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Topic: Abelian varieties

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Abelian group

abelian categories

abelian varieties

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finitely generated abelian groups

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Topological abelian group

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	NationMaster - Encyclopedia: Arithmetic of abelian varieties
		In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those.
		Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
		In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet.
	www.nationmaster.com /encyclopedia/Arithmetic-of-abelian-varieties (2141 words)

	Abelian variety - Wikipedia, the free encyclopedia
		A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure.
		An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety.
		A polarization of an abelian variety is an isogeny from an abelian variety to its dual.
	en.wikipedia.org /wiki/Abelian_variety (1463 words)

	Strongly Abelian Varieties and the Hamiltonian Property - Kiss, Valeriote (ResearchIndex)
		4 Hamiltonian varieties of universal algebras (context) - Klukovits - 1975
		3 Hamiltonian and Abelian properties in locally finite varieti..
		1 A structure theorem for strongly abelian varieties with few..
	citeseer.ist.psu.edu /328813.html (569 words)

	Abelian variety (Site not responding. Last check: 2007-10-26)
		For the purposes of algebraic geometry over the complex numbers an abelian variety is a complex torus (a torus of real dimension 2 n that is a complex manifold) that is also a projective algebraic of dimension n i.e.
		An abelian function is a meromorphic function on an abelian variety which may regarded therefore as a periodic function of n complex variables having 2 n independent periods; equivalently it is a in the function field of an abelian variety.
		The explicit equations defining abelian varieties are general complex: their properties involve the detailed of theta-functions.
	www.freeglossary.com /Abelian_transcendent (657 words)

	PlanetMath: abelian variety
		Proposition 1 The group law on an abelian variety is commutative.
		This example motivated the development of the theory of abelian varieties, and many properties of curves are best understood by looking at the Jacobian.
		This is version 3 of abelian variety, born on 2004-04-05, modified 2004-04-06.
	www.planetmath.org /encyclopedia/AbelianVariety.html (117 words)

	Arithmetic of abelian varieties - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-26)
		In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those.
		Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
		In terms of the ring End(A) there is a definition of abelian variety of CM-type that singles out the richest class.
	encyclopedia.worldsearch.com /arithmetic_of_abelian_varieties.htm (697 words)

	Non-abelian (Site not responding. Last check: 2007-10-26)
		Explicit Approaches to Modular Abelian Varieties William Stein, Berkeley, 2000.
		Explicit Approaches to Modular Abelian Varieties William Stein, Ph.D. thesis, Berkeley, 2000.
		Broue's Abelian Defect Group Conjecture A record of which groups and blocks have been proved to satisfy the conjecture, maintained by Jeremy Rickard.
	www.serebella.com /encyclopedia/article-Non-abelian.html (202 words)

	Abelian variety (Site not responding. Last check: 2007-10-26)
		For the purposes of algebraic geometry over the complexnumbers, an abelian variety is a complex torus (a torus of realdimension 2n that is a complex manifold) that is also a projectivealgebraic variety of dimension n, i.e.
		An abelian function is a meromorphic function on an abelian variety, which may be regardedtherefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is afunction in the function field of an abelian variety.
		In a more geometric language, every algebraic curve C of genus g which is at least 1 is associated with an abelian variety J of dimensiong, by means of an analytic map of C into J.
	www.therfcc.org /abelian-variety-193537.html (554 words)

	Abelian Varieties with Complex Multiplication and Modular Functions - Review 0691016569 (Site not responding. Last check: 2007-10-26)
		This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book.
		The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals.
		In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.
	books.bankhacker.com /Abelian+Varieties+with+Complex+Multiplication+and+Modular+Functions (262 words)

	Curves and Abelian varieties
		Abelian varieties link algebraic geometry with other mathematical disciplines, ranging from number theory to integrable systems.
		Algebraic curves and their moduli are related to Abelian varieties via the Jacobian construction and the Schottky problem.
		Carel Faber and Eduard Looijenga, editors: Moduli of curves and Abelian varieties (The Dutch Intercity seminar on moduli), with papers by Dijkgraaf, Faber, Van der Geer, Hain, Looijenga, Oort.
	euclid.mathematik.uni-kl.de /NEW/node22.html (168 words)

