Systems of orthogonal polynomials arising from the modular \(j\)-function.
Let \(S_p(x)\in \mathbb F_p[x]\) be the supersingular polynomial in characteristic \(p\). In this paper the authors generalize a construction of Atkin which was described in a recent paper of \textit{M. Kaneko} and \textit{D. Zagier} [Computational perspectives on number theory (Chicago, IL 1995), AMS/IP Stud. Adv. Math. 7, 97--126 (1998; Zbl 0955.11018)] which relates the supersingular polynomials to a certain system of orthogonal polynomials. In particular, the authors call \(\psi(x)\in \mathbb R[x]\) \textit{good} if there is a set of monic orthogonal polynomials \(\{P_{n, \psi}\}_{n=1}^\infty\) with respect to a natural inner product \(\langle\;,\;\rangle_\psi\) (it is shown that most \(\psi\) are good). The authors show in many cases that if \(\psi\) is good and \(n\) is large, then \(S_p(x)\) divides \(\psi(x) P_{n, \psi}(x)\) in \(\mathbb F_p[x]\). When \(\psi(x)=1\) the result of Atkin is recovered.
scientific article; zbMATH DE number 2032974