Abstract
Assuming the Mumford–Tate conjecture, we show that the center of the endomorphism ring of an abelian variety defined over a number field can be recovered from an appropriate intersection of the fields obtained from its Frobenius endomorphisms. We then apply this result to exhibit a practical algorithm to compute this center.
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Acknowledgements
The authors would like to thank Andrea Maffei for an enlightening discussion about the Steiner section for reductive groups, the anonymous referees for their critical feedback, Claus Fieker, Mark van Hoeij, Tommy Hofmann, and Jeroen Sijsling for pointers, and David Zywina for helpful guiding discussions. Costa was supported by a Simons Collaboration Grant (550033). Voight was supported by an NSF CAREER Award (DMS-1151047) and a Simons Collaboration Grant (550029).
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Costa, E., Lombardo, D. & Voight, J. Identifying central endomorphisms of an abelian variety via Frobenius endomorphisms. Res. number theory 7, 46 (2021). https://doi.org/10.1007/s40993-021-00264-y
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DOI: https://doi.org/10.1007/s40993-021-00264-y