The Bloch–Kato conjecture for GSp(4)
David Loeffler - University of Warwick
The Bloch--Kato conjecture predicts that the dimension of the Selmer group of a global Galois representation is equal to the order of vanishing of its L-function. In this talk, I will focus on the 4-dimensional Galois representations arising from cohomological automorphic representations of GSp(4) (i.e. from genus two Siegel modular forms). I will show that if the L-function is non-vanishing at some critical value, then the corresponding Selmer group is zero, under a long list of technical hypotheses. The proof of this theorem relies on an Euler system, a p-adic L-function, and a reciprocity law connecting those together. I will also survey work in progress aiming to extend this result to some other classes of automorphic representations.
Password: The order of the permutation group on 9 elements.