Euler systems and the Bloch–Kato conjecture for automorphic Galois representations

Abstract

We survey recent progress on the Bloch–Kato conjecture, relating special values of $L$-functions to cohomology of Galois representations, via the machinery of Euler systems. This includes new techniques for the construction of Euler systems, via the étale cohomology of Shimura varieties, and new methods for proving explicit reciprocity laws, relating Euler systems to critical values of $L$-functions. These techniques have recently been used to prove the Bloch–Kato conjecture for critical values of the degree 4 $L$-function of ${\mathrm{GSp}}_4$, and we survey ongoing work aiming to apply this result to the Birch–Swinnerton-Dyer conjecture for modular abelian surfaces, and to generalise it to a range of other automorphic $L$-functions.