Shimura Varieties and Moduli
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Canonical Heights on Varieties with Morphisms Compositio Mathematica, Tome 89, No 2 (1993), P
COMPOSITIO MATHEMATICA GREGORY S. CALL JOSEPH H. SILVERMAN Canonical heights on varieties with morphisms Compositio Mathematica, tome 89, no 2 (1993), p. 163-205 <http://www.numdam.org/item?id=CM_1993__89_2_163_0> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 89: 163-205,163 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. Canonical heights on varieties with morphisms GREGORY S. CALL* Mathematics Department, Amherst College, Amherst, MA 01002, USA and JOSEPH H. SILVERMAN** Mathematics Department, Brown University, Providence, RI 02912, USA Received 13 May 1992; accepted in final form 16 October 1992 Let A be an abelian variety defined over a number field K and let D be a symmetric divisor on A. Néron and Tate have proven the existence of a canonical height hA,D on A(k) characterized by the properties that hA,D is a Weil height for the divisor D and satisfies A,D([m]P) = m2hA,D(P) for all P ~ A(K). Similarly, Silverman [19] proved that on certain K3 surfaces S with a non-trivial automorphism ~: S ~ S there are two canonical height functions hs characterized by the properties that they are Weil heights for certain divisors E ± and satisfy ±S(~P) = (7 + 43)±1±S(P) for all P E S(K) . -
Models of Shimura Varieties in Mixed Characteristics by Ben Moonen
Models of Shimura varieties in mixed characteristics by Ben Moonen Contents Introduction .............................. 1 1 Shimuravarieties ........................... 5 2 Canonical models of Shimura varieties. 16 3 Integralcanonicalmodels . 27 4 Deformation theory of p-divisible groups with Tate classes . 44 5 Vasiu’s strategy for proving the existence of integral canonical models 52 6 Characterizing subvarieties of Hodge type; conjectures of Coleman andOort................................ 67 References ............................... 78 Introduction At the 1996 Durham symposium, a series of four lectures was given on Shimura varieties in mixed characteristics. The main goal of these lectures was to discuss some recent developments, and to familiarize the audience with some of the techniques involved. The present notes were written with the same goal in mind. It should be mentioned right away that we intend to discuss only a small number of topics. The bulk of the paper is devoted to models of Shimura varieties over discrete valuation rings of mixed characteristics. Part of the discussion only deals with primes of residue characteristic p such that the group G in question is unramified at p, so that good reduction is expected. Even at such primes, however, many technical problems present themselves, to begin with the “right” definitions. There is a rather large class of Shimura data—those called of pre-abelian type—for which the corresponding Shimura variety can be related, if maybe somewhat indirectly, to a moduli space of abelian varieties. At present, this seems the only available tool for constructing “good” integral models. Thus, 1 if we restrict our attention to Shimura varieties of pre-abelian type, the con- struction of integral canonical models (defined in 3) divides itself into two § parts: Formal aspects. -
The Arithmetic Volume of Shimura Varieties of Orthogonal Type
Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties The arithmetic volume of Shimura varieties of orthogonal type Fritz H¨ormann Department of Mathematics and Statistics McGill University Montr´eal,September 4, 2010 Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties 1 Algebraic geometry and Arakelov geometry 2 Orthogonal Shimura varieties 3 Main result 4 Integral models and compactifications of Shimura varieties Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Algebraic geometry X smooth projective algebraic variety of dim d =C. Classical intersection theory Chow groups: CHi (X ) := fZ j Z lin. comb. of irr. subvarieties of codim. ig=rat. equiv. Class map: i 2i cl : CH (X ) ! H (X ; Z) Intersection pairing: i j i+j CH (X ) × CH (X ) ! CH (X )Q compatible (via class map) with cup product. Degree map: d deg : CH (X ) ! Z Fritz H¨ormann Department of Mathematics and Statistics McGill University The arithmetic volume of Shimura varieties of orthogonal type Outline Algebraic geometry and Arakelov geometry Orthogonal Shimura varieties Main result Integral models and compactifications of Shimura varieties Algebraic geometry L very ample line bundle on X . 1 c1(L) := [div(s)] 2 CH (X ) for any (meromorphic) global section s of L. -
