Abstract
We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud–Bosch–Lütkebohmert theory of non-Archimedean theta functions for abelian varieties with semi-abelian reduction.
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Acknowledgements
This project was completed as a part of the Mathematics Research Communities workshop held by the American Mathematical Society in the summer of 2013. The authors would like to thank the AMS for their support, and Matthew Baker and Sam Payne for organizing the workshop and for useful discussions. The third author would like to thank Alberto Bellardini for many helpful conversations. The authors gratefully thank the anonymous referee for carefully reading the paper and for providing some very helpful comments. Foster was supported by NSF RTG Grant DMS-0943832 and by Le Laboratoire d’Excellence CARMIN. Rabinoff was supported by NSF DMS-1601842. Soto would like to thank to the KU Leuven and to the Goethe Universität Frankfurt am Main for the great working conditions.
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Foster, T., Rabinoff, J., Shokrieh, F. et al. Non-Archimedean and tropical theta functions. Math. Ann. 372, 891–914 (2018). https://doi.org/10.1007/s00208-018-1646-3
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DOI: https://doi.org/10.1007/s00208-018-1646-3