	Abelian variety of CM-type (Site not responding. Last check: 2007-10-26)
		In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A).
		There are other cases that reflect that A may not be a simple abelian variety (it might be a cartesian product of elliptic curves, for example).
		The possible types of endomorphism ring have been classified, as rings with involution (the Rosati involution), leading to a classification of CM-type abelian varieties.
	vb.game-host.org /en/Abelian_variety_of_CM-type.htm (467 words)

	Definition of Abelian transcendent
		For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i.e.
		From the point of view of birational geometry, its function field of abelian functions is the fixed field of the symmetric group on g letters acting on the function field of C
		For example there was much interest in the case of hyperelliptic integrals that may be expressed in terms of elliptic integrals: this comes down to asking that J is a product of elliptic curves, up to a finite-to-one mapping (called an isogeny of abelian varieties).
	www.wordiq.com /definition/Abelian_transcendent (636 words)

	Publications of Salman Abdulali
		Abelian varieties and the general Hodge conjecture, Compositio Math.
		Abelian varieties of type III and the Hodge conjecture, Internat.
		Hodge structures on abelian varieties of CM-type, J. Reine Angew.
	personal.ecu.edu /abdulalis/publications.html (155 words)

	Abstracts (Salman Abdulali)
		A Kuga fiber variety f : A → V is an abelian scheme parametrized by an arithmetic variety and constructed from a symplectic representation of an algebraic group.
		The principal result of this paper is that if A → V is a Kuga fiber variety defined by a Q-irreducible representation satisfying a certain rigidity condition, and if the generic fibers are principal abelian varieties, then Bot A is an abelian extension of Bot V.
		We prove the general Hodge conjecture for any complex abelian variety of CM-type such that the Hodge ring of each power of the abelian variety is generated by divisors.
	personal.ecu.edu /abdulalis/abstracts.html (504 words)

	Elements of Modular Abelian Varieties
		Given a point x on a modular abelian variety return the exact order of x, if x is known exactly as a torsion point, and if not the order of an approximation of x by a torsion point, obtained using continued fractions.
		The dimension of the homology of the parent of x, where x is an element of a modular abelian variety.
		If the modular abelian variety element x is defined by an element z in the real homology H_1(A, R), find an element of H_1(A, Q) which approximates z, using continued fractions, and return the corresponding point.
	www.math.lsu.edu /magma/text1329.htm (1049 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Abelian varieties, even abelian surfaces, are more complicated than elliptic curves and there isn't anything general like the Weierstrass cubic equation.
		For instance, not all abelian surfaces can be embedded in P^4, and if you want to know which ones can, and how, you will have to learn about the Horrocks-Mumford bundle.
		There are results about equations defining abelian varieties but they are not simple to state.
	www.math.niu.edu /~rusin/known-math/95/abel.variet (306 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Harvard Room 507 Moduli spaces of abelian varieties in positive characteristic FRANS OORT (Universiteit Utrecht and MIT) Abstract: Abelian varieties in characteristic p have additional structures (the action of Frobenius on the p-divisible group, Newton polygons, structures of the p-kernel) which give access to geometric aspects of these moduli spaces in all characteristics.
		(NP) The stratification by Newton Polygons of the moduli space of abelian varieties, and of deformation spaces of p-divisible groups.
		As a corollary: a new proof of the irreducibility of the moduli space of principally polarized abelian varieties in arbitrary characteristic.
	www-math.mit.edu /~abuch/seminar/abstracts/020305-oort.txt (292 words)

	Semistable abelian varieties with small division fields, by Armand Brumer and Kenneth Kramer (Site not responding. Last check: 2007-10-26)
		Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$.
		We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is ``small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent.
		The Jacobian of the modular curve $X_0(41)$ is a simple semistable abelian variety of dimension 3, with bad reduction only at $p=41$ and the Galois group of its 2-division field is a 2-group.
	www.math.uiuc.edu /Algebraic-Number-Theory/0358 (197 words)

	Archived publications by Kenneth Ribet
		Ribet, Kenneth A. Division fields of abelian varieties with complex multiplication.
		Ribet, Kenneth A. Twists of modular forms and endomorphisms of abelian varieties.
		Ribet, Kenneth A. Kummer theory on extensions of abelian varieties by tori.
	math.berkeley.edu /~ribet/Articles (1108 words)

