Systems of orthogonal polynomials arising from the modular j-function

Abstract

Let S p(x)∈ F p[x] be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over F p . 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 { P n, ψ ( x)} n=0 ∞ with respect to 〈 , 〉 ψ exists. We prove that if n is sufficiently large and ψ( x) P n, ψ ( x) is p-integral, then S p(x)|ψ(x)P n,ψ(x) over F p[x] . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.