Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914546720964608 |
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| author | Groutides, Alexandros |
| author_facet | Groutides, Alexandros |
| contents | Let $π_1,π_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some finite extension $F/\mathbf{Q}_p$ of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on unramified zeta-integrals, we introduce a natural general notion of $(π_1\timesπ_2)$-integral data at which the Rankin-Selberg zeta-integral can be evaluated. We then establish an integral refinement of Jacquet-Langland's GCD-result for this zeta-integral, when evaluated at $(π_1\timesπ_2)$-integral data. This is compatible with the notion of integrality coming from the Fourier coefficients of newforms of even integral weights. Our approach relies on a reinterpretation of the Rankin-Selberg zeta-integral, and works of Assing and Saha on values of $p$-adic Whittaker new vectors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations Groutides, Alexandros Number Theory Representation Theory Let $π_1,π_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some finite extension $F/\mathbf{Q}_p$ of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on unramified zeta-integrals, we introduce a natural general notion of $(π_1\timesπ_2)$-integral data at which the Rankin-Selberg zeta-integral can be evaluated. We then establish an integral refinement of Jacquet-Langland's GCD-result for this zeta-integral, when evaluated at $(π_1\timesπ_2)$-integral data. This is compatible with the notion of integrality coming from the Fourier coefficients of newforms of even integral weights. Our approach relies on a reinterpretation of the Rankin-Selberg zeta-integral, and works of Assing and Saha on values of $p$-adic Whittaker new vectors. |
| title | Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2501.16972 |