A remark on Hecke operators and a theorem of Dwork and Koike
Let \(j(z)\) denote the usual elliptic modular function on \(\Gamma_0(1)\). Consider an infinite class of monic polynomials \(j_m\in \mathbb{Z}[j(z)]\) defined by \(j_0(z)= 1\), \(j_m(z):= J(z)/T_0(m)\), where \(J(z):= j(z)- 744\) is the usual Hauptmodul, and \(T_0(m)\) is the normalized \(m\)th Hecke operator of weight zero. The author shows that the \(j_p(z)\) (for \(p\geq 5\) a prime) are important for studying the \(p\)-adic properties of modular forms (Theorem 1.1): he provides an explicit description of the polynomials \(D_p(x)\) in a theorem of \textit{B. Dwork} [Publ. Math., Inst. Hautes Étud. Sci. 37, 27--115 (1969; Zbl 0284.14008)] and \textit{M. Koike} [J. Fac. Sci., Univ. Tokyo, Sect. I A 20, 129--169 (1973; Zbl 0256.10016)] in terms of the \(j_p\).
scientific article; zbMATH DE number 2060315