Klingen Eisenstein series congruences and modularity

Abstract

We construct a mod $\ell$ congruence between a Klingen Eisenstein series (associated to a classical newform $\varphi$ 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 $H_f^1(\mathbf{Q},{\text{ad}}^0{\rho }_{\varphi }(2-k)\otimes {\mathbf{Q}}_{\ell }/{\mathbf{Z}}_{\ell })$ 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 $H_f^1(\mathbf{Q},{\text{ad}}^0{\overline{\rho }}_{\varphi }(k-2))$ is cyclic. For this we prove a result about extensions of Fontaine-Laffaille modules. We end by formulating conditions for when $H_f^1(\mathbf{Q},{\text{ad}}^0{\overline{\rho }}_{\varphi }(k-2))$ is non-cyclic and the Eisenstein ideal is non-principal.