Andrew Graham (Oxford) : Exceptional zeros for elliptic curves over imaginary quadratic number fields

If E is an elliptic curve over an imaginary quadratic number field with split multiplicative reduction at a prime above p, then the p-adic L-function attached to E vanishes at s=1 for "trivial reasons". In this setting, the exceptional zero formula describes the leading term of the p-adic L-function around s=1 in terms of the complex L-value L(E, 1) and an L-invariant.

In this talk, I'll describe a proof of this formula when E is not the base-change of a curve over Q (under some assumptions), as well as applications to the p-part of the Birch--Swinnerton-Dyer conjecture for E. This is joint work with Daniel Barrera Salazar and Chris Williams.