On the Bloch-Kato conjecture for the symmetric cube (Q6172701)

This paper studies the arithmetic of the \(p\)-adic symmetric cube Galois representation associated to a level-1 modular cusp form. In particular, the authors show in Proposition 2.2 that, for \(p \gg 0\), this representation has a large image. This observation enables the authors to apply [``On the Bloch-Kato conjecture for \(\operatorname{GSp}(4)\)'', Preprint, \url{arXiv:2003.05960}, Theorem D] on the arithmetic of automorphic representations of \(\operatorname{GSp}(4)\) to prove the Bloch-Kato conjecture in analytic rank 0 for the critical twists of this representation. Further, applying [loc. cit., Theorem C], the authors also prove one inclusion of the cyclotomic Iwasawa Main Conjecture for this representation.