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.