Special L-values and Selmer groups of Siegel modular forms of genus 2

Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $\pi$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $\pi$ twisted by the non-trivial quadratic Dirichlet character $\xi$ associated to the extension $\mathbb{Q}(\sqrt{-N})/\mathbb{Q}$ and the $p$-part of the Selmer group of the degree $5$ Galois representation associated to $\pi$ twisted by $\xi$, under certain conditions on the residual Galois representation.