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Cohomology of abelian varieties

Arithmetic Algebraic Geometry - Brian David Conrad

Crystalline cohomology of abelian varieties - MathOverflo

To add a bit more to Brian's comment: the crystalline cohomology of an abelian variety (over a finite field of characteristic p, say) is canonically isomorphic to the Dieudonné module of the p-divisible group of the abelian variety (which is a finite free module over the Witt vectors of the field with a semi-linear Frobenius). If you start with an abelian scheme over the Witt vectors of this field then the crystalline cohomology of the special fibre is canonically isomorphic to the. We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the -representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko

The cohomology of abelian Hessenberg varieties and the

Let κbe a field and let X be an abelian variety over κ. One can then associate to X another abelian variety Xp, called the dual of X, with the following features: piqThe dual pX can be identified with (or defined as) the identity component PicpXq of the Picard variety PicpXq following: after passing to some finite extension of K, the abelian variety Ais isogenous to a principally polarizedone;moreover,thisconjectureisinsensitivetobasechangeandtheconjecturesfortwoisogenous abelianvarietiesareequivalent. Theorem 2.1.6 ([Del82, Theorem 2.11], [Ogu82, Theorem 4.14], [Bla94]). For any abelian variety, ever The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture Megumi Harada, Martha Precup We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson Cohomology of structure sheaf of abelian variety. Let X be an abelian variety over C of dimension n. Consider the structure sheaf O X. It's Euler characteristic is zero, because χ ( O X) = ( O X n) / n!. And the self intersection of O X is 0 of the basic theory of abelian varieties: for ' 6= p, ´et 1 (X)' »= Z2g l, obtained by passing to the limit under the Galois groups of ['n] : A = J(X)! A = J(X), the multiplication by elln maps. 1.2unless ' is the characteristic of k If we look instead at the p-adic cohomology groups Hi(X;Zp) of a (smooth

Now, more can be said. Every abelian variety $A$ over $k$ as a polarization $p$ over $k$, which is in particular a symmetric isogeny $p: A \rightarrow A^\ast$. Now any isogeny induces isomorphism between the $V_\ell$'s (that is $T_\ell \otimes \bf Q_\ell$), hence taking again the wedge power, we get for all $i$ an isomorphism $$H^i_{et}(A,{\bf Z_\ell}) \otimes {\bf Q_\ell} \rightarrow H^i_{et}(A^*,\bf Z_\ell) \otimes \bf Q_\ell.$$ This isomorphism preserves other structures (the $Gal(\bar k. 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 2X (k), i: X!X satisfying the usual group-axiom diagrams. From the functor of points perspective, this is equivalent to R 7!X (R)being a group functor on k-algebras R. Remark 1.1.7 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology

cohomologies. Ogus proved that all the Hodge cycles are absolute Tate for abelian varieties and veri ed his prediction when Xis the product of abelian varieties with complex multiplication, Fermat hypersurfaces, and projective spaces [Ogu82, Theorem 4.16]. When Xis an elliptic curve, Ogus' conjecture follows from Serre{Tate theory In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In.

Notation and conventions. (0.1) In general, k denotes an arbitrary field, k¯ denotes an algebraic closure of k, and k s a separable closure. (0.2) If A is a commutative ring, we sometimes simply write A for Spec(A) Chapter IX. The cohomology of line bundles. In this chapter we study the cohomology of line bundles on abelian varieties. The main results are the Riemann-Roch Theorem (9.11) and the Vanishing Theorem for non-degenerate line bundles (9.14). The key step in deriving these results is the computation of the cohomology of the Poincar´e bundle on X ×Xt

