Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees

Abstract

Let $F$ be a real quadratic field, $p$ be a rational prime inert in $F$, and $N\geq 4$ be an integer coprime to $p$. Consider an overconvergent $p$-adic Hilbert eigenform $f$ for $F$ of weight $(k\_ 1,k\_ 2)\in {\bf Z} ^{2}$ and level ${\it \Gamma} \_ {00}(N)$. We prove that if the slope of $f$ is strictly less than $\min {k\_ 1,k\_ 2}-2$ , then $f$ is a classical Hilbert modular form of level ${\it \Gamma} \_ {00}(N)\cap {\it \Gamma} \_ {0}(p)$ .