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The Breuil-Mézard conjecture for potentially Barsotti-Tate representations. (English) Zbl 1408.11033
Summary: We prove the Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate representations of the absolute Galois group \(G_{K}\), \(K\) a finite extension of \(\mathbb{Q}_{p}\), for any \(p>2\) (up to the question of determining precise values for the multiplicities that occur). In the case that \(K/\mathbb{Q}_{p}\) is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard-Diamond-Jarvis conjecture.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
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