Notes on Shimura Varieties
Notes on Shimura Varieties Patrick Walls September 4, 2010 Notation The derived group G der, the adjoint group G ad and the centre Z of a reductive algebraic group G fit into exact sequences: 1 −! G der −! G −!ν T −! 1 1 −! Z −! G −!ad G ad −! 1 : The connected component of the identity of G(R) with respect to the real topology is denoted G(R)+. We identify algebraic groups with the functors they define. For example, the restriction of scalars S = ResCnRGm is defined by × S = (− ⊗R C) : Base-change is denoted by a subscript: VR = V ⊗Q R. References DELIGNE, P., Travaux de Shimura. In S´eminaireBourbaki, 1970/1971, p. 123-165. Lecture Notes in Math, Vol. 244, Berlin, Springer, 1971. DELIGNE, P., Vari´et´esde Shimura: interpr´etationmodulaire, et techniques de construction de mod`elescanoniques. In Automorphic forms, representations and L-functions, Proc. Pure. Math. Symp., XXXIII, p. 247-289. Providence, Amer. Math. Soc., 1979. MILNE, J.S., Introduction to Shimura Varieties, www.jmilne.org/math/ Shimura Datum Definition A Shimura datum is a pair (G; X ) consisting of a reductive algebraic group G over Q and a G(R)-conjugacy class X of homomorphisms h : S ! GR satisfying, for every h 2 X : · (SV1) Ad ◦ h : S ! GL(Lie(GR)) defines a Hodge structure on Lie(GR) of type f(−1; 1); (0; 0); (1; −1)g; · (SV2) ad h(i) is a Cartan involution on G ad; · (SV3) G ad has no Q-factor on which the projection of h is trivial. These axioms ensure that X = G(R)=K1, where K1 is the stabilizer of some h 2 X , is a finite disjoint union of hermitian symmetric domains. -
Abelian Varieties and Theta Functions Associated to Compact Riemannian Manifolds; Constructions Inspired by Superstring Theory
ABELIAN VARIETIES AND THETA FUNCTIONS ASSOCIATED TO COMPACT RIEMANNIAN MANIFOLDS; CONSTRUCTIONS INSPIRED BY SUPERSTRING THEORY. S. MULLER-STACH,¨ C. PETERS AND V. SRINIVAS MATH. INST., JOHANNES GUTENBERG UNIVERSITAT¨ MAINZ, INSTITUT FOURIER, UNIVERSITE´ GRENOBLE I ST.-MARTIN D'HERES,` FRANCE AND TIFR, MUMBAI, INDIA Resum´ e.´ On d´etailleune construction d^ue Witten et Moore-Witten (qui date d'environ 2000) d'une vari´et´eab´elienneprincipalement pola- ris´eeassoci´ee`aune vari´et´ede spin. Le th´eor`emed'indice pour l'op´erateur de Dirac (associ´e`ala structure de spin) implique qu'un accouplement naturel sur le K-groupe topologique prend des valeurs enti`eres.Cet ac- couplement sert commme polarization principale sur le t^oreassoci´e. On place la construction dans un c^adreg´en´eralce qui la relie `ala ja- cobienne de Weil mais qui sugg`ereaussi la construction d'une jacobienne associ´ee`an'importe quelle structure de Hodge polaris´eeet de poids pair. Cette derni`ereconstruction est ensuite expliqu´eeen termes de groupes alg´ebriques,utile pour le point de vue des cat´egoriesTannakiennes. Notre construction depend de param`etres,beaucoup comme dans la th´eoriede Teichm¨uller,mais en g´en´erall'application de p´eriodes n'est que de nature analytique r´eelle. Abstract. We first investigate a construction of principally polarized abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000. The index theorem for the Dirac operator associated to the spin structure implies integrality of a natural skew pairing on the topological K-group. The latter serves as a principal polarization. -
Mirror Symmetry of Abelian Variety and Multi Theta Functions
