The Birch–Swinnerton-Dyer conjecture relates the behavior of the L-function of an elliptic curve at its central point to the rank of its group of rational points. The Bloch–Kato conjecture generalizes this principle to a broad family of motivic Galois representations, predicting a precise relationship between the order of vanishing of motivic L-functions at integer values and the structure of the associated Selmer groups. Since the foundational work of Kolyvagin in the nineties, Euler systems have played a central role in approaching these conjectures, and in recent years their scope has expanded significantly within the automorphic setting of Shimura varieties.

In this talk, I will focus on unitary Shimura varieties GU(2,2), whose middle-degree cohomology realizes the exterior square of the four-dimensional Galois representations attached to certain automorphic representations of GL_4. The period integral formula of Pollack–Shah for exterior square L-functions has a natural motivic interpretation, suggesting the feasibility of constructing a nontrivial Euler system. A key obstacle to this construction is the failure of a suitable multiplicity-one property, which has long prevented the verification of the certain norm relations required for Euler system methods. I will present a new approach that overcomes this difficulty. The resulting Euler system in the middle-degree cohomology of GU(2,2) provides the first nontrivial evidence toward the Bloch–Kato conjecture for exterior square motives and opens several promising avenues for further arithmetic applications. This is joint work with Andrew Graham and Antonio Cauchi.