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SUMMARY:Yukako Kezuka (Max Planck Institute)
DTSTART:20210507T180000Z
DTEND:20210507T190000Z
DTSTAMP:20260423T024719Z
UID:UCSBsga/22
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCSBsga/22/"
 >Tamagawa number divisibility of central $L$-values</a>\nby Yukako Kezuka 
 (Max Planck Institute) as part of UCSB Seminar on Geometry and Arithmetic\
 n\n\nAbstract\nIn this talk\, I will study the conjecture of Birch and Swi
 nnerton-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 Fermat elliptic curv
 e $x^3+y^3=1$\, and admit complex multiplication by the ring of integers o
 f $\\mathbb{Q}(\\sqrt{-3})$. First\, I will study the $p$-adic valuation o
 f the algebraic part of their central $L$-values\, and exhibit a curious r
 elation between the $3$-part of the Tate--Shafarevich group of $E$ and the
  number of prime divisors of $N$ which are inert in $\\mathbb{Q}(\\sqrt{-3
 })$. In the second part of the talk\, I will study in more detail the case
 s when $N=2p$ or $2p^2$ for an odd prime number $p$ congruent to $2$ or $5
 $ modulo $9$. For these curves\, we establish the $3$-part of the Birch--S
 winnerton-Dyer conjecture and a relation between the ideal class group of 
 $\\mathbb{Q}(\\sqrt[3]{p})$ and the $2$-Selmer group of $E$\, which can be
  used to study non-triviality of the $2$-part of their Tate--Shafarevich g
 roup. The second part of this talk is joint work with Yongxiong Li.\n
LOCATION:https://researchseminars.org/talk/UCSBsga/22/
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