  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.
