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Topic: Arithmetic of abelian varieties

	Arithmetic of abelian varieties - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Arithmetic_of_abelian_varieties (793 words)

	Arithmetic
		Arithmetic Arithmetic is a branch of (or the forerunner of) mathematics which records elementary properties of certain o...
		Arithmetic progression In numbers such that the difference of any two successive members of the sequence is a constant.
		Presburger arithmetic Presburger arithmetic is the Gödel's incompleteness theorem.
	www.brainyencyclopedia.com /topics/arithmetic.html (349 words)

	Abelian variety
		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.
		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.
		For the purposes of number theory the foundations of the theory of abelian varieties are developed over any field, and in fact using a commutative ring, in order to control the process of reduction mod p.
	www.sciencedaily.com /encyclopedia/abelian_variety (635 words)

	Arithmetic of abelian varieties -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		It goes back to the studies of (French mathematician who founded number theory; contributed (with Pascal) to the theory of probability (1601-1665)) Fermat on what are now recognised as (Click link for more info and facts about elliptic curve) elliptic curves; and has become a very substantial area both in terms of results and conjectures.
		Most of these can be posed for an abelian variety A over a (Click link for more info and facts about number field) number field K; or more generally (for (Click link for more info and facts about global field) global fields or more general finitely-generated rings or fields).
		There is some tension here between concepts: integer point belongs in a sense to (The geometery of affine transformations) affine geometry, while abelian variety is inherently defined in (The geometry of properties that remain invariant under projection) projective geometry.
	www.absoluteastronomy.com /encyclopedia/a/ar/arithmetic_of_abelian_varieties.htm (794 words)

	Algebraic groups (Site not responding. Last check: 2007-10-08)
		This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the Jacobian of a hyperelliptic curve of genus 2.
		The Jacobian varieties of curves generalise to the Albanese varieties of varieties in general.
		For the purposes of number theory the foundations of the theory of abelian varieties are developed over any field (mathematics), and in fact using a commutative ring, in order to control the process of reduction mod p.
	read-and-go.hopto.org /Algebraic-groups (436 words)

	Bibliography: J.S. Milne
		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.jmilne.org /math/Personal/mybib.html (693 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)

	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.
		In the second paper, the fields of definition of the Hodge cycles are investigated, and it is shown that the relations between the zeta functions hold over the fields of definition of the canonical models.
		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)

	Read about Arithmetic of abelian varieties at WorldVillage Encyclopedia. Research Arithmetic of abelian varieties and ... (Site not responding. Last check: 2007-10-08)
		In mathematics, the arithmetic of abelian varieties is the study of the number theory of an
		Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures.
		Tate-Néron height function, which is a quadratic form; it has some remarkable properties, amongst all height functions designed to pick of finite sets in A(K) of points of height (roughly, logarithmic size of co-ordinates) at most h.
	encyclopedia.worldvillage.com /s/b/Arithmetic_of_elliptic_curves (656 words)

	Research expertise: (Site not responding. Last check: 2007-10-08)
		Hilbert schemes, the geometry of the theta divisor of Jacobian and Prym varieties.
		Special varieties and related algebraic and computational problems: Moduli spaces of instanton bundles, their cohomology and local equations, computer programs for computing cohomology and local equations of moduli spaces of instanton bundles.
		Abelian varieties and the Schottky problem: Subvarieties of a general Abelian variety.
	euclid.mathematik.uni-kl.de /NEW/node79.html (182 words)

	Math JS Milne Preprints
		For an abelian variety A and its dual B over a local field of prime characteristic, prove that A(K) is dual to the Weil-Chatelet group of B.
		Relates the arithmetic invariants of an abelian variety to those of a Weil restriction of scalars of it.
		-cohomologies of an algebraic variety (with Niranjan Ramachandran)
	www.jmilne.org /math/Preprints (2322 words)

	Weil-Châtelet group - Wikipedia, the free encyclopedia
		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.
		It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
	en.wikipedia.org /wiki/Selmer_group (208 words)

