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

TL;DR

This paper proves a variant of Oesterlé's conjecture concerning $p$-adic periods of a modular curve with added $ ext{Gamma}(2)$-structure, utilizing $p$-adic uniformization and de Shalit's methods.

Contribution

It extends Oesterlé's conjecture to modular curves with $ ext{Gamma}(2)$-structure using advanced $p$-adic techniques.

Findings

Proves a new variant of Oesterlé's conjecture.

Employs $p$-adic uniformization methods.

Analyzes the $p$-adic periods of specific modular curves.

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.