Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local models for Galois deformation rings and applications   We construct projective varieties in mixed characteristic whose singularities
model, in generic cases, those of tamely potentially crystalline Galois
deformation rings for unramified extensions of $\mathbb{Q}_p$ with small
regular Hodge-Tate weights. We establish several significant facts about their
geometry including a unibranch property at special points and a representation
theoretic description of the irreducible components of their special fibers. We
derive from these geometric results a number of local and global consequences:
the Breuil-M\'ezard conjecture in arbitrary dimension for tamely potentially
crystalline deformation rings with small Hodge-Tate weights (with appropriate
genericity conditions), the weight part of Serre's conjecture for $U(n)$ as
formulated by Herzig (for global Galois representations which satisfy the
Taylor-Wiles hypotheses and are sufficiently generic at $p$), and an
unconditional formulation of the weight part of Serre's conjecture for wildly
ramified representations.

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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics