Klingen Eisenstein series congruences and modularity

TL;DR

This paper establishes congruences between Klingen Eisenstein series and cusp forms, linking them to Selmer groups and deformation rings, advancing understanding of modularity and Galois representations.

Contribution

It constructs mod $ ext{ extlbrackdbl} ext{ extrbrackdbl}$ congruences and proves an R=d theorem for Fontaine-Laffaille deformation rings, connecting Eisenstein series to Galois deformation theory.

Findings

Non-vanishing of certain Bloch-Kato Selmer groups under specific conditions

An R=dvr theorem for Fontaine-Laffaille deformation rings

Conditions for non-cyclicity of residual Selmer groups

Abstract

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