On Rankin-Selberg integral structures and Euler systems for $\mathrm{GL}_2\times \mathrm{GL}_2$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910159463251968 |
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| author | Groutides, Alexandros |
| author_facet | Groutides, Alexandros |
| contents | We study how Rankin-Selberg periods and distinction problems interact with integral structures in spherical Whittaker type representations. Using this representation-theoretic framework, we settle a conjecture of Loeffler by showing that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for a product of modular forms are integrally optimal; i.e. any construction of this type with any choice of integral input data in the recipe of Loeffler-Skinner-Zerbes, would give local factors appearing in tame norm relations at $p$, which are integrally divisible by the Euler factor $\mathcal{P}_p^{'}(\mathrm{Frob}_p^{-1})$ modulo $p-1$. We also interpret this as an integrality result on the unramified part of the period associated to the Rankin-Selberg convolution of two modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Rankin-Selberg integral structures and Euler systems for $\mathrm{GL}_2\times \mathrm{GL}_2$ Groutides, Alexandros Number Theory Representation Theory We study how Rankin-Selberg periods and distinction problems interact with integral structures in spherical Whittaker type representations. Using this representation-theoretic framework, we settle a conjecture of Loeffler by showing that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for a product of modular forms are integrally optimal; i.e. any construction of this type with any choice of integral input data in the recipe of Loeffler-Skinner-Zerbes, would give local factors appearing in tame norm relations at $p$, which are integrally divisible by the Euler factor $\mathcal{P}_p^{'}(\mathrm{Frob}_p^{-1})$ modulo $p-1$. We also interpret this as an integrality result on the unramified part of the period associated to the Rankin-Selberg convolution of two modular forms. |
| title | On Rankin-Selberg integral structures and Euler systems for $\mathrm{GL}_2\times \mathrm{GL}_2$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2407.01377 |