Number Theory Seminar

Speaker: Richard Griffon (Leiden)

Title: Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family.

Abstract: L-functions of elliptic curves over global fields conjecturally encode a lot of information about their arithmetic. In general though, even for elliptic curves $E$ over $F_q(t)$, little is known about their L-functions $L(E, s)$. For example, consider the first non-zero coefficient $L*(E,1)$ in the Taylor expansion of $L(E,s)$ around the point $s=1$ (the "special value"): the size of $L*(E,1)$ remains elusive. One expects that $L*(E, 1)$ is “generically” as big as it possibly can when compared to the conductor of $E$, but this has only been proved in a very limited number of cases. In this talk, I will report on a work in progress about an infinite family of elliptic curves $E$ in an Artin-Schreier family over $K=F_q(t)$. I computed their L-functions explicitly, and I am able to deduce a very precise asymptotic bound on $L*(E,1)$ in terms of the conductor of $E$. Via the Birch and Swinnerton-Dyer conjecture (which is a theorem in this case), one can translate this bound into an asymptotic estimate of certain arithmetic invariants of these elliptic curves $E$.

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