Local-global compatibility and the exceptional zero conjecture for GL(3)

TL;DR

This paper proves exceptional zero conjectures for certain automorphic representations of GL(3), establishing local-global compatibility and equating automorphic and Galois invariants, advancing understanding of p-adic L-functions and Galois representations.

Contribution

It unconditionally proves an automorphic exceptional zero conjecture for GL(3) using p-arithmetic cohomology and confirms local-global compatibility at p for GL(n), extending previous results.

Findings

Proved automorphic exceptional zero conjecture for GL(3)

Established local-global compatibility at p for GL(n)

Confirmed equality of automorphic and Fontaine–Mazur L-invariants

Abstract

We prove exceptional zero conjectures for $p$-ordinary regular algebraic cuspidal automorphic representations of $\mathrm{GL}_3(\mathbb{A})$ which are Steinberg at $p$. We make no self-duality assumptions. The paper has two parts. In Part 1, we use $p$-arithmetic cohomology to unconditionally prove an automorphic exceptional zero conjecture in this setting, using Gehrmann's automorphic $\mathcal{L}$-invariant. In Part 2 we prove, under mild assumptions that are expected to always hold, the equality of automorphic and Fontaine--Mazur $\mathcal{L}$-invariants, and thus deduce cases of the full Greenberg--Benois exceptional zero conjecture. As one of the key ingredients for this, we establish local-global compatibility at $\ell = p$ for Galois representations attached to $p$-ordinary torsion classes for $\mathrm{GL}_n$, confirming a conjecture of Hansen in this setting. We prove this for all $n$ following the strategy in the "10-author paper", and use the $n=3$ case to deduce the desired equality of $\mathcal{L}$-invariants.