# On Siegel eigenvarieties at Saito-Kurokawa points

## TL;DR

This paper investigates the local geometry of the $p$-adic Siegel eigenvariety at Saito-Kurokawa points with critical slope, showing smoothness under certain conditions and linking it to Selmer group dimensions and the Bloch-Kato conjecture.

## Contribution

It proves the smoothness of the eigenvariety at specific Saito-Kurokawa points and establishes a connection between geometric properties and Selmer group dimensions, with implications for the Bloch-Kato conjecture.

## Key findings

- Eigenvariety is smooth at the Saito-Kurokawa point under certain Selmer group conditions.
- The irreducible component at the point is not globally endoscopic.
- Failure of smoothness implies the Selmer group dimension is at least two.

## Abstract

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(\Gamma_0(N))$ ordinary at a prime $p\nmid N$ with sign $\epsilon_f=-1$ and write $\alpha$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $\pi_\alpha$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))$ attached to $f$ is at most one and some mild assumptions on the automorphic representation attached to $f$, we show that $\mathcal{E}$ is smooth at the point corresponding to $\pi_\alpha$, and that the irreducible component of $\mathcal{E}$ specializing to $\pi_\alpha$ is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of $\mathcal{E}$ at $\pi_\alpha$ yields that $\dim H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))\geq 2$.