BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Yukako Kezuka (Max-Planck-Institut für Mathematik) DTSTART:20210519T100000Z DTEND:20210519T110000Z DTSTAMP:20260423T022718Z UID:RSVP/2 DESCRIPTION:Title: On the non-triviality of the 2-part of the Tate-Shafarevich group\nby Yuk ako Kezuka (Max-Planck-Institut für Mathematik) as part of Rendez-vous on special values and periods\n\n\nAbstract\nThe conjecture of Birch and Swi nnerton-Dyer concerns a deep connection between the arithmetic of elliptic curves and the behaviour of their associated complex $L$-functions at $s= 1$. \nThe conjecture was formulated in the early 60's\, and much of it rem ains mysterious today.\nIndeed\, the exact Birch-Swinnerton-Dyer formula r emains unknown even for the classical family of elliptic curves $E$ of the form $x^3+y^3=N$\, where $N$ is a positive integer. \n\nIn this talk\, I will study the "$p$-part" of the conjecture for these curves at small prim es $p$. \nThese cases are often eschewed\, but they seem to make up a most significant part of the full conjecture.\n\nFirst\, I will study the $3$- adic valuation of the algebraic part of their central $L$-values\, and use it to show that the "analytic" order of the Tate-Shafarevich group of $E$ is a perfect square for some $N$. \nIn the second part of the talk\, I wi ll explain how we can obtain the $3$-part of the Birch-Swinnerton-Dyer con jecture in certain special cases of $N$ where the rank of $E$ is known to be equal to $0$ or $1$. For the $2$-part of the conjecture\, I will explai n a relation between the ideal class group of a corresponding cubic field extension and the $2$-Selmer group of $E$. \nThis can be used to study non -triviality of the $2$-part of the Tate-Shafarevich group of $E$\, even wh en $E$ has rank $1$. \n\nThe second part of this talk is joint work with Y ongxiong Li.\n LOCATION:https://researchseminars.org/talk/RSVP/2/ END:VEVENT END:VCALENDAR