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Abelian Varieties ABELIAN VARIETIES BRIAN CONRAD LECTURE NOTES BY TONY FENG CONTENTS Note to the reader 3 1. Basic theory 4 1.1. Group schemes 4 1.2. Complex tori 5 1.3. Link between complex abelian varieties and complex tori 6 1.4. The Mordell–Weil Theorem 7 1.5. Commutativity 8 1.6. Torsion 11 1.7. Rigidity 12 2. The Picard functor 17 2.1. Overview 17 2.2. Rigidification 17 2.3. Representability 20 2.4. Properties of PicX =k . 22 3. Line bundles on abelian varieties 25 3.1. Some fundamental tools 25 3.2. The φ construction 27 3.3. The Poincaré bundle 30 3.4. Projectivity of abelian varieties 31 4. Torsion 37 4.1. Multiplication by n 37 4.2. Structure of the torsion subgroup 41 5. The dual abelian variety 46 5.1. Smoothness 46 5.2. Characterization of φ 48 5.3. The Néron–Severi groupL 51 6. Descent 54 6.1. Motivation 54 6.2. fpqc descent 55 7. More on the dual abelian variety 62 7.1. Dual morphisms 62 7.2. Cohomology of the Poincaré bundle 63 7.3. Dual isogenies 67 1 Abelian Varieties Math 249C, 2015 7.4. Symmetric morphisms 69 7.5. Ampleness 71 7.6. Endomorphisms 72 8. The Weil Pairing 78 8.1. Cartier duality 78 8.2. Explicit description of the Weil pairing 81 9. The Mordell–Weil Theorem 85 9.1. Overview 85 9.2. Proof assuming weak Mordell–Weil plus heights 86 9.3. The weak Mordell–Weil Theorem 87 10. Heights 93 10.1. Naïve Heights 93 10.2. Intrinsic theory of heights 97 10.3. Tate’s canonical height 101 References 105 2 Abelian Varieties Math 249C, 2015 NOTE TO THE READER This document consists of lecture notes that Tony Feng “live-TEXed” from a course given by Brian Conrad at Stanford University in the Spring quarter of 2015, which both Feng and Conrad edited afterwards. The material in §10.3 is largely distinct from Conrad’s lectures, and two substitute lectures were delivered (by Akshay Venkatesh and Zhiwei Yun) when Conrad was out of town. For two lectures missed by Feng, we are grateful to Ho Chung Siu and David Sherman for providing notes to fill in the gaps. 3 Abelian Varieties Math 249C, 2015 1. BASIC THEORY 1.1. Group schemes. Definition 1.1.1. Let S be a scheme. An S-group (or group scheme over S) is a group object in the category of S-schemes. In other words, it is an S-scheme G equipped with an S-map m : G S G G (multiplication), an S map i : G G (inversion), and a section e : S G such× that! the usual group axiom diagrams commute:! (1) (Associativity)! 1 m G G G × / G G × × × m 1 m × G G m / G × (2) (Identity) 1 e G S × / G G × × e 1 m × 1 % G G m / G × (3) (Inverse) 1,i G / G G × i ,1 G G % S m × e m # ) G Remark 1.1.2. By Yoneda’s Lemma, it is equivalent to endow G S Hom S ,G with a ( 0) = ( 0 ) group structure functorially in S-schemes S . 0 Exercise 1.1.3. Using the Yoneda interpretation, show that if G ,H are S-groups and f : G H is an S-scheme map that respects the multiplication morphisms, then it au- tomatically! respects the inversion map and identity section. Carry over all other triv- ialities from the beginnings of group theory (such as uniqueness of identity section). Can you do all this by writing huge diagrams and avoiding Yoneda? Exercise 1.1.4. Let f : G H be a homomorphism of S-groups. The fiber product f 1 e G S is the! scheme-theoretic kernel of f , denoted ker f . Prove that it − ( H ) = H ,eH is a locally closed× subscheme of G whose set of S -points (for an S-scheme S ) is the 0 0 subgroup ker G S H S . The situation for cokernels is far more delicate, much ( ( 0) ( 0)) like for quotient sheaves.! The fact that this is a (locally closed) subscheme is not entirely trivial, as sections need not be closed immersions in general! (Consider the affine line with the doubled origin mapping to the affine line by crushing the two origins.) 4 Abelian Varieties Math 249C, 2015 Exercise 1.1.5. For each of the following group-valued functors on schemes, write down a representing affine scheme and the multiplication, inversion, and identity maps at the level of coordinate rings: m Ga (S) = Γ (S, S ),GLn (S) = GLn (Γ (S, S )),µm = ker(t : GL1 GL1). O O ! For a finite group G , do the same for the functor of locally constant G -valued functions (called the constant Z-group associated to G ). Definition 1.1.6. An abelian variety over a field k is a smooth, connected, proper k- group scheme X . In particular, there are morphisms m : X X X , e X (k), i : X X satisfying the usual group-axiom diagrams. × ! 