Year: 2025
Type: article
Abstract: 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.
Source: Canadian Journal of Mathematics
Authors: Tobias Berger, Jim Brown, Krzysztof Klosin
Institutions: University of Sheffield, Occidental College, Princeton University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4414480381
FWCI:
Cites:
Cited by:
Related to: