Theta Operator Equals Fontaine Operator on Modular Curves

Inspired by [Pan22], we give a new proof that for an overconvergent modular eigenform $f$ of weight $1+k$ with $k\in {\mathbb{Z}}_{\ge 1}$, assuming that its associated global Galois representation ${\rho }_f$ is irreducible, then $f$ is classical if and only if ${\rho }_f$ is de Rham at $p$. For the proof, we prove that theta operator ${\theta }^k$ coincides with Fontaine operator in a suitable sense.