Parity of Selmer ranks in quadratic twist families

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• Adam Morgan (KCL)
• Tuesday 11 October 2016, 14:30-15:30
• MR13.

If you have a question about this talk, please contact Jack Thorne.

The Shafarevich—Tate group of an elliptic curve over a number field has square order (if finite) as a consequence of the Cassels—Tate pairing. For general principally polarised abelian varieties, however, this can fail to be the case. We examine how this phenomenon behaves under quadratic twist and derive consequences for the behaviour of 2-Selmer ranks in quadratic twist families. Specifically, we prove results about the proportion of twists of a fixed principally polarised abelian variety having odd (resp. even) 2-Selmer rank, generalising work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. We exhibit several new features of the statistics which were not present in these settings.

This talk is part of the Number Theory Seminar series.

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