Systems of orthogonal polynomials arising from the modular j-function


Journal Article

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.

Full Text

Duke Authors

Cited Authors

  • Basha, S; Getz, J; Nover, H; Smith, E

Published Date

  • January 1, 2004

Published In

Volume / Issue

  • 289 / 1

Start / End Page

  • 336 - 354

International Standard Serial Number (ISSN)

  • 0022-247X

Digital Object Identifier (DOI)

  • 10.1016/j.jmaa.2003.09.067

Citation Source

  • Scopus