A Rank-Two Case of Local-Global Compatibility for $l = p$

Abstract: We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the automorphic side comes from a twist of Steinberg at $v | l$, then the Galois side has nontrivial monodromy at $v$. Based on this observation, we will give a definition of the Fontaine-Mazur $\mathcal{L}$-invariants attached to certain automorphic representations.