Systems of orthogonal polynomials arising from the modular j-function |
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Authors: | Stephanie Basha Harris Nover |
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Institution: | a Department of Mathematics, Santa Clara University, Santa Clara, CA 95053, USA b 4404 South Ave. W, Missoula, MT 59804, USA c Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA d 5849 Sand Rd, Bellingham, WA 98226, USA |
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Abstract: | Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . 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 on for every . Suppose a system of orthogonal polynomials {Pn,ψ(x)}n=0∞ with respect to exists. We prove that if n is sufficiently large and ψ(x)Pn,ψ(x) is p-integral, then over . 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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