##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Fucheng Tan

#### Michigan State University

## The Breuil-Mezard conjecture for split residual representations

##### Abstract:

I will explain how to prove the Breuil-Mezard conjecture for split (non-scalar) residual representations by local methods. Combined with the cases previously proved by Kisin and Paskunas, this completes the proof of the conjecture for $\mathrm{GL}_2(\mathbb{Q}_p)$. As an application, we can prove the Fontaine-Mazur conjecture in the cases that the global residual representation restricts to the decomposition group at $p$ as an extension of the trivial character by the mod $p$ cyclotomic character. These are the cases complementary to Kisin's. This is a joint work with Yongquan Hu.

Host: Kiran Kedlaya

### May 23, 2013

### 2:00 PM

### AP&M 7321

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