DEFORMATIONS OF SAITO-KUROKAWA TYPE AND THE PARAMODULAR CONJECTURE

We study short crystalline, minimal, essentially self-dual deformations of a mod p non-semisimple Galois representation (sigma) over bar with (sigma) over bar (ss) = chi(k-2) circle plus rho circle plus 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-f(1) (Q, rho(f)(1-k) circle plus Q(p)/Z(p)) and H-f(1) (Q, rho(2 - k)) have order p, and there exists a characteristic zero absolutely irreducible deformation of Tr 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-red = T theorem showing modularity of all such deformations of (sigma) over bar.