2 ! From the functor of points perspective, this is equivalent to R X (R) being a group functor on k-algebras R. 7! Remark 1.1.7. By smoothness, it suffices to check the group-scheme axioms for such given data on X on k-points. This is sometimes a useful fact. Example 1.1.8. In dimension 1, an abelian variety is an elliptic curve (a genus-1 curve with a rational point e X (k)). 2 Exercise 1.1.9. Let G be a group scheme locally of finite type over a field k, and let the map m : G G G be the multiplication morphism. Prove that the tangent map × ! d m(e ,e ) : Te (G ) Te (G ) Te (G ) ⊕ ! is addition. This is very useful! 1.2. Complex tori. Definition 1.2.1. A complex torus is a connected compact Lie group over C. These are the analytic analogues of abelian varieties over C. Example 1.2.2. If V Cg and Λ V is a lattice (a discrete, co-compact subgroup) then V =Λ is a complex torus.' ⊂ Example 1.2.3. Let C be a connected compact Riemann surface. Then C X an, where X is a smooth proper connected curve over C (i.e. “comes from” algebraic' geometry). We now construct a natural complex torus from C , studied by Abel and Jacobi long before the advent of the algebraic theory of abelian varieties (and in fact a big impetus for its development by Weil and others). 1 The space Ω (C ) of holomorphic 1-forms on C is a g -dimensional C=vector space. We have H C ,Z Z2g , and there is a natural map H C ,Z 1 C via integration 1( ) 1( ) Ω ( )∗ along cycles. Here' is a crucial property: ! Exercise 1.2.4. Show that the image of H C ,Z in 1 C is a lattice. One approach is to 1( ) Ω ( )∗ 1 1 1 use the fact from Hodge theory that the natural C-linear map Ω (C ) Ω (C ) H (C ,C) is an isomorphism. (There are proofs which do not use Hodge theory.)⊕ ! 1 R If we pick bases w j for Ω C and σi for H1 C ,Z , then !j is a 2g g matrix, ( )∗ ( ) ( σi ) called the period matrixf g . f g × 5 Abelian Varieties Math 249C, 2015 Definition 1.2.5. The analytic Jacobian of C is J 1 C H C ,Z , a g -dimensional C = Ω ( )∗= 1( ) complex torus. Remark 1.2.6. Notice that J is covariant in C . Namely, a map f : C C induces C 0 C C and f : H C ,Z H C ,Z . One can check that the diagram! Ω( 0)∗ Ω( )∗ 1( 0 ) 1( ) ! ∗ ! C H C ,Z Ω( 0) o 1( 0 ) Ω(C ) o H1(C ,Z) is commutative, hence we obtain an induced map on analytic Jacobians. If g > 0, then there is an almost canonical way of embedding C in JC . For a basepoint R c c0 C , we have a map ιc : C JC defined by c mod H1 C ,Z . The integral 0 c0 ( ( )) depends2 on the choice of path,! and integrating a holomorphic7! differential is unaffected by homotopy, so the integral is well-defined modulo integrals along loops. Exercise 1.2.7. If g > 0, then prove that the map of sets ιc0 : C JC is complex-analytic ! and has smooth image over which C is a finite analytic covering space. Deduce that ic0 is a closed embedding when g > 1, and prove that ic0 is an isomorphism when g = 1 by identifying H1(ic0 ,Z) with the identity map when g = 1. This is a powerful tool for studying curves using knowledge of complex tori. We would like to replicate this in algebraic geometry over C, and then over general fields. Before we discuss the algebraic theory, we remark on the ubiquity of the preceding construction. Theorem 1.2.8. Every complex torus A is commutative (contrast with connected com- pact Lie groups over R!) and the holomorphic exponential map expA : T0(A) A is a surjective homomorphism with kernel ΛA T0(A) a lattice. Hence A T0(A)=ΛA!. ⊂ ' Proof. See pp. 1-2 of [Mum]. The key is to study the adjoint representation of A act- ing on T A . This map a dc e (with c x a x a 1) is a holomorphic map A 0( ) a ( ) a ( ) = − GL(T0(A)) from a connected7! compact complex manifold into an open submanifold of! a Euclidean space, so it must be constant (by the maximum principle in several complex variables).
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