Fernando Trejos Suárez: New Constructions of Euler Systems via Eisenstein Cohomology

In recent years, Sangiovanni and Skinner have announced a new general method for constructing (cyclotomic) Euler systems using Eisenstein series and boundary cohomology of Shimura varieties, sometimes referred to as the “pullback method.” Their first application was a construction of an Euler system for the adjoint of a modular form (for which a parallel construction was also announced by Loeffler—Zerbes using a degenerate version of their Asai—Flach elements), a case of historical interest due to its connection to Wiles’s proof of Fermat’s Last Theorem. In this talk, we explain the key ideas of this new method in a simple case: we re-construct the Euler systems of cyclotomic units and elliptic units using classical Eisenstein series on the modular curve (based on joint expository work with Shang and Skinner). Time permitting, we describe a new application of this method: we construct an Euler system for the triple product, i.e. the p-adic Galois representation V_1 x V_2 x V_3, where V_i is the Galois representation associated to a cuspform f_i.