Classicality of overconvergent Hilbert eigenforms: case of quadratic residue degrees (Q481033)

Summary: 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 \( (k_ 1,k_ 2)\in \mathbb{Z} ^{2}\) 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 \(\Gamma_{00}(N)\cap \Gamma_ {0}(p)\) .