Joint IAS/PU Number Theory

We discuss the construction of an Euler system for the triple product, the p-adic Galois representation $V = V_1 x V_2 x V_3$, where $V_1,V_2,V_3$ are the 2-dimensional representations attached to cuspforms $f_1,f_2,f_3$. Applications include new cases of the Bloch—Kato conjecture: e.g. in the case that $f_i$ is an elliptic curve $E_i$ over the rationals, and if $χ$ denotes a non-trivial Dirichlet character, we prove (under some technical assumptions) that the non-vanishing of $L(E_1 x E_2 x E_3 x χ, s)$ at the central value $s=2$ implies the vanishing of the relevant Selmer group for $V$. The construction is geometric, taking place on the Shimura variety for $GSp(4)$, and uses the pullback method pioneered by Sangiovanni—Skinner. (This talk will not assume any prior knowledge of Euler systems.)