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Given a connected reductive algebraic group G over a number field F, the global Langlands (reciprocity) conjecture roughly predicts that, there should be a correspondence between (automorphic side) the isomorphism classes of  (cuspidal, cohomological) automorphic representations of G and (Galois side) the isomorphism classes of (irreducible, locally de-Rham) Galois representations for Gal(\bar{F} / F) taking values in the Langlands dual group of G.

In the first part of this talk, I will sketch the main argument of our expected theorem/proof for (automorphic to Galois) direction of this conjecture, for G = GSpin(n,2), n odd and F to be totally real, under 3 technical assumptions (for time being), namely we assume that automorphic representations are additionally “non-CM” and “non-endoscopic” and “std-regular”.

In the second part, mainly following works of Loeffler and Chenevier on overconvergent p-adic automorphic forms,  I will present an idea to remove the std-regular assumption on the theorem via the theory of eigenvarieties.