Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Congruences involving the $U_{\ell}$ operator for weakly holomorphic
  modular forms   Let $\lambda$ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a
weakly holomorphic modular form of weight $\lambda+\frac 12$ on $\Gamma_0(4)$
with integral coefficients. Let $\ell\geq 5$ 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 \[ (f|U_{\ell^m})(z) \equiv b(0) +
\sum_{i=1}^{t}\sum_{n=1}^{\infty} b(d_i n^2) q^{d_i n^2} \pmod{\ell}, \] where
$d_1, \ldots, d_t$ are square-free positive integers, and the operator $U_\ell$
on formal power series is defined by \[ \left( \sum_{n=0}^\infty a(n)q^n
\right) \bigg| U_\ell = \sum_{n=0}^\infty a(\ell n)q^n. \] 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 \[
  \biggl\{ \lim_{m \rightarrow \infty} \tilde{f}|U_{\ell^{2m}}, \lim_{m
\rightarrow \infty} \tilde{f}|U_{\ell^{2m+1}} \biggr\} = \biggl\{
a(0)\theta(z), a(0)\theta^\ell(z) \in \mathbb{F}_{\ell}[[q]] \biggr\}, \] 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.

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The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys