The image and kernel of Atkin’s 𝑈_{𝑝} operator modulo 𝑝

Abstract

We compute the image of Atkin’s U p U_{p} operator on reduced modular forms. If A ≥ 1 A\geq 1 and 2 ≤ B ≤ p + 1 2\leq B \leq p+1 , then U p : M ~ A p + B ( Γ 1 ( N ) ) ↠ M ~ A + B ( Γ 1 ( N ) ) U_{p}: \widetilde M_{Ap+B}\left ( \Gamma _{1}(N) \right ) \twoheadrightarrow \widetilde M_{A+B} \left (\Gamma _{1}(N) \right ) is a surjection. In particular, the dimension of ker ⁡ U p \ker U_{p} is known for weights at least p + 2 p+2 .