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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2308.02708 |
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| _version_ | 1866916548802772992 |
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| author | Huang, Xiaoyu |
| author_facet | Huang, Xiaoyu |
| contents | In this paper, we study Fontaine-Laffaille, self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions regarding the orders of certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke algebra, we also prove an R = T theorem in the general context. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. Assuming the Bloch-Kato conjecture, our result identifies special L-value conditions for the existence of a unique abelian surface isogeny class and an R = T theorem for certain 6-dimensional Galois representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02708 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On deformation rings of residual Galois representations with three Jordan-Holder factors and modularity Huang, Xiaoyu Number Theory 11F80, 11F55 In this paper, we study Fontaine-Laffaille, self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions regarding the orders of certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke algebra, we also prove an R = T theorem in the general context. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. Assuming the Bloch-Kato conjecture, our result identifies special L-value conditions for the existence of a unique abelian surface isogeny class and an R = T theorem for certain 6-dimensional Galois representations. |
| title | On deformation rings of residual Galois representations with three Jordan-Holder factors and modularity |
| topic | Number Theory 11F80, 11F55 |
| url | https://arxiv.org/abs/2308.02708 |