The corollary shows that the group structure on an abelian variety is uniquely determined by the choice of a zero element. COROLLARY 2.4. The group law on an abelian variety is commutative. PROOF. Commutative groups are distinguished among all groups by the fact that the map taking an element to its inverse is a homomorphism. The preceding corollary shows tha to apply Iwasawa's method to the study of Abelian varieties defined over cyclotomic fields. This article has a twofold purpose. Firstly it gives an exposition of the basic results of Mazur's theory in a slightly more elementary setting (Galois cohomology rather than the full generality of flat quasi-compact Grothendieck cohomology). Secondly, I supplement the theory with hypo Review of cohomology theory on schemes. In order to prove the Theorem of the Cube, we need to digress to review some result on the cohomology of vector bundles over a flat family of varieties in this section. Let be a scheme. The category of quasicoherent sheaves on is an abelian category. If is further noetherian, we also consider the category of coherent sheaves on . Let be a morphism of. Abelian varieties over the complex numbers. Rational maps into abelian varieties. Review of cohomology. The theorem of the cube. Abelian varieties are projective. Isogenies; The dual abelian variety; The dual exact sequence. Endomorphisms. Polarizations and invertible sheaves. The etale cohomology of an abelian variety. Weil pairings. The Rosati involution

Yunqing Tang | Institute for Advanced StudyModuli of K3 surfaces and irreducible symplectic manifolds

The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture Megumi Harada 1 and Martha Precupy2 1Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada 2Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, Missouri, USA Abstract. We define a subclass of Hessenberg varieties called abelian. View Calendar April 15, 2021 4:30 PM - 5:30 PM via Zoom Video Conferencing I'll discuss recent work using tropical techniques to find new rational cohomology classes in moduli spaces A_g of abelian varieties, building on previous joint work with Galatius and Payne on M_g. I will try to take a broad view. Joint work with Madeline Brandt, [

This is a complex of abelian groups whose terms are coherent sheaves on X. The algebraic de Rham cohomology of Xis by de nition the hyper cohomology of this complex: H dR (X) := H(X; X=k): The hypercohomology of a bounded below complex of abelian sheaves is de ned in the appendix. Theorem. Assume khas characteristic 0. Algebraic de Rham cohomology is a Weil cohomology theory with coe cients in. In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author's previous work on local duality and Grothendieck's duality conjecture. It generalizes the perfectness of the Cassels-Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin-Milne's global finite flat.

Cohomology of structure sheaf of abelian variet

We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients An overview of abelian varieties in homotopy theory 183 3 Quillen's theorem There is a cohomology theory MU associated to complex bordism and equipped with an orientation u. There is also a smash product cohomology theory MU ^MU coming equipped with two orientations u and v, one per factor of MU , and henc

This is a survey on various aspects of the cohomology of the moduli space of abelian varieties abelian varieties was previously obtained in [39]. In the coherent setting, a similar propagation property was proved, e.g., in [30, Proposition 3.14], for cohomology support loci of GV-sheaves on abelian varieties. Moreover, in the abelian context, codimension lower bounds have been obtained in [31, Theorem 1.1] for cohomology An overview of abelian varieties in homotopy theory 3 One of the applications in mind has been the construction of fi nite resolutions of the K(n)-local sphere. Henn has given finite length algebraic resol utions allowing compu-tation of the cohomology of the Morava stabilizer group in terms of the cohomology of finite subgroups [ 22. ALGEBRAIC COMBINATORICS Megumi Harada & Martha E. Precup The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture Volume 2, issue 6 (2019), p. 1059-1108

Cycles in the de Rham cohomology of abelian varieties 1.2The main results Our rst result is as follows. Theorem 1.1 (Theorem3.2.4). If Ais a polarized abelian variety over Q and its '-adic algebraic monodromy group G ' is connected, then the Mumford{Tate conjecture for Aimplies that the absolute Tate cycles of Acoincide with its Hodge cycles Fundamentals of (Abelian) Group Cohomology. In this post we will talk about the basic theory of group cohomology, including the cohomology of profinite groups. We will assume that the reader is familiar with the basic theory of derived functors as in, say, Weibel's Homological Algebra The dual abelian variety (any characteristic) is constructed and the duality theorem is proved. Finally (a highlight!), the cohomology of line bundles is treated. The Riemann-Roch and vanishing theorems are proved, a method of computing the index of a line bundle is demonstrated, and it is shown that the cube of any ample line bundle is very ample. In the final chapter, the possible structures.

