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

TL;DR

This paper constructs nontrivial elements in the Bloch--Kato Selmer group for symmetric cube Galois representations using $G_2$ Eisenstein series deformations, linking automorphic forms to special values of L-functions.

Contribution

It introduces a novel method of using deformations of $G_2$ Eisenstein series to study the Bloch--Kato conjecture for symmetric cube Galois representations.

Findings

Constructed a Selmer group element under certain hypotheses

Linked automorphic forms on $G_2$ to Galois representations

Provided detailed analysis of conjectures involved

Abstract

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.