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SUMMARY:Céline Maistret (University of Bristol)
DTSTART:20220124T140000Z
DTEND:20220124T150000Z
DTSTAMP:20260423T021400Z
UID:GANT/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GANT/2/">Par
 ity of ranks of abelian surfaces</a>\nby Céline Maistret (University of B
 ristol) as part of Greek Algebra & Number Theory Seminar\n\n\nAbstract\nLe
 t $K$ be a number field and $A/K$ an abelian surface. By the Mordell-Weil 
 theorem\, the group of $K$-rational points on $A$ is finitely generated an
 d as for elliptic curves\, its rank is predicted by the Birch and Swinnert
 on-Dyer conjecture. A basic consequence of this conjecture is the parity c
 onjecture: the sign of the functional equation of the L-series determines 
 the parity of the rank of $A/K$. \n\nAssuming finiteness of the Shafarevic
 h-Tate group\, we prove the parity conjecture for principally polarized ab
 elian surfaces under suitable local constraints. Using a similar approach 
  we show that for two elliptic curves $E_1$ and $E_2$ over $K$ with isomor
 phic $2$-torsion\, the parity conjecture is true for $E_1$ if and only if 
 it is true for $E_2$. In both cases\, we prove analogous unconditional res
 ults for Selmer groups.\n
LOCATION:https://researchseminars.org/talk/GANT/2/
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