Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916672223313920 |
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| author | Groutides, Alexandros |
| author_facet | Groutides, Alexandros |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals Groutides, Alexandros Number Theory Representation Theory 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. |
| title | Integral structures in smooth $\mathrm{GL}_2(\mathbf{Q}_p)$-representations and zeta integrals |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2401.10870 |