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arXiv · 2308.02708

On deformation rings of residual Galois representations with three Jordan-Holder factors and modularity

Xiaoyu Huang
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Abstract

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.

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Xiaoyu Huang. 2023-08-04. On deformation rings of residual Galois representations with three Jordan-Holder factors and modularity. https://arxiv.org/abs/2308.02708

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