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

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.

33 pages

References in corpus (1)

• Siegel Modular Forms

Cited by in corpus (1)

• Pullbacks of Siegel Eisenstein Series and Weighted Averages of Critical $L$-values