Arithmetic moduli of elliptic curves. (English) Zbl 0576.14026

Katz, Nicholas M.; Mazur, Barry

Annals of Mathematics Studies, 108. Princeton, New Jersey: Princeton University Press. XIV, 514 p. hbk: $ 78.00; pbk: $ 22.50 (1985).

This book gives a full account of the theory of moduli for elliptic curves order \({\mathbb{Z}}\) from a scheme-theoretical point of view. In order to give representable moduli problems elliptic curves need to be rigidified by introducing supplementary structures not preserved by the automorphism group. The \(\Gamma\) (N)-problem, or full level-N problem, is determined by a choice of generators of the group of division points with order dividing N. For elliptic curves over an arbitrary base ring this has to be stated in terms of subgroup schemes exactly generated by two divisors. There are similar statements for the \(\Gamma_ 0(N)-\), \(\Gamma_ 1(N)\)-problems and their variants. The authors introduce functorial machinery to treat these problems in a uniform way and avoid algebraic stacks by the device of relative representability.
The first four chapters give a self-contained account of the elementary theory of elliptic curves over a scheme, their finite subgroup and moduli problems. The reader is referred to other sources for deeper results such as the Serre-Tate theorem on deformations and the construction of Tate curves.
The subsequent chapters deal with the intricate details involved in completing the moduli scheme at infinity and describing its structure at points corresponding to super-singular curves. In particular two chapters cover problems specific to characteristic p and Igusa curves. The culminating result is a general good-reduction theorem for Jacobians.
Altogether this book is an essential complement to the many works on the transcendental theory of elliptic curves.

Reviewer: G.Horrocks

MSC:

14H10	Families, moduli of curves (algebraic)
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14D22	Fine and coarse moduli spaces
14H52	Elliptic curves
14D20	Algebraic moduli problems, moduli of vector bundles
11F03	Modular and automorphic functions
14H45	Special algebraic curves and curves of low genus

Keywords:

full level-N problem; elliptic curves; moduli scheme at infinity; characteristic p; Igusa curves; good-reduction theorem for Jacobians