Let K be a number field and let V be a continuous p-adic representation of the absolute Galois group of K. The Iwasawa main conjectures attach arithmetic significance to the special values of the p-adic L-function attached to V (assuming one exists). Euler systems are a powerful tool for proving such conjectures. Constructing Euler systems is a difficult problem, though, and examples exist only for a small handful of representations. In this talk we discuss applications of a new method for constructing Euler systems, first pioneered by Sangiovanni and Skinner. To explain the basic ideas we will focus on a simple example (based on joint work with Shang and Skinner): we reconstruct the two simplest Euler systems, the cyclotomic units and elliptic units (which correspond to V=\Q_p(1), and K=\Q or K=an imaginary quadratic field, respectively). The classes forming the Euler system are defined as extensions of Galois representations within the étale cohomology of the modular curve, relative to a collection of points (cusps or CM points, respectively). We use Eisenstein series to construct distinguished classes in cohomology; these Eisenstein series are naturally connected to the relevant p-adic L-functions in both cases. Time permitting, we discuss new applications of this method, including forthcoming work of the speaker to construct an Euler system for the “triple product.”
Thackeray 427