Deformations of Saito-Kurokawa type and the Paramodular Conjecture

Abstract

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{{\rm ss}}=\chi^{k-2}\oplus\rho\oplus\chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f({\bf Q},\rho_f(1-k)\otimes{\bf Q}_p/{\bf Z}_p)$ and $H^1_f({\bf Q},\rho(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\overline{\sigma}$ then the universal deformation ring is a dvr. When $k=2$ this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of an abelian surface of conductor 731. When $k>2$, we obtain an $R^{{\rm red}}=T$ theorem showing modularity of all such deformations of $\overline{\sigma}$.