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arXiv · 2108.04094

Degenerating products of flag varieties and applications to the Breuil--Mezard conjecture

Robin Bartlett
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Abstract

We consider closed subschemes in the affine grassmannian obtained by degenerating $e$-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For $G= \operatorname{GL}_2$, and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of $G$. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton--Gee stack classifying $p$-adic representations of the Galois group of a finite extension of $\mathbb{Q}_p$. As an application we prove new cases of the Breuil--M\'ezard conjecture in dimension two.

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Robin Bartlett. 2021-08-09. Degenerating products of flag varieties and applications to the Breuil--Mezard conjecture. https://arxiv.org/abs/2108.04094

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