Abelian Varieties - James Milne

1 Abelian MilneVersion 16, 2008 These notes are an introduction to the theory of Abelian Varieties , including the arithmeticof Abelian Varieties and Faltings s proof of certain finiteness theorems. The orginal versionof the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough information@misc{milneAV,author={ Milne , James S.},title={ Abelian Varieties ( )},year={2008},note={Available at },pages={166+vi}} (July 27, 1998). First version on the web, 110 (March 17, 2008). Corrected, revised, and expanded; 172 at send comments and corrections to me at the address on my web photograph shows the Tasman Glacier, New 1998, 2008 paper copies for noncommercial personal use may be made without explicit permis-sion from the copyright Abelian Varieties : Geometry71 Definitions; Basic Properties.

2 72 Abelian Varieties over the Complex Numbers.. 103 Rational Maps Into Abelian Varieties .. 154 Review of cohomology .. 205 The Theorem of the Cube.. 216 Abelian Varieties are Projective .. 277 Isogenies .. 328 The Dual Abelian Variety.. 349 The Dual Exact Sequence.. 4110 Endomorphisms .. 4211 Polarizations and Invertible Sheaves .. 5312 The Etale Cohomology of an Abelian Variety .. 5413 Weil Pairings .. 5714 The Rosati Involution .. 6115 Geometric Finiteness Theorems .. 6316 Families of Abelian Varieties .. 6717 N eron models; Semistable Reduction .. 6918 Abel and Jacobi.

3 71II Abelian Varieties : Arithmetic751 The Zeta Function of an Abelian Variety .. 752 Abelian Varieties over Finite Fields .. 783 Abelian Varieties with complex multiplication .. 83 III Jacobian Varieties851 Overview and definitions .. 852 The canonical maps fromCto its Jacobian variety .. 913 The symmetric powers of a curve .. 944 The construction of the Jacobian variety .. 985 The canonical maps from the symmetric powers ofCto its Jacobian variety 1016 The Jacobian variety as Albanese variety; autoduality .. 1047 Weil s construction of the Jacobian variety .. 1088 Generalizations.

4 1109 Obtaining coverings of a curve from its Jacobian .. 113iii10 Abelian Varieties are quotients of Jacobian Varieties .. 11611 The zeta function of a curve .. 11712 Torelli s theorem: statement and applications .. 12013 Torelli s theorem: the proof .. 12214 Bibliographic notes .. 125IV Finiteness Theorems1291 Introduction .. 1292 The Tate Conjecture; Semisimplicity.. 1353 Finiteness I implies Finiteness II.. 1394 Finiteness II implies the Shafarevich Conjecture.. 1445 Shafarevich s Conjecture implies Mordell s Conjecture.. 1456 The Faltings Height.. 1497 The Modular Height.. 1538 The Completion of the Proof of Finiteness I.

5 158 Bibliography161 Index165ivNotationsWe use the standard (Bourbaki) notations:ND f0;1;2;:::g,ZDring of integers,QDfield of rational numbers,RDfield of real numbers,CDfield of complex numbers,FpDZ=pZDfield ofpelements,pa prime number. Given an equivalence relation, denotes the equivalence class containing . A family of elements of a setAindexed by asecond setI, , is a functioni7!aiWI! fieldkis said to be separably closed if it has no finite separable extensions of degree> 1. We useksepandkalto denote separable and algebraic closures ofkrespectively. Fora vector spaceNover a fieldk,N_denotes the dual vector ;k/.

6 All rings will be commutative with1unless it is stated otherwise, and homomorphismsof rings are required to map1to1. Ak-algebra is a ringAtogether with a homomorphismk!A. For a ringA,A is the group of units inA:A Dfa2 Ajthere exists ab2 Asuch thatabD1g:XdfDY Xis defined to beY, or equalsYby definition;X Y Xis a subset ofY(not necessarily proper, ,Xmay equalY);X Y XandYare isomorphic;X'Y XandYare canonically isomorphic (or there is a given or unique isomorphism).Conventions concerning algebraic geometryIn an attempt to make the notes as accessible as possible, and in order to emphasize thegeometry over the commutative algebra, I have based them as far as possible on my notesAlgebraic Geometry (AG).

7 Experts on schemes need only note the following. Analgebraic varietyover a fieldkis a geometrically reduced separated scheme of finite type overkexcept that we omitthe nonclosed points from the base space. It need not be connected. Similarly, analgebraicspaceover a fieldkis a scheme of finite type overk, except that again we omit the more detail, an affine algebra over a fieldkis a finitely generatedk-algebraRsuchthatR kkalhas no nonzero nilpotents for one (hence every) algebraic such ak-algebra, we associate a ring (topological space endowedwith a sheaf ofk-algebras), and an affine variety overkis a ringed space isomorphic toone of this form.

8 Analgebraic varietyoverkis a ringed ;OV/admitting a finiteopen coveringVDSU isuch ;OVjUi/is an affine variety for eachiand whichsatisfies the separation axiom. IfVis a variety overkandK k, the set ofpoints ofVwith coordinates inKandVKorV=Kis the variety overKobtained fromVbyextension of spaceis similar, except an algebraic space foranyfinitely generatedk-algebra and we drop the separatedness often describe regular maps by their actions on points. Recall that a regular map WV!Wofk- Varieties is determined by the map of ! itdefines. Moreover, to give a regular mapV!Wofk- Varieties is the same as to givenatural !

9 Over the affinek-algebras (AG ).Throughoutkis a a minimum, the reader is assumed to be familiar with basic algebraic geometry, as forexample in my notes AG. Some knowledge of schemes and algebraic number theory willalso be addition to the references listed at the end, I refer to the following of my course notes:GTGroup Theory ( , 2007).FTFields and Galois Theory ( , 2008).AGAlgebraic Geometry ( , 2008).ANTA lgebraic Number Theory ( , 2008).LECL ectures on Etale Cohomology ( , 1998).CFTC lass Field Theory ( , 2008).AcknowledgementsI thank the following for providing corrections and comments on earlier versions of thesenotes: Holger Deppe, Frans Oort, Bjorn Poonen (and Berkeley students), Vasily Shabat,Olivier Wittenberg, and easiest way to understand Abelian Varieties is as higher-dimensional analogues of ellip-tic curves.

10 Thus we first look at the various definitions of an elliptic curve. Fix a groundfieldkwhich, for simplicity, we take to be algebraically closed. An elliptic curve overkcan be defined, according to taste, as:(a) ( 2;3) a projective plane curve overkof the formY2 ZDX3 CaXZCbZ3; 4a3C27b2 0I(1)(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 regularmaps, or(d) (kDC/an algebraic curveEsuch C= (as a complex manifold) forsome lattice briefly sketch the proof of the equivalence of these definitions (see also Milne 2006,Chapter II).)