	Selected Matches for: Items Authored by Stein, William
		Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989), Birkh"auser Boston, Boston, MA, 1991, pp.
		Agashe and W. Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank 0, to appear in Math.
		Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989), Birkhäuser Boston, Boston, MA, 1991, pp.
	modular.fas.harvard.edu /papers/stein-msn.html (5542 words)

	Abelian variety
		For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i.e.
		In case n is 1 thisn'tion is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for n > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.
		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.
	news-server.org /a/ab/abelian_variety.html (564 words)

	Paula Cohen Lecture, Lille University (Site not responding. Last check: 2007-10-08)
		These are moduli varieties for polarised abelian varieties of a given dimension and with their Shimura subvarieties, parametrising abelian varieties with some extra structure.
		These play a crucial role in the arithmetic theory of abelian varieties and the transcendence theory of modular functions.
		A missing ingredient in equidistribution ideas in the anabelian situation is an adequate notion of height applied to modular varieties and a link between CM points and special properties of the height.
	www.math.uci.edu /~mfried/htmlfiles/cohen.html (413 words)

	Invariants of modular abelian varieties (Site not responding. Last check: 2007-10-08)
		Now that the Shimura-Taniyama conjecture has been proved, the main outstanding problem in the field is the Birch and Swinnerton-Dyer conjecture (BSD conjecture), which ties together the arithmetic invariants of an elliptic curve.
		Approaches to the BSD conjecture that rely on congruences between modular forms are likely to require a deeper understanding of the analogous conjecture for higher-dimensional abelian varieties.
		As a first step, I have obtained theorems that make possible explicit computation of some of the arithmetic invariants of modular abelian varieties.
	modular.fas.harvard.edu /job/Prop/node2.html (99 words)

	Modular abelian varieties (Site not responding. Last check: 2007-10-08)
		This is the order of the group of /line(F)_p-points of the component group of the reduction modulo p of the N'eron model of the abelian variety attached to M. At present, it is necessary that p exactly divides the level.
		The abelian varieties B and C below correspond to the Jacobians labeled 65B and 65A in [ Flynn, Lepr'evost, Schaeffer, Stein, Stoll, Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves.], respectively.
		When the Atkin-Lehner involution W_p acts as +1 on a modular abelian variety A, the order of the component group can be larger than the Tamagawa number c_p=[A(Q_p):A_0(Q_p)] that appears in the conjecture of Birch and Swinnerton-Dyer.
	www.dtr.isy.liu.se /Magma/text799.html (995 words)

	Course Notes --- J.S. Milne (Site not responding. Last check: 2007-10-08)
		This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts.
		An introduction to both the geometry and the arithmetic of abelian varieties.
		It includes a discussion of the theorems of Honda and Tate concerning abelian varieties over finite fields and the paper of Faltings in which he proves Mordell's Conjecture.
	www.jmilne.org /math/CourseNotes (360 words)

	Arithmetic of Abelian Varieties (Spring 1997) (Site not responding. Last check: 2007-10-08)
		Families of abelian varieties: reduction type and local monodromy.
		Moduli spaces of abelian varieties: compactifications, heights, Tate conjecture.
		Stratification of the moduli space of abelian varieties in positive characteristic.
	www.mccme.ru /ium/s97/abvar.html (29 words)

	math lessons - Mordell-Weil theorem
		In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group.
		The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posed by Poincaré around 1908; it was proved by Louis Mordell in 1922.
		For an abelian variety, there is no a priori preferred representation, though, as a projective variety.
	www.mathdaily.com /lessons/Mordell-Weil_theorem (423 words)

	Elliptic curve (Site not responding. Last check: 2007-10-08)
		Elliptic curves are by definition non-singular, meaning they don't have cusps or self-intersections, and a binary operation can be defined for their points in a natural geometric fashion, thus turning the set of points into an abelian group.
		One can check that this turns the curve into an abelian group, and thus into an abelian variety.
		For further developments see arithmetic of abelian varieties.
	www.free-download-soft.com /info/ms-proxy.html (842 words)

