New Constructions of Euler Systems via Eisenstein Series and the Cohomology of Shimura Varieties

Let K be a number field and let V be a "nice" p-adic representation of the absolute Galois group of K. The Iwasawa main conjectures attach arithmetic significance to special values of the p-adic L-function attached to V. Euler systems are a powerful tool for proving such conjectures; constructing Euler systems is difficult, though, and few examples exist. We will discuss applications of a new method for constructing Euler systems, first pioneered by Sangiovanni and Skinner. To explain the key ideas we focus on a simple example (based on joint work with Shang and Skinner): we reconstruct the two simplest Euler systems, the cyclotomic and elliptic units (corresponding to V=\Q_p(1), and K=\Q or K=an imaginary quadratic field, respectively). The Euler system classes 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 special classes in cohomology; these Eisenstein series are naturally connected to the relevant p-adic L-functions in both cases. Afterwards, we discuss new applications of this method, including forthcoming work of the speaker to construct an Euler system for the “triple product” within the cohomology of the Siegel threefold.