Computing the local $2$-component of a non-selfdual automorphic representation of $\mathrm{GL}_3$

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Abstract

In this paper, we explicitly determine the local $2$-adic component of a non-selfdual automorphic representation $\Pi$ of $\mathrm{GL}_3$ constructed by van Geemen and Top. We prove that $\Pi_2$ is a parabolically induced representation of $\mathrm{GL}_3(\mathbb{Q}_2)$ given by $\Pi_2 = \mathrm{Ind}_P^{\mathrm{GL}_3(\mathbb{Q}_2)}(\pi\boxtimes \chi)$, where $P$ is the standard parabolic subgroup of $\mathrm{GL}_3$ with Levi subgroup $\mathrm{GL}_2 \times \mathrm{GL}_1$, $\chi$ is an unramified character of $\mathbb{Q}_2^\times$ satisfying $\chi(2) = -2\sqrt{-1}$, and $\pi$ is a supercuspidal representation of $\mathrm{GL}_2(\mathbb{Q}_2)$. Furthermore, we describe $\pi$ explicitly as a compactly induced representation $\pi = \mathrm{c-Ind}_{J_\alpha}^{\mathrm{GL}_2(\mathbb{Q}_2)} \Lambda$ and determine the representation $\Lambda$ explicitly. The proof relies on explicit computations of Hecke eigenvalues using computer calculations. The automorphic representation $\Pi$ is realized in the cuspidal cohomology of the congruence subgroup $\Gamma_0(128) \subset \mathrm{SL}_3(\mathbb{Z})$. By computing the Hecke eigenvalues of an associated Hecke eigenvector, we are able to uniquely identify the local structure of $\Pi_2$. As an application, we obtain an explicit description of the $2$-adic local component of the Galois representation $\rho_{\mathrm{vGT},\ell}$ associated with $\Pi$.