Joint IAS/Princeton University Number Theory Seminar

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs nontrivial elements in the Selmer groups of these symmetric cube Galois representations; this works by $p$-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group $G_2$. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.