Systems of orthogonal polynomials arising from the modular j-function

Let G-fraktur signp(x) ε double struck F sign p[x] be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over double struck F signp. 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) ε 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 G-fraktur signp(x)\ψ(x)Pn(x) over double struck F signp[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.

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Author Jayce Robert Getz Mathematics