1 Mirror symmetry of Abelian variety and Multi Theta functions by Kenji FUKAYA (深谷賢治) Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan Table of contents § 0 Introduction. § 1 Moduli spaces of Lagrangian submanifolds and construction of a mirror torus. § 2 Construction of a sheaf from an affine Lagrangian submanifold. § 3 Sheaf cohomology and Floer cohomology 1 (Construction of a homomorphism). § 4 Isogeny. § 5 Sheaf cohomology and Floer cohomology 2 (Proof of isomorphism). § 6 Extension and Floer cohomology 1 (0 th cohomology). § 7 Moduli space of holomorphic vector bundles on a mirror torus. § 8 Nontransversal or disconnected Lagrangian submanifolds. ∞ § 9 Multi Theta series 1 (Definition and A formulae.) § 10 Multi Theta series 2 (Calculation of the coefficients.) § 11 Extension and Floer cohomology 2 (Higher cohomology). § 12 Resolution and Lagrangian surgery. 2 § 0 Introduction In this paper, we study mirror symmetry of complex and symplectic tori as an example of homological mirror symmetry conjecture of Kontsevich [24], [25] between symplectic and complex manifolds. We discussed mirror symmetry of tori in [12] emphasizing its “noncom- mutative” generalization. In this paper, we concentrate on the case of a commutative (usual) torus. Our result is a generalization of one by Polishchuk and Zaslow [42], [41], who studied the case of elliptic curve. The main results of this paper establish a dictionary of mirror symmetry between symplectic geometry and complex geometry in the case of tori of arbitrary dimension. We wrote this dictionary in the introduction of [12]. We present the argument in a way so that it suggests a possibility of its generalization. -
Some Contemporary Problems with Origins in the Jugendtraum R.P
Some Contemporary Problems with Origins in the Jugendtraum R.P. Langlands The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate developments. The first of these, the theory of class fields or of abelian extensions of number fields, attained what was pretty much its final form early in this century. The second, the algebraic theory of elliptic curves and, more generally, of abelian varieties, has been for fifty years a topic of research whose vigor and quality shows as yet no sign of abatement. The third, the theory of automorphic functions, has been slower to mature and is still inextricably entangled with the study of abelian varieties, especially of their moduli. Of course at the time of Hilbert these subjects had only begun to set themselves off from the general mathematical landscape as separate theories and at the time of Kronecker existed only as part of the theories 1 of elliptic modular functions and of cyclotomic fields. It is in a letter from Kronecker to Dedekind of 1880, in which he explains his work on the relation between abelian extensions of imaginary quadratic fields and elliptic curves with complex multiplication, that the word Jugendtraum appears. Because these subjects were so interwoven it seems to have been impossible to disintangle the different kinds of mathematics which were involved in the Jugendtraum, especially to separate the algebraic aspects from the analytic or number theoretic. Hilbert in particular may have been led to mistake an accident, or perhaps necessity, of historical development for an “innigste gegenseitige Ber¨uhrung.” We may be able to judge this better if we attempt to view the mathematical content of the Jugendtraum with the eyes of a sophisticated contemporary mathematician. -
Abelian Varieties with Complex Multiplication (For Pedestrians)
ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION (FOR PEDESTRIANS) J.S. MILNE Abstract. (June 7, 1998.) This is the text of an article that I wrote and dissemi- nated in September 1981, except that I’ve updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a simplified exposition of Deligne’s extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The addendum discusses some additional topics in the theory of complex multiplication. Contents 1. Statement of the Theorem 2 2. Definition of fΦ(τ)5 3. Start of the Proof; Tate’s Result 7 4. The Serre Group9 5. Definition of eE 13 6. Proof that e =1 14 7. Definition of f E 16 8. Definition of gE 18 9. Re-statement of the Theorem 20 10. The Taniyama Group21 11. Zeta Functions 24 12. The Origins of the Theory of Complex Multiplication for Abelian Varieties of Dimension Greater Than One 26 13. Zeta Functions of Abelian Varieties of CM-type 27 14. Hilbert’s Twelfth Problem 30 15. Algebraic Hecke Characters are Motivic 36 16. Periods of Abelian Varieties of CM-type 40 17. Review of: Lang, Complex Multiplication, Springer 1983. 41 The main theorem of Shimura and Taniyama (see Shimura 1971, Theorem 5.15) describes the action of an automorphism τ of C on a polarized abelian variety of CM-type and its torsion points in the case that τ fixes the reflex field of the variety. In his Corvallis article (1979), Langlands made a conjecture concerning conjugates of All footnotes were added in June 1998. -
