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

Dewar, Michael

doi:10.1090/S0002-9939-2011-11115-8

The image and kernel of Atkin’s $U_{p}$ operator modulo $p$
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Proc. Amer. Math. Soc. 140 (2012), 1931-1938 Request permission

Abstract:

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

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