Abstract

If E is an elliptic curve over Q with split multiplicative reduction at p, then the p-adic L-function associated with E vanishes at s=1 independently of whether the complex L-function vanishes. In this case, one has an “exceptional zero formula” relating the first derivative of the p-adic L-function to the complex L-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 p-adic local Langlands correspondence for GL(2)(Q_p).

In this talk, I will discuss an exceptional zero formula for (not necessarily essentially self-dual) regular algebraic, cuspidal automorphic representations of GL(3) which are Steinberg at p. The formula involves an automorphic L-invariant constructed by Gehrmann. Joint work with Daniel Barrera and Chris Williams.