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SUMMARY:Yukako Kezuka (MPI (Bonn))
DTSTART:20210422T130000Z
DTEND:20210422T140000Z
DTSTAMP:20260423T024724Z
UID:UCDANT/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCDANT/11/">
 Tamagawa number divisibility of central L-values</a>\nby Yukako Kezuka (MP
 I (Bonn)) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbstrac
 t\nIn this talk\, I will report on some recent progress on the conjecture 
 of Birch and Swinnerton-Dyer for elliptic curves $E$ of the form $x^3+y^3=
 N$ for cube-free positive integers $N$. They are cubic twists of the Ferma
 t elliptic curve $x^3+y^3=1$\, and admit complex multiplication by the rin
 g of integers of $\\mathbb{Q}(\\sqrt{-3})$. First\, I will explain the Tam
 agawa number divisibility satisfied by the central $L$-values\, and exhibi
 t a curious relation between the $3$-part of the Tate$-$Shafarevich group 
 of $E$ and the number of prime divisors of $N$ which are inert in $\\mathb
 b{Q}(\\sqrt{-3})$. I will then explain my joint work with Yongxiong Li\, s
 tudying in more detail the cases when $N=2p$ or $2p^2$ for an odd prime nu
 mber $p$ congruent to $2$ or $5$ modulo $9$. For these curves\, we establi
 sh the $3$-part of the Birch$-$Swinnerton-Dyer conjecture and a relation b
 etween the ideal class group of $\\mathbb{Q}(\\sqrt[3]{p})$ and the $2$-Se
 lmer group of $E$\, which can be used to study non-triviality of the $2$-p
 art of their Tate$-$Shafarevich group.\n
LOCATION:https://researchseminars.org/talk/UCDANT/11/
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