Slope--Weight Coincidence Loci in $p$-adic Hodge Theory and Applications to Modularity Lifting

Rahul Thakurdas Kundnani,Shri  Kant,Khurheed  Alam

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Slope--Weight Coincidence Loci in $p$-adic Hodge Theory and Applications to Modularity Lifting

R. T. Kundnani,

Sh. Kant,

K. Alam

2025-11-07

https://doi.org/10.24108/preprints-3113858

We develop a geometric and cohomological framework for analyzing $p$-adic Galois representations through the equality of Hodge and Newton data. For a semistable $p$-adic representation, we define the \emph{slope--weight coincidence locus} as the subspace where the Hodge polygon and Newton polygon coincide at every subobject level. We prove that this condition forces equality of Frobenius slopes and Hodge--Tate weights, thereby linking filtered $(\varphi,N)$-module structures with geometric Hodge data. Locally, the coincidence locus is described by polyhedral and determinantal equations inside Kisin-type deformation spaces, and it contains the full ordinary crystalline stratum. On this locus, local Euler factors admit canonical expressions in terms of Hodge weights, and the resulting compatibility enables an $R=T$ theorem after imposing coincidence constraints at all $p$-adic places. The framework thus unifies semistable comparison, deformation theory, and arithmetic geometry, providing an explicit bridge between geometric filtrations and automorphic weights.

Ссылка для цитирования:

Kundnani R. T., Kant Sh., Alam K. 2025. Slope--Weight Coincidence Loci in $p$-adic Hodge Theory and Applications to Modularity Lifting. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113858

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