Cohomology ring of a projective variety. Let X be a projective variety with ample sheaf O X ( 1). Then H ∗ ( ⊕ n O X ( n)) is a graded algebra via the cup product: H i ( O ( n)) ⊗ H j ( O ( m)) → H i + j ( O ( n) ⊗ O ( m)) ≅ H i + j ( O ( n + m)). Remark that each individual homogeneous component is a graded abelian group (via n ) (as in Remark15.13) they can be used to prove that varieties are not isomorphic. The cohomology of sheaves is a very general concept. It can not only be defined for sheaves of modules on a scheme, but even for an arbitrary sheaf of Abelian groups on a topological space, and consequently it plays a big role in topology as well. Nevertheless, in order to avoid technicalities we will restrict to. MATH 731: TOPICS IN ALGEBRAIC GEOMETRY I { ABELIAN VARIETIES BHARGAV BHATT Course Description. The goal of the rst half of this class is to introduce and study the basic structure theory of abelian varieties, as covered in (say) Mumford's book. In the second half of the course, we shall discuss derived categories and the Fourier{Mukai transform, and give some geometric applications. Contents. Contents Introduction iii Moduli spaces of curves and abelian varieties . . . . . . . . . . . . iii Cohomological investigations. gof principally polarized abelian varieties of genus g and its compacti cations. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in.

etale cohomology of an abelian variety and its dual

Cohomology - Wikipedi

  1. In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the étale cohomology groups with finite coefficients
  2. In: Moduli of abelian varieties (Texel Island, 1999), 255-298, Progress in Math. 195, Birkhäuser, 2001. Corrigendum. Bas Edixhoven, Ben Moonen and Frans Oort Open problems in algebraic geometry. Bull. Sci. Math. 125 (2001), 1- Ben Moonen and Yuri Zarhin Hodge classes on abelian varieties of low dimension. Math. Ann. 315 (1999), 711-733
  3. from abelian varieties over to abelian varieties over , is a well-behaved duality theory. In particular, any abelian variety is canonically isomorphic to its bidual. (This explains why the double Picard functor on a general variety is the universal abelian variety generated by that variety, the so-called Albanese variety.) In fact, we won't quite finish the proof in this post, but we will.
  4. 1 ABELIAN VARIETIES 3 1.2 Examples of abelian varieties 1.2.1 Let k = C and consider a k-vector space V together with a lattice U. Then X = V/U is an analytic abelian group. In general it does not come from an algebraic variety. Theorem 1.2 X comes from an algebraic variety iff there exists a positive definite her

Cycles in the de Rham cohomology of abelian varieties over

5.Abelian varieties are projective. 6.Isogenies 7.The dual abelian variety 8.The dual exact sequence. 9.Endomorphisms. 10.Polarizations and invertible sheaves. 11.The etale cohomology of an abelian variety. 12.Weil pairings. 13.The Rosati involution. 14.The zeta function of an abelian variety. 15.Families of abelian varieties. 16.Abelian. THE COHOMOLOGY OF ABELIAN VARIETIES OVER A NUMBER FIELD M I Bashmakov-PARABOLIC POINTS AND ZETA-FUNCTIONS OF MODULAR CURVES Ju I Manin-THE RATIONAL POINTS ON THE JACOBIANS OF MODULAR CURVES V G Berkovi-Recent citations Fine Selmer groups, Heegner points and anticyclotomic p-extensions Ahmed Matar -On the structure of Selmer groups of p - ordinary modular forms over Z p - extensions Keenan. COHOMOLOGY OF THE UNIVERSAL ABELIAN SURFACE WITH APPLICATIONS TO ARITHMETIC STATISTICS SERAPHINA EUN BI LEE ABSTRACT.The moduli stack A 2 of principally polarized abelian surfaces comes equipped with the universal abelian surface ˇ: X 2!A 2. The fiber of ˇover a point corresponding to an abelian surface Ain A 2 is Aitself. We determine the '-adic cohomology of X 2 as a Galois.

