Congruences involving the $$U_{\ell }$$ operator for weakly holomorphic modular forms

Abstract

Let $$\lambda $$ be an integer, and $$f(z)=\sum _{n\gg -\infty } a(n)q^n$$ be a weakly holomorphic modular form of weight$$\lambda +\frac{1}{2}$$ on $$\Gamma _0(4)$$ with integral coefficients. Let $$\ell \ge 5$$ be a prime. Assume that the constant term a(0) is not zero modulo $$\ell $$. Further, assume that, for some positive integer m, the Fourier expansion of $$(f|U_{\ell ^m})(z) = \sum _{n=0}^\infty b(n)q^n$$ has the form $$\begin{aligned} (f|U_{\ell ^m})(z) \equiv b(0) + \sum _{i=1}^{t}\sum _{n=1}^{\infty } b(d_i n^2) q^{d_i n^2} \pmod {\ell }, \end{aligned}$$where $$d_1, \ldots , d_t$$ are square-free positive integers, and the operator $$U_\ell $$ on formal power series is defined by $$\begin{aligned} \left( \sum _{n=0}^\infty a(n)q^n \right) \bigg | U_\ell = \sum _{n=0}^\infty a(\ell n)q^n. \end{aligned}$$Then, $$\lambda \equiv 0 \pmod {\frac{\ell -1}{2}}$$. Moreover, if $${\tilde{f}}$$ denotes the coefficient-wise reduction of f modulo $$\ell $$, then we have $$\begin{aligned} \biggl \{ \lim _{m \rightarrow \infty } {\tilde{f}}|U_{\ell ^{2m}}, \lim _{m \rightarrow \infty } {\tilde{f}}|U_{\ell ^{2m+1}} \biggr \} = \biggl \{ a(0)\theta (z), a(0)\theta ^\ell (z) \in \mathbb {F}_{\ell }[[q]] \biggr \}, \end{aligned}$$where $$\theta (z)$$ is the Jacobi theta function defined by $$\theta (z) = \sum _{n\in \mathbb {Z}} q^{n^2}$$. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.