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Abelian Varieties Gerard van der Geer Ben Moonen ABELIAN VARIETIES (PRELIMINARY VERSION OF THE FIRST CHAPTERS) Contents Notation and conventions ................................ 1 1. Definitions and basic examples ........................... 5 2. Line bundles and divisors on abelian varieties .................. 17 1. Thetheoremofthesquare............................ 17 § 2. Projectivityofabelianvarieties . 22 § 3. Projective embeddings of abelian varieties . 26 § 3. Basic theory of group schemes ........................... 29 1. Definitionsandexamples............................ 29 § 2. Elementary properties of group schemes . 34 § 3. Cartierduality ................................... 41 § 4. The component group of a group scheme . 43 § 4. Quotients by group schemes ............................. 49 1. Categorical quotients . 49 § 2. Geometric quotients, and quotients by finite group schemes . 52 § 3. FPPFquotients.................................... 61 § 4. Finitegroupschemesoverafield . 66 § 5. Isogenies ......................................... 72 1. Definition of an isogeny, and basic properties . 72 § 2. FrobeniusandVerschiebung . 76 § 3. Densityoftorsionpoints........................... 83 § 6. The Picard scheme of an abelian variety ..................... 87 1. RelativePicardfunctors ........................... 87 § 2. Digression on graded bialgebras . 91 § 3. The dual of an abelian variety . 95 § 7. Duality .......................................... 98 1. Formation of quotients and the descent of coherent sheaves . 98 § 2. Twodualitytheorems ............................... 100 § 3. Further properties of Pic0 .............................. 101 § X/k 4. Applications to cohomology . 108 § 5. The duality between Frobenius and Verschiebung . 110 § 8. The Theta group of a line bundle ......................... 113 1. The theta group G (L) ................................. 113 § 2. Descent of line bundles over homomorphisms . 116 § 3. Theta groups of non-degenerate line bundles . 118 § 4. Representation theory of non-degenerate theta groups . 122 § 9. The cohomology of line bundles .......................... 126 10. Tate modules, p-divisible groups, and the fundamental group ........ 141 1. Tate-!-modules ..................................... 141 § i 2. The p-divisiblegroup.................................. 145 § 3. The algebraic fundamental group—generalities . 149 § 4. The fundamental group of an abelian variety . 154 § 11. Polarizations and Weil pairings ........................... 159 1. Polarizations .................................... 159 § 2. Pairings......................................... 162 § 3. Existence of polarizations, and Zarhin’s trick . 169 § 4. Polarizations associated to line bundles on torsors . 174 § 5. Symmetriclinebundles ............................. 178 § 12. The endomorphism ring ............................... 180 1. First basic results about the endomorphism algebra . 180 § 2. The characteristic polynomial of an endomorphism . 184 § 3. TheRosatiinvolution .............................. 188 § 4. TheAlbertclassification . 190 § 13. The Fourier transform and the Chow ring .................... 192 1. TheChowring ..................................... 192 § 2. TheHodgebundle ................................... 196 § 3. The Fourier transform of an abelian variety . 199 § 4. Decomposition of the diagonal . 204 § 5. Motivic decomposition . 210 § 14. Jacobian Varieties ................................... 218 1. The Jacobian variety of a curve . 218 § 2. Comparison with the g-th symmetric power of C . 221 § 3. UniversallinebundlesandtheThetadivisor . 224 § 4. Riemann’sTheoremontheThetaDivisor . 231 § 5. Examples ........................................ 233 § 6. A universal property—the Jacobian as Albanese . 235 § 7. Any Abelian Variety is a Factor of a Jacobian . 236 § 8. TheTheoremofTorelli.............................. 237 § 9. The Criterion of Matsusaka-Ran . 239 § 15. Dieudonn´etheory ................................... 245 1. Dieudonn´etheory for finite commutative group schemes and for p-divisible groups 245 § 2. Classification up to isogeny . 246 § 3. The Newton polygon of an abelian variety . 262 § 16. Abelian Varieties over Finite Fields ........................ 265 1. The eigenvalues of Frobenius . 265 § 2. TheHasse-Weil-Serreboundforcurves . 272 § 3. ThetheoremofTate ................................. 275 § 4. Corollaries of Tate’s theorem, and the structure of the endomorphism algebra . 281 § 5. Abelian varieties up to isogeny and Weil numbers . 289 § 6. Isomorphism classes contained in an isogeny class . 293 § 7. Ellipticcurves................................... 296 § 8. Newton polygons of abelian varieties over finite fields . 302 § 9. Ordinaryabelianvarietiesoverafinitefield . 303 § ii Appendix A. Algebra ................................... 308 References .......................................... 314 Index ............................................. 