A complex abelian variety is a smooth projective variety which happens to be a complex torus. This simpli es many things compared to general varieties, but it also means that one can ask harder questions. Abelian varieties are indeed abelian groups (unlike elliptic curves which aren't ellipses), however the use \abelian here comes about from the connection with abelian integrals which. cohomology of any abelian variety [5, Theorem 5.5, p. 926]. We now outline the contents and organization of this paper. We begin in x2 with a summary of the de nitions and basic properties of the Hodge and Lefschetz groups of an abelian variety, and the Kuga ber varieties associated with them. In x3 we formulate our problem in representation theoretic form. In the following four sections we. ABELIAN VARIETIES by A. Silverberg & Yu.G. Zarhin Abstract.— Inthispaperwestudythe´etale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of the absolute inertia group on the ´etale. ON THE VANISHING OF WEIGHT ONE KOSZUL COHOMOLOGY OF ABELIAN VARIETIES MARIAN APRODU AND LUIGI LOMBARDI Abstract. In this Note we prove the vanishing of (twisted) Koszul cohomology groups K p;1 of an abelian variety with values in powers of an ample line bundle. It complements the work of G. Pareschi on the property (N p) [Pa]. Let Lbe an ample line bundle on a complex abelian variety X. The. Abelian varieties are a very interesting class of varietes, subject of current re-search study. On the other hand, they provide many good examples and interesting results that can be discussed in an introduction to algebraic geometry. The goal of these notes is to highlight a few of these. We will prefer the analytic approach, since more intuitive and gurative. The material for these notes is.

PERVERSE SHEAVES ON SEMI-ABELIAN VARIETIES 3 2.1. Cohomology jump loci. Let Xbe a smooth connected complex quasi-projective variety with positive rst Betti number, i.e., b 1(X) >0. The character variety Char(X) of Xis the identity component of the moduli space of rank-one C-local systems on X, i.e., Char(X) := Hom(H 1(X;Z)=Torsion;C ) ˘=(C )b 1(X): De nition 2.1. The i-th cohomology jump. The projective coordinate ring of abelian varieties. In: Igusa, J.I. (ed.) Algebraic Analysis, Geometry and Number Theory, pp. 225-236. Baltimore: Johns Hopkins Press 1989 . Google Scholar [Kim] Kim, S.-O.: Noether-Lefschetz locus for surfaces. Trans. Am. Math. Soc.324, 369-384 (1991) Google Scholar [L1] Lazarsfeld, R.: A sharp Castelnuovo bound for smooth surfaces. Duke Math. J.55, 423. the Shimura variety S, which is a moduli space of abelian varieties (with extra structures), to the corresponding moduli space Mof p-divisible groups (with extra structures), ˇ: S!M:4 One could then analyze the cohomology of the Shimura variety in terms of a Leray spectral sequence. Note that the bres of ˇshould be a moduli space of abelian Non-abelian Hodge theory and cohomology jump loci The non-abelian Hodge theory due to Simpson says \the space of local systems is equal to the space of Higgs bundles. In the rank one case, it says when X is a compact K ahler manifold, M B(X) by de nition== Hom(ˇ 1(X);C) ˘=Pic˝(X) H0(X; 1 X): Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will.

