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SUMMARY:Richard Griffon (Universität Basel)
DTSTART:20200813T173000Z
DTEND:20200813T183000Z
DTSTAMP:20260423T024506Z
UID:MAGIC/17
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MAGIC/17/">E
 lliptic curves with large Tate-Shafarevich groups over $\\mathbb F_q(t)$</
 a>\nby Richard Griffon (Universität Basel) as part of MAGIC (Michigan - A
 rithmetic Geometry Initiative - Columbia)\n\n\nAbstract\nTate-Shafarevich 
 groups are important arithmetic invariants of elliptic curves\, which rema
 in quite mysterious: for instance\, it is conjectured that they are finite
 \, but this is only known in a limited number of cases. Assuming finitenes
 s of $\\operatorname{Sha}(E)$\, work of Goldfeld and Szpiro provides upper
  bounds on $\\#\\operatorname{Sha}(E)$ in terms of the conductor or the he
 ight of $E$. I will talk about a recent work (joint with Guus de Wit) wher
 e we investigate whether these upper bounds are optimal\, in the setting o
 f elliptic curves over $\\mathbb F_q(t)$. More specifically\, we construct
  an explicit family of elliptic curves over $\\mathbb F_q(t)$ which have `
 `large'' Tate-Shafarevich groups. In this family\, $\\operatorname{Sha}(E)
 $ is indeed essentially as large as it possibly can\, according to the abo
 ve mentioned bounds. In contrast with similar results for elliptic curves 
 over $\\mathbb Q$\, our result is unconditional. We also provide additiona
 l information about the structure of the Tate-Shafarevich groups under stu
 dy. The proof combines various interesting intermediate results\, includin
 g an explicit expression for the relevant $L$-functions\, a detailed study
  of the distribution of their zeros\, and the proof of the BSD conjecture 
 for the elliptic curves in the sequence.\n
LOCATION:https://researchseminars.org/talk/MAGIC/17/
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