On the $I(1)$-invariants: Non-abelian Hecke algebra case
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913053897916416 |
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| author | Chitrao, Anand Jana, Arindam Soneji, Asfak |
| author_facet | Chitrao, Anand Jana, Arindam Soneji, Asfak |
| contents | Let $F$ be a finite extension of $\mathbb{Q}_p$. The so-called supersingular representations are the basic building blocks in the theory of mod $p$ representations of ${\rm GL}_2(F)$. The space of pro-$p$-Iwahori invariants of a universal module played a crucial role in the construction of the supersingular representations of ${\rm GL}_2(\mathbb{Q}_p)$. In this paper, we give an explicit description of the pro-$p$-Iwahori invariants of the universal module $π_r$ for $r = 0, q - 1$ using the Iwahori-Hecke model. We also determine the action of the pro-$p$-Iwahori-Hecke algebra on these newly found invariants. As an application, we recover $π_r$ functorially from its space of $I(1)$-invariants and extend a theorem of Ollivier for any totally ramified extension of $\mathbb{Q}_p$ other than itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20218 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the $I(1)$-invariants: Non-abelian Hecke algebra case Chitrao, Anand Jana, Arindam Soneji, Asfak Number Theory 11F70, 20G05, 22E50 Let $F$ be a finite extension of $\mathbb{Q}_p$. The so-called supersingular representations are the basic building blocks in the theory of mod $p$ representations of ${\rm GL}_2(F)$. The space of pro-$p$-Iwahori invariants of a universal module played a crucial role in the construction of the supersingular representations of ${\rm GL}_2(\mathbb{Q}_p)$. In this paper, we give an explicit description of the pro-$p$-Iwahori invariants of the universal module $π_r$ for $r = 0, q - 1$ using the Iwahori-Hecke model. We also determine the action of the pro-$p$-Iwahori-Hecke algebra on these newly found invariants. As an application, we recover $π_r$ functorially from its space of $I(1)$-invariants and extend a theorem of Ollivier for any totally ramified extension of $\mathbb{Q}_p$ other than itself. |
| title | On the $I(1)$-invariants: Non-abelian Hecke algebra case |
| topic | Number Theory 11F70, 20G05, 22E50 |
| url | https://arxiv.org/abs/2604.20218 |