321 iii Notation and conventions. (0.1) In general, k denotes an arbitrary field, k¯ denotes an algebraic closure of k, and ks a separable closure. (0.2) If A is a commutative ring, we sometimes simply write A for Spec(A). Thus, for instance, by an A-scheme we mean a scheme over Spec(A). If A B is a homomorphism of rings and X → is an A-scheme then we write X = X B rather than X Spec(B). B ×A ×Spec(A) (0.3) If X is a scheme then we write X for the topological space underlying X and O for its | | X structure sheaf. If f: X Y is a morphism of schemes we write f : X Y and f !: O → | | | |→| | Y → f∗OX for the corresponding map on underlying spaces, resp. the corresponding homomorphism of sheaves on Y .Ifx X we write k(x) for the residue field. If X is an integral scheme we ∈| | write k(X) for its field of rational functions. If S is a scheme and X and T are S-schemes then we write X(T ) for the set of T -valued points of X, i.e., the set of morphisms of S-schemes T X.OftenwesimplywriteX for → T the base change of X to T ,i.e.,X := X T , to be viewed as a T -scheme via the canonical T ×S morphism X T . T → (0.4) If k is a field then by a variety over k we mean a separated k-scheme of finite type which is geometrically integral. Recall that a k-scheme is said to be geometrically integral if for some algebraically closed field K containing k the scheme XK is irreducible and reduced. By EGA IV, (4.5.1) and (4.6.1), if this holds for some algebraicallyclosedoverfieldK then XK is integral for every field K containing k. A variety of dimension 1 (resp. 2, resp. n ! 3) is called a curve (resp. surface,resp.n-fold). By a line bundle (resp. a vector bundle of rank d) on a scheme X we mean a locally free OX -module of rank 1 (resp. of rank d). By a geometric vector bundle of rank d on X we mean a group scheme π: V X over X for which there exists a affine open covering X = Uα such that → d ∪ the restriction of V to each Uα is isomorphic to Ga over Uα. In particular this means that we have isomorphisms of U -schemes ϕ : π−1(U ) ∼ U Ad, such that all transition morphisms α α α −→ α × ϕ ◦ ϕ−1 t : U Ad β α U Ad α,β α,β × −− − − − −→ α,β × are linear automorphisms of U Ad over U := U U ; this last condition means that α,β × α,β α ∩ β tα,β is given by a O(Uα,β)-linear automorphism of O(Uα,β)[x1,...,xd]. For d = 1 we obtain the notion of a geometric line bundle. If V is a geometric vector bundle of rank d on X then its sheaf of sections is a vector bundle of rank d. Conversely, if E is a vector bundle of rank d on X then the scheme V := Spec Sym(E ∨) has a natural structure of a geometric vector bundle of rank d. These two constructions are ! " quasi-inverse to each other and establish an equivalence between vector bundles and geometric vector bundles. (0.5) In our definition of an ´etale morphism of schemes we follow EGA; this means that we only require the morphism to be locally of finite type. Note that in some literature ´etale morphisms are assumed to be quasi-finite. Thus, for instance, if S is a scheme and I is an index set, the disjoint union i∈I S is ´etale over S according to our conventions, also if the set I is infinite. # –1– (0.6) If K is a number field then by a prime of K we mean an equivalence class of valuations of K. See for instance Neukirch [1], Chap. 3. The finite primes of K are in bijection with the maximal ideals of the ring of integers OK . An infinite prime corresponds either to a real embedding K$ R or to a pair ι, ¯ι of complex embeddings K$ C. → { } → If v is a prime of K, we have a corresponding homomorphism ord : K∗ R and a normal- v → ized absolute value .Ifv is a finite prime then we let ord be the corresponding valuation, || || v v normalized such that ord (K∗)=Z, and we define by v || || v −ordv (x) (qv) if x = 0, x v := ' || || 0ifx = 0, $ where qv is the cardinality of the residue field at v.Ifv is an infinite prime then we let ι(x) if v corresponds to a real embedding ι: K R, x = | | → || || v ι(x) 2 if v corresponds to a pair of complex embeddings ι, ¯ι , $ | | { } and we define ord by the rule ord (x):= log ι(x) .Here : C R! is given by a + bi = v v − | | || → 0 | | √a2 + b2. ! " –2– Definition. Let p be a prime number. We say that a scheme X has characteristic p if the unique morphism X Spec(Z) factors through Spec(F ) $ Spec(Z). This is equivalent to the → p → requirement that p f = 0 for every open U X and every f O (U). We say that a scheme X · ⊂ ∈ X has characteristic 0 if X Spec(Z) factors through Spec(Q) $ Spec(Z). This is equivalent to → → the requirement that n O (U)∗ for every n Z 0 and every open U X. ∈ X ∈ \{ } ⊂ Note that if X Y is a morphism of schemes and Y has characteristic p (with p aprime → number or p = 0) then X has characteristic p, too. The absolute Frobenius. Let p be a prime number. Let Y be a scheme of characteristic p. Then we have a morphism Frob : Y Y , called the absolute Frobenius morphism of Y ;itis Y → given by (a) Frob is the identity on the underlying topological space Y ; Y | | (b) Frob! : O O is given on sections by f f p. Y Y → Y *→ To describe Frob in another way, consider a covering U of Y by affine open subsets, say Y { α} U =Spec(A ).
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