Bence Hevesi: Congruences between automorphic forms and reciprocity

I will discuss the role of congruences between Eisenstein series and cusp forms in verifying reciprocity beyond the self-dual case. In particular, I will attempt to give a gentle introduction to Scholze's construction of automorphic Galois representations for $GL_n$ over an imaginary CM field and the approach of the 10 author (Allen—Calegari--Caraiani—Gee—Helm—Le Hung—Newton—Scholze—Taylor—Thorne) to prove local-global compatibility at $p$ for such Galois representations.