	Some Questions in Algebraic Geometry (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: Introduction In June 1995 several mathematicians will gather in Utrecht for a conference on "Arithmetic and geometry of abelian varieties," and this seems a good occasion to share with them some of the questions that have occupied my mind over the years.
		6 Oort - Moduli of supersingular abelian varieties (context) - Li Oort - Moduli of abelian varieties (context) - Norman - 1980
		1 Ueno - Principally polarized abelian varieties of dimension..
	citeseer.ist.psu.edu /oort95some.html (592 words)

	Amazon.ca: Books: Abelian Varieties with Complex Multiplication and Modular Functions (Site not responding. Last check: 2007-10-08)
		In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions.
		This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book.
		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.
	www.amazon.ca /exec/obidos/ASIN/0691016569 (548 words)

	Price Check: Abelian Varieties (Site not responding. Last check: 2007-10-08)
		Abelian varieties and their subvarieties The following theorem was proved by Abramovich and the author [AV1], under restrictive hypotheses and then by Hrushovski [H], in general.
		We do not write A+B for the direct sum, since it is already used for the sum of A and B inside a common ambient abelian variety.
		Explicit approaches to modular abelian varieties William A. Stein Doctor of Philosophy in Mathematics University of California at Berkeley Download a PDF File of Thesis I investigate the Birch and.
	deals.dunning-marketing.com /abelian-varieties.41007.html (286 words)

	Abelian Varieties with Complex Multiplication and Modular Functions \| Fan Blurb (Site not responding. Last check: 2007-10-08)
		Used Abelian Varieties with Complex Multiplication and Modular Functions are in stock for only $45.00.
		Abelian l-adic Representations and Elliptic Curves (Research Notes in Mathematics (a K Peters), Vol 7)
		Used Abelian l-adic Representations and Elliptic Curves (Research Notes in Mathematics (a K Peters), Vol 7) are in stock for only $33.33.
	fanblurb.com /amazon/asin.0691016569.Book_Abelian_Varieties_with_Complex_Multiplication_and_Modular_Functions.html (800 words)

	Universal Norms on Abelian Varieties Over Global Function Fields - Papanikolas (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		We examine the Mazur-Tate canonical height pairing defined between an abelian variety over a global field and its dual.
		By expressing local factors of this pairing in terms of nonarchimedean theta functions, we show in the case of global function fields that certain of these pairings are annihilated by universal norms coming from Carlitz cyclotomic extensions.
		1 Bi-extensions associated to divisors on abelian varieties an..
	citeseer.ist.psu.edu /379880.html (468 words)

	Publication list (Site not responding. Last check: 2007-10-08)
		Hodge classes on self-products of a variety with an automorphism, Comp.
		Cyclic covers of branched along v+2 hyperplanes and the generalized Hodge conjecture for certain abelian varieties, in Arithmetic of Complex Manifolds, Erlangen 1988, Springer Lecture Notes in Math.
		Addendum to: Hodge classes on self-products of a variety with an automorphism, Compositio Math.
	www.math.duke.edu /~schoen/publicationlist.html (333 words)

	The Cassels-Tate Pairing On Polarized Abelian Varieties - Poonen, Stoll (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Let (A; ) be a principally polarized abelian variety de ned over a global eld k, and let X(A) be its Shafarevich{Tate group.
		Poonen, B.; Stoll, M. The Cassels-Tate pairing on polarized abelian varieties.
		6 Arithmetic on curves of genus (context) - Cassels - 1962
	citeseer.ist.psu.edu /poonen98casselstate.html (865 words)

	Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions.
		Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions.
		F.A.Q. Abelian Varieties with Complex Multiplication and Modular Functions
		The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals.
	pup.princeton.edu /titles/6242.html (289 words)

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