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SUMMARY:Ari Shnidman (Hebrew University of Jerusalem)
DTSTART:20210325T163000Z
DTEND:20210325T173000Z
DTSTAMP:20260422T055654Z
UID:SFUQNTAG/27
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SFUQNTAG/27/
 ">Selmer groups of abelian varieties with cyclotomic multiplication</a>\nb
 y Ari Shnidman (Hebrew University of Jerusalem) as part of SFU NT-AG semin
 ar\n\n\nAbstract\nLet $A$ be an abelian variety over a number field $F$\, 
 with complex multiplication by the $n$-th cyclotomic field $\\mathbb{Q}(\\
 zeta)$.  If $n = 3^m$\, we show that the average size of the $(1-\\zeta)
 $-Selmer group of $A_d$\, as $A_d$ varies through the twist family of $A$\
 , is equal to 2.  As a corollary\, the average $\\mathbb{Z}[\\zeta]$-rank
  of $A_d$ is at most 1/2\, and at least 50% of $A_d$ have rank 0.    Mor
 e generally\, we prove average rank bounds for various twist families of a
 belian varieties with "cyclotomic" multiplication (not necessarily CM) ove
 r $\\bar F$\, such as sextic twist families of trigonal Jacobians over $\\
 mathbb{Q}$. These results have application to questions of  "rank gain" f
 or a fixed elliptic curve over a family of sextic fields\, as well as the 
 distribution of $\\#C_d(F)$\, as $C_d$ varies through twists of a fixed cu
 rve $C$ of genus $ g > 1$.  This is joint work with Ariel Weiss.\n
LOCATION:https://researchseminars.org/talk/SFUQNTAG/27/
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