
Andrew Graham: The exceptional zero conjecture for GL(3)
Abstract: If EEE is an elliptic curve over Q\mathbb{Q}Q with split multiplicative reduction at ppp, then the ppp-adic LLL-function associated with EEE vanishes at s=1s=1s=1 independently of whether the complex LLL-function vanishes. In this case, one has an “exceptional zero formula” relating the first derivative of the ppp-adic LLL-function to the complex LLL-function multiplied by a certain L-invariant. This L-invariant can be interpreted in several ways — on the automorphic side for example, L-invariants parameterise part of the ppp-adic local Langlands correspondence for GL2(Qp)\mathrm{GL}_2(\mathbb{Q}_p)GL2(Qp).
In this talk, I will discuss an exceptional zero formula for (not necessarily essentially self-dual) regular algebraic, cuspidal automorphic representations of GL3\mathrm{GL}_3GL3 which are Steinberg at ppp. The formula involves an automorphic L-invariant constructed by Gehrmann. Joint work with Daniel Barrera and Chris Williams.
More details can be found on https://researchseminars.org/seminar/LNTS