Eisenstein series for G₂ and the symmetric cube Bloch--Kato conjecture

2022/02/08 by Mundy, Sam
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2202.03585

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 ρ_F be the p-adic Galois representation attached to F. Assume that the L-function attached to the symmetric cube of ρ_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 ρ_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 G₂ in a generically cuspidal family and then studying a lattice in the corresponding family of G₂-Galois representations. We also make a detailed study of the specific conjectures used and explain how one might try to prove them.

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