On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture

TL;DR

This paper links the properties of Siegel Eisenstein series pullbacks to the Bloch-Kato conjecture, establishing bounds on algebraic L-values via congruences and Galois representations.

Contribution

It introduces a novel congruence between Saito-Kurokawa lifts and non-CAP Siegel cusp forms to study Selmer groups and L-values.

Findings

Proves an inequality relating p-adic valuations of L-values and Selmer groups.

Constructs a new congruence between lifts and cusp forms.

Uses Galois representations to derive bounds on Selmer groups.

Abstract

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $\rho_{f} \otimes \epsilon^{1-k}$ with $\epsilon$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.