On the $p$-adic periods of the modular curve $X(\Gamma_0(p) \cap \Gamma(2))$

Abstract

We prove a variant of Oesterl\'e's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $\Gamma(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of curves whose reduction is semi-stable.