Abstract :
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.
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