Deformations of Saito-Kurokawa type and the Paramodular Conjecture (with an appendix by Cris Poor, Jerry Shurman, and David S. Yuen)

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\bar{\sigma }$ with ${\bar{\sigma }}^{\mathrm{s}\mathrm{s}}={\chi }^{k-2}\oplus \rho \oplus {\chi }^{k-1}$, 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 $H^1_f($\mathbf{Q}$, \rho_f(1-k)\otimes $\mathbf{Q}$_p/$\mathbf{Z}$_p)$and$H^1_f($\mathbf{Q}$, \rho(2-k))$haveorder$p$, and there exists a characteristic zero absolutely irreducible deformation of $\bar{\sigma }$ 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 the abelian surface of conductor 731. When $k>2$, we obtain an $R^{\mathrm{r}\mathrm{e}\mathrm{d}}=T$ theorem showing modularity of all such deformations of $\bar{\sigma }$.