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Basic Theory of Abelian Varieties 1. Definitions

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Basic Theory of Abelian Varieties 1. Definitions 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 c 1998 J.S. Milne. You may make one copy of these notes for your own personal use. 1 0J.S.MILNE 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 df Y 2Z = X3 + aXZ + bZ3,∆=4a3 +27b2 =0 . (*) (char =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. 0.4. (k = C) An elliptic curve is complex manifold of the form C/ΛwhereΛisa lattice in C. We briefly sketch the equivalence of these definitions (see also my notes on Elliptic Curves, especially §§5,10). (0.1) =⇒(0.2). The condition ∆ = 0 implies that the curve is nonsingular; take the distinguished point to be (0 : 1 : 0). (0.2) =⇒(0.1). Let ∞ be the distinguished point on the curve E of genus 1. The Riemann-Roch theorem says that dim L(D)=deg(D)+1− g =deg(D) where L(D)={f ∈ k(E) | div(f)+D ≥ 0}. On taking D =2∞ and D =3∞ successively, we find that there is a rational function x on E with a pole of exact order 2 at ∞ and no other poles, and a rational function y on E with a pole of exact order 3 at ∞ and no other poles. The map P → (x(P ):y(P ):1),∞ →(0:1:0) defines an embedding E→ P2. On applying the Riemann-Roch theorem to 6∞, we find that there is relation (*) between x and y, and therefore the image is a curve defined by an equation (*). (0.1,2) =⇒(0.3): Let Div0(E) be the group of divisors of degree zero on E,andlet Pic0(E) be its quotient by the group of principal divisors; thus Pic0(E) is the group of divisor classes of degree zero on E. The Riemann-Roch theorem shows that the map P → [P ] − [∞]: E(k) → Pic0(E) is a bijection, from which E(k) acquires a canonical group structure. It agrees with the structure defined by chords and tangents, and hence is defined by polynomials, i.e., it is defined by regular maps. 2J.S.MILNE (0.3) =⇒(0.2): We have to show that the existence of the group structure implies that the genus is 1. Our first argument applies only in the case k = C.TheLefschetz trace formula states that, for a compact oriented manifold X and a continuous map α : X → X with only finitely many fixed points, each of multiplicity 1, number of fixed points = Tr(α|H0(X, Z)) − Tr(α|H1(X, Z)) + ··· If X has a group structure, then, for any nonzero point a ∈ X, the translation map ta : x → x + a has no fixed points, and so df i i Tr(ta) =Σ(−1) Tr(ta|H (X, Q)) = 0. The map a → Tr(ta): X → Z is continuous, and so Tr(ta)=0alsofora =0.Butt0 is the identity map, and Tr(id) = (−1)i dim Hi(X, Q)=χ(X) (Euler-Poincar´e characteristic). Since the Euler-Poincar´e characteristic of a complete nonsingular curve of genus g is 2 − 2g,weseethatifX has a group structure then g =1.[Thesameargumentworks over any field if one replaces singular cohomology with ´etale cohomology.] We now give an argument that works over any field. If V is an algebraic variety with a group structure, then the sheaf of differentials is free. For a curve, this means that the canonical divisor class has degree zero. But this class has degree 2g − 2, and so again we see that g =1. (0.4) =⇒(0.2). The Weierstrass ℘-function and its derivative define an embedding z → (℘(z):℘(z):1):C/Λ → P2, whose image is a nonsingular projective curve of genus 1 (in fact, with equation of the form (*)). (0.2) =⇒(0.4). This follows from topology. Abelian varieties. Definition (0.1) simply doesn’t generalize — there is no simple description of the equations defining an abelian variety of dimension1 g>1. In general, it is not possible to write down explicit equations for an abelian variety of dimension > 1, and if one could, they would be too complicated to be of use. I don’t know whether (0.2) generalizes. Abelian surfaces are the only minimal surfaces with the Betti numbers 1, 4, 6, 4, 1 1The case g = 2 is something of an exception to this statement. Every abelian variety of dimension 2 is the Jacobian variety (see below) of a curve of genus 2, and every curve of genus 2 has an equation of the form 2 4 6 5 6 Y Z = f0X + f1X Z + ···+ f6Z . Flynn (Math. Proc. Camb. Phil. Soc. 107, 425–441) has found the equations of the Jacobian variety of such a curve in characteristic =2 , 3, 5 — they form a set 72 homogeneous equations of degree 2 in 16 variables (they take 6 pages to write out). See: Cassels, J.W.S., and Flynn, E.V., Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, Cambridge, 1996. ABELIAN VARIETIES 3 and canonical class linearly equivalent to zero. In general an abelian variety of di- mension g has Betti numbers 2g 2g 1, ,... , ,... ,1. 1 r Definition (0.3) does generalize: we can define an abelian variety to be a nonsingular connected projective2 variety with a group structure defined by regular maps. Definition (0.4) does generalize, but with a caution. If A is an abelian variety over C,then A(C) ≈ Cg/Λ for some lattice Λ in Cg (isomorphism simultaneously of complex manifolds and of groups). However, when g>1, for not all lattices Λ does Cg/Λ arise from an abelian variety. In fact, in general the transcendence degree over C of the field of meromorphic functions Cg/Λis<g, with equality holding if and only if Cg/Λ is an algebraic (hence abelian) variety. There is a very pleasant criterion on Λ for when Cg/Λ is algebraic (§2). Abelian varieties and elliptic curves. As we noted, if E is an elliptic curve over an algebraically closed field k, then there is a canonical isomorphism P → [P ] − [0]: E(k) → Pic0(E). This statement has two generalizations. (A). Let C be a curve and choose a point Q ∈ C(k); then there is an abelian variety J, called the Jacobian variety of C, canonically attached to C, and a regular map ϕ : C → J such that ϕ(Q)=0and 0 Σ niPi → Σ ni ϕ(Pi): Div (C) → J(k) induces an isomorphism Pic0(C) → J(k). The dimension of J is the genus of C. (B). Let A be an abelian variety. Then there is a “dual abelian variety” A∨ such that Pic0(A)=A∨(k)andPic0(A∨)=A(k) (we shall define Pic0 in this context later). In the case of an elliptic curve, E∨ = E. In general, A and A∨ are isogenous, but they are not equal (and usually not even isomorphic). Appropriately interpreted, most of the statements in Silverman’s books on elliptic curves hold for abelian varieties, but because we don’t have equations, the proofs are more abstract. In fact, every (reasonable) statement about elliptic curves should have a generalization that applies to all abelian varieties. However, for some, for example, the Taniyama conjecture, the correct generalization is difficult to state3.Topassfrom a statement about elliptic curves to one about abelian varieties, replace 1 by g (the dimension of A), and half the copies of E by A and half by A∨.
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