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Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials.

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Kaneko,  M.
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier,  D.
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Kaneko, M., & Zagier, D. (1998). Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials. In Computational perspectives on number theory. Proceedings of a conference in honor of A. O. L. Atkin, Chicago, IL, USA, September 1995 (pp. 97-126). Providence, RI: American Mathematical Society.

Cite as: https://hdl.handle.net/21.11116/0000-0004-34AB-1
Abstract
An elliptic curve E over a field K of characteristic pgt;0 is called supersingular if the group E(\overlineK) has no p-torsion. This condition depends only on the j-invariant of E and it is well known that there are only finitely many supersingular j-invariants in \bbfFp. \par The authors of the paper under review describe several different ways of constructing canonical polynomials in \bbfQ [j] whose reduction modulo p gives the supersingular polynomial ssp(j):= \prod\Sb E/\overline\bbfFp\\ E\text supersingular \endSb (j-j(E))\in \bbfFp[j]. These polynomials are of three kinds: \par A. Polynomials coming from modular forms of weight p-1. \par Four special modular forms of weight p-1 are defined and, if f is one of these four forms, the coefficients of the associated polynomial \widetildef are p-integral and ssp(j)= \pm j^δ (j-1728)^\varepsilon \widetildef(j) \bmod p\qquad (δ\in 0,1,2,\ \varepsilon\in 0,1). B. The Atkin orthogonal polynomials. \par This description was found by Atkin more than ten years ago but proofs have never been published. Atkin has defined a sequence of polynomials An(j)\in \bbfQ[j], one in each degree n, as the orthogonal polynomials with respect to a special scalar product. The coefficients of An are rational numbers in general but they are p-integral for primes pgt; 2n. In particular if np is the degree of the supersingular polynomial ssp, then Anp has p-integral coefficients and we have the congruence ssp(j) \equiv Anp(j) \pmod p as well as recursion relation, closed formula and differential equation of An. \par The proofs here are simpler than those of Atkin. \par C. Other orthogonal polynomials coming from hyperelliptic series. \par This is a partially expository paper.