Elliptic and modular curves over finite fields and related computational issues. (English) Zbl 0915.11036

Buell, D. A. (ed.) et al., Computational perspectives on number theory. Proceedings of a conference in honor of A. O. L. Atkin, Chicago, IL, USA, September 1995. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 7, 21-76 (1998).
In this wonderful paper the author explains various methods to compute explicit equations for the modular curves \(X_0(n)\). He explains how these equations can be used to explicitly compute isogenies between elliptic curves. These explicit isogenies are then in turn applied to the problem of counting points on elliptic curves over finite fields. In a separate section the author discusses methods to compute the number of points on curves over finite fields of genus \(g>1\). More generally, he discusses the problem of computing the characteristic polynomial of Frobenius acting on the \(l\)-adic cohomology groups of varieties over finite fields. Along the way the author illustrates the methods of this article by giving various explicit computationally non-trivial examples. The appendix contains a detailed description of a variety of techniques to compute explicit models for the curves \(X_0(n)\) for \(n=37\), \(75\), \(161\), \(191\) and \(239\).
This paper is a great source for anyone who wants to compute explicit models for modular curves.
For the entire collection see [Zbl 0881.00035].


11G20 Curves over finite and local fields
11Y16 Number-theoretic algorithms; complexity
14Q05 Computational aspects of algebraic curves
14H52 Elliptic curves