Algebra, Geometry, & Number Theory Seminar

We discuss the construction of an Euler system for the (split) triple product, that is, the p-adic Galois representation V = V_1 x V_2 x V_3, where V_i is the Galois representation associated to a cuspform f_i of weight k_i. Applications of this construction include new cases of the Bloch—Kato conjecture: for example, in the case that f_i corresponds to an elliptic curve E_i over \Q and if \chi denotes a non-trivial Dirichlet character, we prove (under some technical assumptions) that the non-vanishing of L(E_1 x E_2 x E_3 x \chi, s) at the central value s=2 implies the vanishing of the relevant Selmer group for V. This complements the Bloch—Kato results of Yifeng Liu (for the twisted triple product) and Haining Wang (for the case E_1=E_2=E_3 and \chi=1). Our Euler system incorporates the cyclotomic variation not seen by the Euler system "base class" defined by Darmon—Rotger using diagonal cycles. We use the pullback method pioneered by Sangiovanni—Skinner in their construction of an Euler system for the adjoint of a modular form. The main technical step is the construction of distinguished Klingen—Eisenstein classes in the cohomology of the Siegel threefold. (This talk will not assume prior knowledge of Euler systems.)