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 GL2 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 L-invariants attached to certain automorphic representations.
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