Eisenstein series for $G_2$ and the symmetric cube Bloch--Kato conjecture

Let $F$ be a cuspidal eigenform of even weight and trivial nebentypus, let $p$ be a prime not dividing the level of $F$, and let ${\rho }_F$ be the $p$-adic Galois representation attached to $F$. Assume that the $L$-function attached to the symmetric cube of ${\rho }_F$ vanishes to odd order at its central point. Then under some mild hypotheses, and conditional on certain consequences of Arthur’s conjectures, we construct a nontrivial element in the Bloch–Kato Selmer group of an appropriate twist of the symmetric cube of ${\rho }_F$, in accordance with the Bloch–Kato conjectures. Our technique is based on the method of Skinner and Urban. We construct a class in the appropriate Selmer group by $p$-adically deforming Eisenstein series for the exceptional group $G_2$ in a generically cuspidal family and then studying a lattice in the corresponding family of $G_2$-Galois representations. We also make a detailed study of the specific conjectures used and explain how one might try to prove them.