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arXiv · 2401.10870

Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals

Alexandros Groutides
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Abstract

Using zeta-integrals and lattices of functions on a spherical variety, we study integral structures in spherical representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ and their interaction with the unique linear functional invariant under an unramified maximal torus. Within this framework, we reformulate and prove the first instance of optimality of abstract integral norm-relations as proposed by Loeffler. We also interpret this as a form of integrality for toric periods associated to modular forms, where part of it can be regarded as an arithmetic integral analogue of Waldspurger's multiplicity one in the unramified setting.

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Alexandros Groutides. 2025-04-03. Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals. https://arxiv.org/abs/2401.10870

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