Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108 (Annals of Mathematics Studies)

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld. TABLE OF CONTENTS INTRODUCTION Chapter 1: GENERALITIES ON "A-STRUCTURES" AND "A-GENERATORS" 1.1 Review of relative Cartier divisors 1.2 Relative Cartier divisors in curves 1.3 Existence of incidence schemes 1.4 Points of "exact order N" and cyclic subgroups 1.5 A mild generalization: A -structures and A -generators 1.6 General representability theorems for A-structures and A-generators 1.7 Factorization into prime powers of A-structures and A-generators 1.8 Full sets of sections 1.9 Intrinsic A-structures and A-generators 1.10 Relation to Cartier divisors 1.11 Extensions of an etale group 1.12 Roots of unity 1.13 Some open problems Chapter 2: REVIEW OF ELLIPTIC CURVES 2.1 The group structure 2.2 Generalized Weierstrass equations, and some elementary universal families 2.3 The structure of [N] 2.4 Rigidity 2.5 Manifestations of autoduality 2.6 Hasse's theory 2.7 Applications to rigidity 2.8 Pairings 2.9 Deformation theory Chapter 3: THE FOUR BASIC MODULI PROBLEMS FOR ELLIPTIC CURVES: SORITES 3.1 Γ(N)-structures 3.2 Γ1(N)-structures 3.3 Balanced Γ1(N)-structures 3.4 Γ0(N)-structures 3.5 Factorization into prime powers 3.6 Relative representability 3.7 The situation when N is invertible Chapter 4: THE FORMALISM OF MODULI PROBLEMS 4.1 The category (Ell) 4.2 Moduli problems 4.3 Representable moduli problems 4.4 Rigid moduli problems 4.5 Geometric properties of moduli problems 4.6 Some examples 4.7 A basic result: representability and rigidity 4.8 Another example 4.9 Yet another example 4.10 Lemmas on group-schemes 4.11 Modular families 4.12 More geometric properties of moduli problems 4.13 The category (Ell/R) 4.14 Moduli problems of finite level APPENDIX: MORE ON RIGIDITY AND REPRESENTABILITY Chapter 5: Regularity theorems 5.1 First main theorem 5.2 Axiomatics 5.3 End of the proof 5.4 Summary of parameters at supersingular points 5.5 First applications 5.6 Pairings Chapter 6: CYCLICITY 6.1 The main theorem 6.2 Axiomatics 6.3 End of the proof 6.4 Cyclicity as a closed condition 6.5 The moduli problem [N–Isog] 6.6 The moduli problem [Γ0(N)]: proof of the First Main Theorem 6.7 Detailed theory of cyclic isogenies and cyclic subgroups; standard factorizations 6.8 More on [N-Isog] Chapter 7: QUOTIENTS BY FINITE GROUPS 7.1 The general situation 7.2 A descent situation 7.3 Quotients of product problems 7.4 Applications to the four basic moduli problems 7.5 Axiomatics 7.6 Applications to regularity 7.7 Summary of parameters at supersingular points 7.8 More parameters for [Γ0(p^n)] at supersingular points 7.9 Detailed study of the congruence quotients [Γ0(p^n); a,b] of [bal. Γ1p^n)] APPENDIX: BASE CHANGE FOR RINGS OF INVARIANTS Chapter 8: COARSE MODULI SCHEMES, CUSPS, AND COMPACTIFICATIONS 8.1 Coarse moduli schemes 8.2 The j-line as a coarse moduli scheme 8.3 Localization of moduli problems over the j-line 8.4 The j-invariant as a fine modulus, coarse moduli schemes as fine moduli schemes (!) 8.5 Base change for coarse moduli schemes 8.6 Cusps by normalization near infinity; compactified coarse moduli schemes 8.7 Interlude: The groups T[N] and T 8.8 Relation to the Tate curve 8.9 Relation with ordinary elliptic curves via the Serre-Tate parameter 8.10 Other universality properties of the groups T[N] 8.11 Computation of Cusps(P) via the Tate curve Chapter 9: MODULI PROBLEMS VIEWED OVER CYCLOTOMIC INTEGER RINGS 9.1 Generalities 9.2 A descent situation 9.3 The situation near infinity 9.4 Applications to the basic moduli problems Chapter 10: THE CALCULUS OF CUSPS AND COMPONENTS VIA THE GROUPS T[N] AND THE GLOBAL STRUCTURE OF THE BASIC MODULI PROBLEMS 10.1 Motivation 10.2 Analysis of (Γ(N)] 10.3 Group action 10.4 Canonical problems 10.5 Explication for T[N] 10.6 Cusp-labels and component-labels 10.7 Some combinatorial lemmas 10.8 Application to structure near infinity 10.9 Applications to the four basic moduli problems 10.10 Detailed analysis at a prime p, balanced subgroups 10.11 Basic examples of balanced subgroups 10.12 Application to the moduli problem [ΓQ(P^n); a,a] 10.13 The numerology of moduli schemes, via the line bundle ω Chapter 11 : INTERLUDE: EXOTIC MODULAR MORPHISMS AND ISOMORPHISMS 11.1 Motivation 11.2 "Abstract" morphisms 11.3 Some basic examples Chapter 12: NEW MODULI PROBLEMS IN CHARACTERISTIC p; IGUSA CURVES 12.1 Frobenius 12.2: Basic lemmas 12.3 Igusa structures 12.4 The Hasse invariant 12.5 Ordinary curves 12.6 First analysis of the Igusa curve 12.7 Analysis of cusps 12.8 The equation defining Ig(p), and a theorem of Serre 12.9 Numerology of Igusa curves 12.10 "Exotic" projections from Igusa curves; "exotic" Igusa structures Chapter 13: REDUCTIONS mod p OF THE BASIC MODULI PROBLEMS 13.1 Some general considerations on crossings at supersingular points 13.2 Modular schemes as examples 13.3 Analysis of p-power isogenies between elliptic curves 13.4 Global structure of the moduli spaces W(P,[Γ0,(p^n)]), M(P,[p^n-Isog]) 13.5 Analysis of [Γ1(p^n)] in characteristic p 13.6 Explicit calculations via the groups T[p^n] of [Γ0(p^n)], [Γ1(p^n)] 13.7 The reduction mod p of [Γ(p^n)]^can 13.8 Complete local ring of [Γ(p^n)]^can at supersingular points; intersection numbers 13.9 Distribution of the cusps on [Γ(p^n)]^can 13.10 The reduction mod p of a general p-power level moduli problem 13.11 The reduction mod p of [bal. Γ1(p^n)]^can 13.12 The reduction mod p of quotients of [bal. Γ1(p^n)] by subgroups of (Z/p^nZ)^× × (Z/p^nZ)^× Chapter 14: APPLICATION TO THEOREMS OF GOOD REDUCTION 14.1 General review of vanishing cycles 14.2 Application to curves 14.3 Application to modular curves: explicitation of the numerical criterion 14.4 Characters and conductors 14.5 The Good Reduction Theorem 14.6 Explicitation of the Good Reduction Theorem 14.7 Application to Jacobians NOTES ADDED IN PROOF REFERENCES