We study analytic continuation of overconvergent modular forms on Siegel varieties. We first analyze the dynamic of Hecke correspondances at $p$ over Siegel varieties with parahoric-level structure. We then concentrate on genus $2$ and prove a classicity criterion: a Siegel overconvergent modular form, of weight $(k_1,k_2)$ , eigen for $U_p$ with eigenvalue $a_p$ , such that $k_2 > v(a_p)+3$ is classical. This implies that genus $2$ cuspidal ordinary $p$ -adic modular forms of weight $(k_1,k_2)$ with $k_1 \geq k_2 \geq 4$ are classical.