On a conjecture of Atkin for the primes 13, 17, 19, and 23 (Q2471522)

\textit{A. O. L. Atkin} [Comput. Math. Res., 8--19 (1968; Zbl 0186.36302) and Number Theory, Proc. Sympos. Pure Math. 12, 33--40 (1969; Zbl 0222.12019)] formulated conjectures about certain divisibility properties of the \(q\)-expansion coefficients of the modular invariant \(j\) (and a more general class of meromorphic modular functions on \(\Gamma_0(p^N)\) with rational Fourier coefficients). In particular, it turns out that the more one applies the \(U\)-operator to \(j\), the closer \(p\)-adically it becomes a Hecke eigenform (\(p= 1,17,19\), or \(23\)). The author proves Atkin's conjecture for an infinite set of meromorphic modular forms, including certain polynomials in \(j\) with integer coefficients (Theorem 2). The essential ingredients of the proof are \textit{J.-P. Serre's} theory of \(p\)-adic modular forms [Modular Functions of one Variable III, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 350, 191--268 (1973; Zbl 0277.12014)] and \textit{H. Hida}'s control theorem [Elementary theory of L-functions and Eisenstein series. Cambridge: Cambridge University Press (1993; Zbl 0942.11024)].