Systems of orthogonal polynomials arising from the modular \(j\)-function.
type: Item
instance of: scholarly article
title: Systems of orthogonal polynomials arising from the modular \(j\)-function.
zbMATH Open document ID: 1109.11025
author: D. Massart
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-01-26T00:00:00Z
review text: Let \(S_p(x)\in \mathbb F_p[x]\) be the supersingular polynomial in characteristic \(p\). In this paper the authors generalize a construction of Atkin which was described in a recent paper of \textit{M. Kaneko} and \textit{D. Zagier} [Computational perspectives on number theory (Chicago, IL 1995), AMS/IP Stud. Adv. Math. 7, 97--126 (1998; Zbl 0955.11018)] which relates the supersingular polynomials to a certain system of orthogonal polynomials. In particular, the authors call \(\psi(x)\in \mathbb R[x]\) \textit{good} if there is a set of monic orthogonal polynomials \(\{P_{n, \psi}\}_{n=1}^\infty\) with respect to a natural inner product \(\langle\;,\;\rangle_\psi\) (it is shown that most \(\psi\) are good). The authors show in many cases that if \(\psi\) is good and \(n\) is large, then \(S_p(x)\) divides \(\psi(x) P_{n, \psi}(x)\) in \(\mathbb F_p[x]\). When \(\psi(x)=1\) the result of Atkin is recovered.
reviewed by: Scott Ahlgren
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 33E50
Mathematics Subject Classification ID: 11F85
zbMATH DE Number: 2032974
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: supersingular j-invariants
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.09.067
OpenAlex ID: W1976864234
cites work: Weierstrass points on \(X_0(p)\) and supersingular \(j\)-invariants
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DOI: 10.1016/J.JMAA.2003.09.067
description: scientific article; zbMATH DE number 2032974