Cyclotomic Fields and Modular the Cohomology of Abelian

  1. Singular cohomology of qfh-sheaves.. 82 8. Singular homology of varieties over C.. 86 9. Algebraic Lawson homology.. 88 10. Appendix: h-cohomology.. 90 1. Introduction The main objective of the present paper is to construct a reasonable singular homology theory on the category of schemes of finite type over an arbitrary field k. Let X be a CW-complex. The theorem of Dold and.
  2. bodied in the theory of the jacobian of curves, and as concretized by the theory of abelian varieties, to treat cohomology of all dimensions. Equally important, just as in the theory * Compare the notions of a geometric cohomology theory in [M], and the slightly more restricted version of this called a Weil cohomology theory in [K]. 3. of group representations where the irreducible.
  3. cohomology R ψf(C)x of f at some For every abelian variety Bover a field F, we denote its dual abelian variety by B ∨. For every abelian K-variety Awith N´eron model A, we denote by t(A), u(A) and a(A) the reductive, resp. unipotent, resp. abelian rank of Ao s. We call these values the toric, resp. unipotent, resp. abelian rank of A. Obviously, their sum equals the dimension of A. By.
  4. variety satisfying the following universal property: for any regular map a0: X!Ato an abelian variety there exists a unique map ˚: A!Alb(X) such that a0= ˚ a. For example, when Xis a smooth algebraic curve, Alb(X) coincides with the Jacobian variety Jac(X). If we take G= Aut(X)0 in (1.1) and consider a natural homomorphism Aut(X)0!Aut(Alb(X)), then we obtain that Lin(Aut(X)0) is mapped to.
  5. Speaker: Margarida Melo (University of Coimbra and University of Roma Tre)Title: On the top weight cohomology of the moduli space of abelian varietiesAbstrac..
  6. Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the.
  7. 7 Cohomology of a Line Bundle on a Complex Torus: Mirror Symmetry Approach 89 PartII. AlgebraicTheory 8 Abelian Varieties and Theorem of the Cube 99 9 Dual Abelian Variety 109 10 Extensions, Biextensions, and Duality 122 11 Fourier-Mukai Transform 134 12 Mumford Group and Riemann's Quartic Theta Relation 150 13 More on Line Bundles 166 14 VectorBundles on Elliptic Curves 175 15.

We establish a—and conjecture further—relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of the cohomology of twisted ideal sheaves. The main ingredient is the Generic Vanishing criterion established in Pareschi G. and Popa M. (GV-sheaves, Fourier-Mukai transform, and Generic. continuous Galois cohomology which is fundamental to the conjecture. In particular, we will need to discuss the distinguished subspaces H1 f as they are the local building blocks for the motivic construct. Seeing that semi-abelian varieties unite affine and projective algebraic groups, a proper understanding of the according basic concepts is. Bücher bei Weltbild: Jetzt Complex Abelian Varieties von Christina Birkenhake versandkostenfrei online kaufen bei Weltbild, Ihrem Bücher-Spezialisten An abelian variety A 0 over F is said to be nondegenerate if all the ℓ-adic Tate classes (see §1) on A 0 are generated by divisor classes in the ℓ-adic étale cohomology ring of A 0. If A 0 is nondegenerate, then the Tate conjecture holds for A 0 TORSION POINTS OF ABELIAN VARIETIES AND F-ISOCRYSTALS MARCO D'ADDEZIO Today I will talk about a joint work with Emiliano Ambrosi. We found some new properties of the category of F-isocrystals. Our goal was to prove a certain extension of the theorem of Lang{N eron for abelian varieties. The starting point was my previous work on the monodromy groups of overconvergent F-isocrystals, [D'Ad17.

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Abelian Varieties - lccs - Columbia Universit

A an abelian variety defined over K, of dimension not equal to 3 S a finite set of primes of OK, including all places of bad reduction for A ˚an ample class in the Neron-Severi group of A. Then there are at most finitely manyhypersurfaces in A belonging to the class ˚, defined over K and having good reduction outside S. Brian Lawrence (joint work with Will Sawin) Shafarevich Conjecture. 3.5 Cohomology rings of direct products and abelian groups 32 4 Relations to cohomology of subgroups 35 4.1 Restriction and the Eckmann-Shapiro Lemma 35 4.2 Transfer or corestriction 38 5 Cohomology of wreath products 45 5.1 Tensor induced modules 45 5.2 Wreath products and the monomial representation 46 5.3 Cohomology of wreath products 49 5.4 Odd degree and other variations on the theme 54 6. The Shafarevich conjecture for hypersurfaces in abelian varieties. Authors: Brian Lawrence, Will Sawin. Download PDF. Abstract: Faltings proved that there are finitely many abelian varieties of genus of a number field , with good reduction outside a finite set of primes . Fixing one of these abelian varieties , we prove that there are finitely.

