THE WEIGHT PART OF SERRE’S CONJECTURE FOR $\text{GL}(2)$ - TOBY GEE; TONG LIU; DAVID SAVITT (Forum of Mathematics, Pi, 2015)
Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.