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arXiv · 2411.16381

Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

Tristan Ricoul
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Abstract

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

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Tristan Ricoul. 2024-11-25. Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations. https://arxiv.org/abs/2411.16381

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