Galois representations associated with a non-selfdual automorphic representation of GL(3)

TL;DR

This paper proves the full local $L$-factor coincidence for a non-selfdual automorphic representation of GL(3), confirming a conjecture and establishing local-global compatibility using recent Galois representation constructions.

Contribution

It establishes the coincidence of local $L$-factors at all primes for a specific non-selfdual automorphic representation of GL(3), confirming a conjecture and proving local-global compatibility.

Findings

Confirmed local $L$-factor coincidence at all primes.

Proved local-global compatibility at $p=2$.

Utilized recent advances in Galois representation theory.

Abstract

In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$ of level $\Gamma_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Greni\'e's results to compare three-dimensional $2$-adic Galois representations. We also prove the local-global compatibility at $p = 2$, including the case $p = \ell$.