Euler Systems and Bloch—Kato for the Triple Product

We discuss the construction of an Euler system for the triple product, the p-adic Galois representation 𝑉=𝑉1𝑥𝑉2𝑥𝑉3, where 𝑉1,𝑉2,𝑉3 are the 2-dimensional representations attached to cuspforms 𝑓1,𝑓2,𝑓3. Applications include new cases of the Bloch—Kato conjecture: e.g. in the case that 𝑓𝑖 is an elliptic curve 𝐸𝑖 over the rationals, and if χ denotes a non-trivial Dirichlet character, we prove (under some technical assumptions) that the non-vanishing of 𝐿(𝐸1𝑥𝐸2𝑥𝐸3𝑥χ,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.)