We construct a mod $\ell $ congruence between a Klingen Eisenstein series (associated with a classical newform $\phi $ of weight k) and a Siegel cusp form f with irreducible Galois representation. We use this congruence to show non-vanishing of the Bloch–Kato Selmer group $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0\rho _{\phi }(2-k)\otimes \mathbf {Q}_{\ell }/\mathbf {Z}_{\ell })$ under certain assumptions and provide an example. We then prove an $R=dvr$ theorem for the Fontaine–Laffaille universal deformation ring of ${\overline {\rho }}_f$ under some assumptions, in particular, that the residual Selmer group $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }(k-2))$ is cyclic. For this, we prove a result about extensions of Fontaine–Laffaille modules. We end by formulating conditions for when $H^1_f(\mathbf {Q}, \operatorname {\mathrm {ad}}^0{\overline {\rho }}_{\phi }(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.

For K, an imaginary quadratic field with discriminant −DK, and associated quadratic Galois character χK, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level DK and nebentypus χK via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group U(2, 2) split over K. We generalize this (under certain conditions on K and p) to the case of p-oldforms of level pDK and character χK. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space of special Hermitian Jacobi forms. We then show how to adapt this construction to lift a Hida family of modular forms to a p-adic analytic family of automorphic forms in the Maass space of level p.

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.