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Creation and Basic Functions The dimension of the modular abelian variety A. DirichletCharacter(A) : ModAbVar -> GrpDrchElt A morphism with finite kernel from the modular abelian variety A to a modular abelian variety attached to modular symbols. A surjective morphism to the modular abelian variety A from an abelian variety attached to modular symbols. www.math.lsu.edu /magma/text1321.htm   (5832 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. A polarization of an abelian variety is an isogeny from an abelian variety to its dual. A morphism of polarized abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A. en.wikipedia.org /wiki/Abelian_variety   (1547 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. planetmath.org /encyclopedia/AbelianVariety.html   (128 words)

Jacobian - Wikipedia, the free encyclopedia In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. In this sense, the Jacobian is akin to a derivative of a multivariate function. The absolute value of the Jacobian determinant at p gives us the factor by which the function F expands or shrinks volumes near p; this is why it occurs in the general substitution rule. en.wikipedia.org /wiki/Jacobian   (672 words)

Springer Online Reference Works Sometimes a Jacobi variety is simply considered to be a commutative algebraic group. The significance of Jacobi varieties in the theory of algebraic curves is clear from the Torelli theorem (cf. This property of Jacobi varieties is characteristic if one considers only principally polarized Abelian varieties belonging to a neighbourhood of the Jacobian of a general curve. eom.springer.de /j/j054140.htm   (654 words)

Algebraic curve In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections. For higher genus g some of those carry over to the Jacobian variety[?], an abelian variety of dimension g www.ebroadcast.com.au /lookup/encyclopedia/al/Algebraic_curve.html   (251 words)

Springer Online Reference Works 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 study of the action of endomorphisms of Abelian varieties, in particular of the Frobenius endomorphism on its Tate module, makes it possible to give a proof of the Riemann hypothesis (for algebraic curves over finite fields, cf. Its principal result is the Mordell–Weil theorem: The group of rational points of an Abelian variety, defined over a finite extension of the field of rational numbers, is finitely generated. eom.springer.de /a/a010260.htm   (763 words)

[No title] However, algebraic varieties may also have complicated singular sets and may be parametrized with rings other than the complex numbers. Jacobian variety A space associated to a Riemann surface defined most succinctly as the complex cohomology (as a vector space) divided by the integral cohomology (as a lattice). It is simultaneously a complex manifold, an algebraic variety, and a Lie group. www.ornl.gov /ortep/topology/defs.txt   (5717 words)

Services Jacobian Technologies is able to develop custom “turn-key” metrology equipment from a need to measure some product attribute. We utilize our education and experience as well as a variety of design tools, including Zemax-EE to design superior optics and optical systems. Chances are, we have direct experience, have some familiarity with, know an expert in the area, or can apply our closely related skills and knowledge to the task to get it done. www.jacobiantech.com /html/services.html   (229 words)

The Jacobian in Thermodynamics - Introduction & Background The use of the Jacobian in thermodynamics was first introduced by Dr. Kali Mukerjee at Michigan State University. The Jacobian is a useful way to find a relation between two different vector spaces. Building off from this idea it is possible to formulate a set of equations that can be applied to a wide variety of thermodynamic problems including multicomponent systems. www.egr.msu.edu /classes/msm851/details.html   (206 words)

Discrete logarithm-based cryptography using the shafarevich-tate group patent invention A method as recited in claim 1, wherein the abelian variety is an elliptic curve or a Jacobian variety of a higher genus curve. A computer-readable medium as recited in claim 9, wherein the abelian variety is an elliptic curve or a Jacobian variety of a higher genus curve. A computing device as recited in claim 17, wherein the abelian variety is an elliptic curve or a Jacobian variety of a higher genus curve. www.freshpatents.com /Discrete-logarithm-based-cryptography-using-the-shafarevich-tate-group-dt20060518ptan20060104447.php?type=claims   (1044 words)

UW Algebra Seminar In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly. Here, a generalised arrangement consists of a (possibly reducible) variety X which is a degeneration of projective space together with a collection D_1,..,D_n of codimension 1 subvarieties obtained as limits of hyperplanes. Prym varieties, the natural extension of the classical Prym map and properties of this map are the topic of this second talk. www.math.washington.edu /~smith/Seminar/F03abs.html   (785 words)

Selected Matches for: Items Authored by Stein, William Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Analyze the kernel of this isogeny and of the natural map from the Jacobian of $X_0(p)$ to that of $X_1(p)$, coupled with the fact that $E_0$ is $X_0(p)$-optimal \ref[J.-F. Mestre and J. Oesterlé, op. The proof involves the study of the component groups at $p$ of Jacobians of intermediate curves between $X_1(p)$ and $X_0(p)$. modular.fas.harvard.edu /papers/stein-msn.html   (5542 words)

UC Davis Math: profile_files Two important related structures are the Jacobian variety, which is the torus obtained by dividing a sheaf cohomology (as a vector space) by an integral cohomology (as a lattice), and the theta function of the Riemann matrix. Among other things, they arrived at a new understanding of Prym varieties, the spaces which in this setting correspond to and generalize Jacobian varieties. Cohomological structure in soliton equations and Jacobian varieties, J. Differential Geom. www.math.ucdavis.edu /research/profiles/mulase   (498 words)

Jacobian - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-21) The Jacobian determinant at a given point gives important information about the behavior of F near that point. Furthermore, if the Jacobian determinant at p is positive, then F preserves orientation near p; if it is negative, F reverses orientation. The Jacobian determinant is used when making a change of variables when integrating a function over its domain. en.wikipedia.org.cob-web.org:8888 /wiki/Jacobian   (681 words)

Mathematics Jacobian   (Site not responding. Last check: 2007-10-21) Brazil's ethanol production is likely to go for a downturn as a result of reduced domestic supply of sugar for this crop season. In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. Polynomial Automorphisms : and the Jacobian Conjecture (Progress in Mathematics) by Arno van den Essen 3764363509... www.jackup.info /info/Mathematics-Jacobian   (326 words)

Math Department Colloquia A classical theorem of Kolchin asserts that X_{\infty}, the space of formal arcs in X is an irreducible infinite dimenional variety. We also prove that the category of graded D-branes of type B in such Landau-Ginzburg models is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W=0. Abstract: In the 1990's, Lusztig introduced the totally nonnegative part of an arbitrary flag variety, calling it a "remarkable polyhedral subspace." He proved that this space is contractible, but stronger geometric results have been elusive: for example, it is unknown (but expected) that this space is homeomorphic to a ball. www.math.miami.edu /anno/colloquium.htm   (2950 words)

[No title]   (Site not responding. Last check: 2007-10-21) These notes are a review on computational methods that allow us to use computers as a tool in the research of Riemann surfaces, algebraic curves and Jacobian varieties. It is well known that compact Riemann surfaces, projective algebraic curves and Jacobian varieties are only different views to the same object, i.e., these categories are equivalent. Vice versa, we want to be able to compute the uniformization for a given algebraic plane curve, or a Riemann surface corresponding to a given Jacobian variety. www.math.fsu.edu /~seppala/papers/Taniguchi/Taniguchi.html   (367 words)

Abel's theorem: Encyclopedia - Abel's theorem (via CobWeb/3.1 planetlab2.tamu.edu)   (Site not responding. