Systems of orthogonal polynomials arising from the modular j -function

Let Sp(x) ∈ Fp[x] be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over Fp. Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97–126), we define an inner product , ψ on R[x] for every ψ(x) ∈ Q[x]. Suppose a system of orthogonal polynomials {Pn,ψ(x)}∞n=0 with respect to , ψ exists. We prove that if n is sufficiently large and ψ(x)Pn,ψ(x) is p-integral, then Sp(x)|ψ(x)Pn,ψ (x) over Fp[x]. Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.  2003 Elsevier Inc. All rights reserved