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arXiv · 1512.06380

Potentially crystalline deformation rings and Serre weight conjectures

Daniel Le·Bao V. Le Hung·Brandon Levin·Stefano Morra
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Abstract

We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.

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Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra. 2017-10-30. Potentially crystalline deformation rings and Serre weight conjectures. https://arxiv.org/abs/1512.06380

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