mathématiques de Jussieu for hospitality during academic visits in the frame of the IRSES Moduli and LIA. The authors would also like to thank the referee for helpful suggestions.
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Research of the second author is also supported by SERB-DST-NPDF Grant vide PDF/2016/002938.
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Philippon, P., Saha, B. & Saha, E. An abelian analogue of Schanuel’s conjecture and applications. Ramanujan J 52, 381–392 (2020). https://doi.org/10.1007/s11139-019-00173-w
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DOI: https://doi.org/10.1007/s11139-019-00173-w