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ABELIAN VARIETIES

J.S. MILNE Abstract. These are the notes for Math 731, taught at the University of Michigan in Fall 1991, somewhat revised from those handed out during the course. They are available at www.math.lsa.umich.edu/,jmilne/.

Please send comments and corrections to me at [email protected]. v1.1 (July 27, 1998). First version on the web. These notes are in more primitive form than my other notes -- the reader should view them as an unfinished painting in which some areas are only sketched in.

Contents Introduction 1

Part I: Basic Theory of Abelian Varieties

1. Definitions; Basic Properties. 7 2. Abelian Varieties over the Complex Numbers. 9 3. Rational Maps Into Abelian Varieties 15 4. The Theorem of the Cube. 19 5. Abelian Varieties are Projective 25 6. Isogenies 29 7. The Dual Abelian Variety. 32 8. The Dual Exact Sequence. 38 9. Endomorphisms 40 10. Polarizations and Invertible Sheaves 50 11. The Etale Cohomology of an Abelian Variety 51 12. Weil Pairings 52 13. The Rosati Involution 53 14. The Zeta Function of an Abelian Variety 54 15. Families of Abelian Varieties 57 16. Abelian Varieties over Finite Fields 60 17. Jacobian Varieties 64 18. Abel and Jacobi 67

Part II: Finiteness Theorems

19. Introduction 70 20. N'eron models; Semistable Reduction 76 21. The Tate Conjecture; Semisimplicity. 77

cfl1998 J.S. Milne. You may make one copy of these notes for your own personal use.

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22. Geometric Finiteness Theorems 82 23. Finiteness I implies Finiteness II. 87 24. Finiteness II implies the Shafarevich Conjecture. 92 25. Shafarevich's Conjecture implies Mordell's Conjecture. 94 26. The Faltings Height. 98 27. The Modular Height. 102 28. The Completion of the Proof of Finiteness I. 106 Appendix: Review of Faltings 1983 (MR 85g:11026) 107 Index 110

ABELIAN VARIETIES 1

Introduction The easiest way to understand abelian varieties is as higher-dimensional analogues of elliptic curves. Thus we first look at the various definitions of an elliptic curve.

Fix a ground field k which, for simplicity, we take to be algebraically closed.

0.1. An elliptic curve is the projective curve given by an equation of the form

Y 2Z = X3 + aXZ + bZ3, \Delta df= 4a3 + 27b2 6= 0: (*) (char 6= 2; 3).

0.2. An elliptic curve is a nonsingular projective curve of genus one together with a distinguished point.

0.3. An elliptic curve is a nonsingular projective curve together with a group structure defined by regular maps.