AV -- J.S. Miln

  1. aries 1 Let X be a complex abelian variety. An abelian arrangement in X is a set A= fY1;:::;Y'gof codimension-one abelian subvarieties. Denote the complement of an abelian arrangement by M(A) := X r [Y2A Y: 2A component of the arrangement is a connected component of an intersection \Y2SY for some S A.
  2. This site uses cookies. By continuing to use this site you agree to our use of cookies. To find out more, see our Privacy and Cookies policy
  3. Title: The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture. Authors: Megumi Harada, Martha Precup (Submitted on 20 Sep 2017 , last revised 24 Dec 2017 (this version, v2)) Abstract: We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give.
  4. ed by that of Aexcept in the middle dimension g 1. The primitive cohomology of , in the sense of Lefschetz, is Hg 1 pr ( ;Z) := Ke
  5. skeptical of William V.D. Hodge's famous conjecture describing the cohomology classes of algebraic cycles, and suggested that it might be time to search for a counter-example. David Mumford had found some interesting candidates in the Hodge ring of abelian varieties with complex multiplication, and Weil observed that these Hodge classes actually existed on a continuous family of abelian.
  6. Abstract: In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian.
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On the cohomology of moduli of abelian varieties - Harvard

The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151) This book aims first to prove the local Langlands conjecture for GL n over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the simple Shimura varieties. These two problems go hand in hand Étale cohomology and reduction of abelian varieties A. Silverberg; Yu. G. Zarhin. Bulletin de la Société Mathématique de France (2001) Volume: 129, Issue: 1, page 141-157; ISSN: 0037-9484; Access Full Article top Access to full text Full (PDF) Abstract top In this paper we study the étale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for. sending any abelian variety to its underlying pointed variety. The right adjoint U U is faithful, but more remarkably it is also full: any basepoint-preserving map of varieties between abelian varieties is automatically a group homomorphism.(A proof of this fact is outlined in the article abelian variety.)Moreover, U U is monadic.As a consequence the composite functo Thus, an Abelian variety can be imbedded as a closed subvariety in a projective space; each rational mapping of a non-singular variety into an Abelian variety is regular; the group law on an Abelian variety is commutative. The theory of Abelian varieties over the field of complex numbers $\C$ is, in essence, equivalent to the theory of Abelian functions founded by C.G.J. Jacobi, N.H. Abel and.

Cohomology of line bundles on abelian varieties 12 2.3. An application of G.I.T 13 2.4. Spreading the abelian scheme structure 16 2.5. Smoothness 17 2.6. Adelic description and Hecke correspondences 18 3. Shimura varieties of PEL type 20 3.1. Endomorphisms of abelian varieties 20 3.2. Positive definite Hermitian forms 23 3.3. Skew-Hermitian modules 23 3.4. Shimura varieties of type PEL 25 3.5. \'Etale cohomology and reduction of abelian varieties . By A. Silverberg and Yu. G. Zarhin. Abstract. In this paper we study the \'etale cohomology groups associated to abelian varieties. We obtain necessary and sufficient conditions for an abelian variety to have semistable reduction (or purely additive reduction which becomes semistable over a quadratic extension) in terms of the action of. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this paper I present a number of results on cycles on the moduli space Ag of principally polarized abelian varieties of dimension g. The results on the tautological ring are my own work, the results on the torsion of λg and on the cycle classes of the Ekedahl-Oort stratification are joint work with Torsten Ekedahl. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic $0$ cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\mathrm{GL}_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties

Duality for Cohomology of Curves With Coefficients in

Étale cohomology and reduction of abelian varieties - NASA/AD

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The cohomology of the moduli space of abelian varieties

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