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 $N\ge 4$ be an integer coprime to $p$. Consider an overconvergent $p$-adic Hilbert eigenform $f$ for $F$ of weight $(k_1,k_2)\in {\mathbf{Z}}^2$ and level ${\Gamma }_{00}(N)$. 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 ${\Gamma }_{00}(N)\cap {\Gamma }_0(p)$ .