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CATEGORIES:Number Theory Seminar
SUMMARY:Parity of Selmer ranks in quadratic twist families
- Adam Morgan (KCL)
DTSTART;TZID=Europe/London:20161011T143000
DTEND;TZID=Europe/London:20161011T153000
UID:TALK67451AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/67451
DESCRIPTION:The Shafarevich--Tate group of an elliptic curve o
ver a number field has square order (if finite) as
a consequence of the Cassels--Tate pairing. For g
eneral principally polarised abelian varieties\, h
owever\, this can fail to be the case. We examine
how this phenomenon behaves under quadratic twist
and derive consequences for the behaviour of 2-Sel
mer ranks in quadratic twist families. Specificall
y\, we prove results about the proportion of twist
s of a fixed principally polarised abelian variety
having odd (resp. even) 2-Selmer rank\, generalis
ing work of Klagsbrunâ€“Mazurâ€“Rubin for elliptic cur
ves and Yu for Jacobians of hyperelliptic curves.
We exhibit several new features of the statistics
which were not present in these settings.
LOCATION:MR13
CONTACT:Jack Thorne
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