Abelian Varieties
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 . -
Torsion Subgroups of Abelian Varieties
Torsion Subgroups of Abelian Varieties Raoul Wols April 19, 2016 In this talk I will explain what abelian varieties are and introduce torsion subgroups on abelian varieties. k is always a field, and Vark always denotes the category of varieties over k. That is to say, geometrically integral sepa- rated schemes of finite type with a morphism to k. Recall that the product of two A; B 2 Vark is just the fibre product over k: A ×k B. Definition 1. Let C be a category with finite products and a terminal object 1 2 C. An object G 2 C is called a group object if G comes equipped with three morphism; namely a \unit" map e : 1 ! G; a \multiplication" map m : G × G ! G and an \inverse" map i : G ! G such that id ×m G × G × G G G × G m×idG m G × G m G commutes, which tells us that m is associative, and such that (e;id ) G G G × G idG (idG;e) m G × G m G commutes, which tells us that e is indeed the neutral \element", and such that (id ;i)◦∆ G G G × G (i;idG)◦∆ m e0 G × G m G 1 commutes, which tells us that i is indeed the map that sends \elements" to inverses. Here we use ∆ : G ! G × G to denote the diagonal map coming from the universal property of the product G × G. The map e0 is the composition G ! 1 −!e G. id Now specialize to C = Vark. The terminal object is then 1 = (Spec(k) −! Spec(k)), and giving a unit map e : 1 ! G for some scheme G over k is equivalent to giving an element e 2 G(k). -
Mostow Type Rigidity Theorems
Handbook of c Higher Education Press Group Actions (Vol. IV) and International Press ALM41, Ch. 4, pp. 139{188 Beijing-Boston Mostow type rigidity theorems Marc Bourdon ∗ Abstract This is a survey on rigidity theorems that rely on the quasi-conformal geometry of boundaries of hyperbolic spaces. 2010 Mathematics Subject Classification: 20F65, 20F67, 20F69, 22E40, 30C65, 30L10. Keywords and Phrases: Hyperbolic groups and nonpositively curved groups, asymptotic properties of groups, discrete subgroups of Lie groups, quasiconformal mappings. 1 Introduction The Mostow celebrated rigidity theorem for rank-one symmetric spaces states that every isomorphism between fundamental groups of compact, negatively curved, lo- cally symmetric manifolds, of dimension at least 3, is induced by an isometry. In his proof, Mostow exploits two major ideas: group actions on boundaries and reg- ularity properties of quasi-conformal homeomorphisms. This set of ideas revealed itself very fruitful. It forms one of the bases of the theory of Gromov hyperbolic groups. It also serves as a motivation to develop quasi-conformal geometry on metric spaces. The present text attempts to provide a synthetic presentation of the rigidity theorems that rely on the quasi-conformal geometry of boundaries of hyperbolic spaces. Previous surveys on the subject include [74, 34, 13, 102, 80]. The orig- inality of this text lies more in its form. It has two objectives. The first one is to discuss and prove some classical results like Mostow's rigidity in rank one, Ferrand's solution of Lichn´erowicz's conjecture, the Sullivan-Tukia and the Pansu ∗Universit´eLille 1, D´epartement de math´ematiques, Bat. -
Perspectives on Geometric Analysis
Surveys in Differential Geometry X Perspectives on geometric analysis Shing-Tung Yau This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I[731] worked on fundamental groups of manifolds with non-positive curva- ture. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much inter- ested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S.Y. Cheng told me about these inspiring lec- tures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [563] published his results right be- fore us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about Monge-Amp`ere equations started to broaden in these years.