Number Theory Seminar

Title: Perrin-Riou's conjecture and rational points on supersingular elliptic curves.

Speaker: Kazim Buyukboduk

Abstract: I will report on some work in progress towards Perrin-Riou's conjecture (that predicts the non-vanishing of the $p$-adic Beilinson-Kato elements when the analytic rank of an elliptic curve $E$ equals one). We will simultaneously treat all $p$-semistable and $p$-supersingular elliptic curves, and in the case of the latter, we will also show how one can then construct a $\mathbb{Q}$-rational point on $E$ of infinite order, starting off with the special values of $p$-adic L-functions. Our approach is altogether different from those of Bertolini and Darmon (who handle elliptic curves with good ordinary reduction) and it is based on the general theory of $\Lambda$-adic Kolyvagin systems.