	Bibliography: J.S. Milne (Site not responding. Last check: 2007-10-26)
		Hodge cycles and abelian varieties (notes of a seminar of P. Deligne), in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math.
		In Automorphic Forms, Shimura Varieties, and $L$-Functions (Proceedings of a Conference held at the University of Michigan, Ann Arbor, July 6-16, 1988), Perspectives in Mathematics Vols 10, 11, Academic Press, 1990, pp 283-414.
		The conjecture of Langlands and Rapoport for Siegel modular varieties.
	www.math.lsa.umich.edu /~jmilne/Personal/mybib.html (704 words)

	Alice Silverberg -- Bibliography
		Polarizations on abelian varieties and self-dual l-adic representations of inertia groups, A.
		Etale cohomology and reduction of abelian varieties, A.
		Semistable reduction and torsion subgroups of abelian varieties, A.
	www.math.uci.edu /~asilverb/bibliography/index.html (1056 words)

	SPRING SCHOOL ON ABELIAN VARIETIES
		The Spring School is organized by the Mathematics Research Institute (MRI) in the Netherlands, in collaboration with the Thomas Stieltjes Institute.
		Gunther Cornelissen: The Mordell-Weil Theorem for Abelian Varieties
		There is a new textbook on Abelian Varieties in preparation by Gerard van der Geer and Ben Moonen.
	staff.science.uva.nl /~bmoonen/springsch/SprSch.html (476 words)

	Liste de publications (Rutger Noot) (Site not responding. Last check: 2007-10-26)
		Abelian varieties - Galois representations and properties of ordinary reduction.
		Abelian varieties with l-adic Galois representation of Mumford's type.
		Hodge classes, Tate classes, and local moduli of abelian varieties.
	www-irma.u-strasbg.fr /~noot/publications/liste.html (123 words)

	Arithmetic of abelian varieties (Site not responding. Last check: 2007-10-26)
		In mathematics, the arithmetic of abelian varieties isthe study of the number theory of an abelian variety, or family of those.
		Most of these can be posed for an abelian variety Aover a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
		In termsof the ring End(A) there is a definition of abelian variety of CM-type that singles out the richest class.
	www.therfcc.org /arithmetic-of-abelian-varieties-218259.html (681 words)

	Errata (Site not responding. Last check: 2007-10-26)
		Theorem 5.1 remains true (with minor modifications to the proof) provided that abelian varieties of type III are excluded.
		Errata: Filtrations on the cohomology of abelian varieties
		Errata: Hodge structures on abelian varieties of CM-type
	personal.ecu.edu /abdulalis/Errata.html (198 words)

	SMF - Publications - Cours Spécialisés - Parutions - 6 (Site not responding. Last check: 2007-10-26)
		This book takes the classical theory of complex tori and complex abelian varieties as an excuse to go through more modern aspects of complex algebraic and analytic geometry.
		Starting with complex elliptic curves, it moves on to the higher-dimensional case, giving characterizations from different points of view of those complex tori which are abelian varieties, i.e.
		Standard theorems about abelian varieties are proved, and moduli spaces are discussed.
	smf.emath.fr /Publications/CoursSpecialises/1999/6/html/smf_cours-spec_6.html (256 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		We characterize affine algebras as precisely those abelian algebras which generate a variety satisfying an idempotent Mal'cev condition which fails to hold in the variety of semilattices.
		In the second part of the talk we discuss the local structure of algebras in locally finite abelian varieties, and we explain why such varieties must be finitely generated.
		We give an example showing that locally finite abelian varieties may be nonfinitely based, but we go on to show that they cannot be inherently nonfinitely based.
	www.math.u-szeged.hu /confer/algebra/1996/kearnes.htm (152 words)

	Isogeny classes of abelian varieties with no principal polarizations (Site not responding. Last check: 2007-10-26)
		We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization.
		We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class.
		Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field k have positive characteristic and that there be a Galois extension of k with a certain non-abelian Galois group.
	www.math.uiuc.edu /Algebraic-Number-Theory/0225 (147 words)

	Amazon.ca: Books: Complex Abelian Varieties (Site not responding. Last check: 2007-10-26)
		The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties.
		Moreover several moduli spaces of abelian varieties with additional structure are constructed.
		Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture.
	www.amazon.ca /exec/obidos/ASIN/3540204881 (245 words)

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