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Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . 67 17 Neron´ models; Semistable Reduction . 69 18 Abel and Jacobi . 71 II Abelian Varieties: Arithmetic 75 1 The Zeta Function of an Abelian Variety . 75 2 Abelian Varieties over Finite Fields . 78 3 Abelian varieties with complex multiplication . 83 III Jacobian Varieties 85 1 Overview and definitions . 85 2 The canonical maps from C to its Jacobian variety . 91 3 The symmetric powers of a curve . 94 4 The construction of the Jacobian variety . 98 5 The canonical maps from the symmetric powers of C to its Jacobian variety 101 6 The Jacobian variety as Albanese variety; autoduality . 104 7 Weil’s construction of the Jacobian variety . 108 8 Generalizations . 110 9 Obtaining coverings of a curve from its Jacobian . 113 iii 10 Abelian varieties are quotients of Jacobian varieties . 116 11 The zeta function of a curve . 117 12 Torelli’s theorem: statement and applications . 120 13 Torelli’s theorem: the proof . 122 14 Bibliographic notes . 125 IV Finiteness Theorems 129 1 Introduction . 129 2 The Tate Conjecture; Semisimplicity. 135 3 Finiteness I implies Finiteness II. 139 4 Finiteness II implies the Shafarevich Conjecture. 144 5 Shafarevich’s Conjecture implies Mordell’s Conjecture. 145 6 The Faltings Height. 149 7 The Modular Height. 153 8 The Completion of the Proof of Finiteness I. 158 Bibliography 161 Index 165 iv Notations We use the standard (Bourbaki) notations: N 0; 1; 2; : : : , Z ring of integers, Q D f g D D field of rational numbers, R field of real numbers, C field of complex numbers, D D Fp Z=pZ field of p elements, p a prime number. Given an equivalence relation, Œ D D denotes the equivalence class containing . A family of elements of a set A indexed by a second set I , denoted .ai /i I , is a function i ai I A. 2 7! W ! A field k is said to be separably closed if it has no finite separable extensions of degree > 1. We use ksep and kal to denote separable and algebraic closures of k respectively. For a vector space N over a field k, N _ denotes the dual vector space Homk.N; k/. All rings will be commutative with 1 unless it is stated otherwise, and homomorphisms of rings are required to map 1 to 1.A k-algebra is a ring A together with a homomorphism k A. For a ring A, A is the group of units in A: ! A a A there exists a b A such that ab 1 : D f 2 j 2 D g df X YX is defined to be Y , or equals Y by definition; D X YX is a subset of Y (not necessarily proper, i.e., X may equal Y ); X YX and Y are isomorphic; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism). ' Conventions concerning algebraic geometry In an attempt to make the notes as accessible as possible, and in order to emphasize the geometry over the commutative algebra, I have based them as far as possible on my notes Algebraic Geometry (AG). Experts on schemes need only note the following. An algebraic variety over a field k is a geometrically reduced separated scheme of finite type over k except that we omit the nonclosed points from the base space. It need not be connected. Similarly, an algebraic space over a field k is a scheme of finite type over k, except that again we omit the nonclosed points. In more detail, an affine algebra over a field k is a finitely generated k-algebra R such that R kal has no nonzero nilpotents for one (hence every) algebraic closure kal of k. ˝k With such a k-algebra, we associate a ring space Specm.R/ (topological space endowed with a sheaf of k-algebras), and an affine variety over k is a ringed space isomorphic to one of this form. An algebraic variety over k is a ringed space .V; OV / admitting a finite S open covering V Ui such that .Ui ; OV Ui / is an affine variety for each i and which D j satisfies the separation axiom. If V is a variety over k and K k, then V .K/ is the set of points of V with coordinates in K and VK or V=K is the variety over K obtained from V by extension of scalars. An algebraic space is similar, except that Specm.R/ is an algebraic space for any finitely generated k-algebra and we drop the separatedness condition. We often describe regular maps by their actions on points. Recall that a regular map V W of k-varieties is determined by the map of points V .kal/ W.kal/ that it W ! ! defines. Moreover, to give a regular map V W of k-varieties is the same as to give ! natural maps V .R/ W.R/ for R running over the affine k-algebras (AG 4.37). ! Throughout k is a field. v Prerequisites As a minimum, the reader is assumed to be familiar with basic algebraic geometry, as for example in my notes AG. Some knowledge of schemes and algebraic number theory will also be helpful. References. In addition to the references listed at the end, I refer to the following of my course notes: GT Group Theory (v3.00, 2007). FT Fields and Galois Theory (v4.20, 2008). AG Algebraic Geometry (v5.10, 2008). ANT Algebraic Number Theory (v3.00, 2008). LEC Lectures on Etale Cohomology (v2.01, 1998). CFT Class Field Theory (v4.00, 2008). Acknowledgements I thank the following for providing corrections and comments on earlier versions of these notes: Holger Deppe, Frans Oort, Bjorn Poonen (and Berkeley students), Vasily Shabat, Olivier Wittenberg, and others. vi Introduction The easiest way to understand abelian varieties is as higher-dimensional analogues of ellip- tic 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. An elliptic curve over k can be defined, according to taste, as: (a) (char.k/ 2; 3) a projective plane curve over k of the form ¤ Y 2Z X 3 aXZ bZ3; 4a3 27b2 0 (1) D C C C ¤ I (b) a nonsingular projective curve of genus one together with a distinguished point; (c) a nonsingular projective curve together with a group structure defined by regular maps, or (d) (k C/ an algebraic curve E such that E.C/ C= (as a complex manifold) for D some lattice in C. We briefly sketch the proof of the equivalence of these definitions (see also Milne 2006, Chapter II). (a) (b). The condition 4a3 27b2 0 implies that the curve is nonsingular. Since ! C ¤ it is defined by an equation of degree 3, it has genus 1. Take the distinguished point to be .0 1 0/. W W (b) (a). Let be the distinguished point on the curve E of genus 1. The Riemann- ! 1 Roch theorem says that dim L.D/ deg.D/ 1 g deg.D/ D C D where L.D/ f k.E/ div.f / D 0 : D f 2 j C g On taking D 2 and D 3 successively, we find that there exists a rational function D 1 D 1 x on E with a pole of exact order 2 at and no other poles, and a rational function y on 1 E with a pole of exact order 3 at and no other poles. The map 1 P .x.P / y.P / 1/, P ; 7! W W ¤ 1 .0 1 0/ 1 7! W W defines an embedding 2 E, P : ! On applying the Riemann-Roch theorem to 6 , we find that there is relation (1) between 1 x and y, and therefore the image is a curve defined by an equation (1). 1 2 INTRODUCTION (a,b) (c): Let Div0.E/ be the group of divisors of degree zero on E, and let 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.
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