The exceptional zero conjecture for $\mathrm{GL}(3)$

Abstract: If $E$ is an elliptic curve over $\mathbb{Q}$ 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 $\mathrm{GL}_2(\mathbb{Q}_p)$.

In this talk, I will discuss an exceptional zero formula for (not necessarily essentially self-dual) regular algebraic, cuspidal automorphic representations of $\mathrm{GL}_3$ which are Steinberg at $p$. The formula involves an automorphic L-invariant constructed by Gehrmann. Joint work with Daniel Barrera and Chris Williams.

number theory

Audience: researchers in the topic

London number theory seminar

Series comments: For reminders, join the (very low traffic) mailing list at mailman.ic.ac.uk/mailman/listinfo/london-number-theory-seminar

For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html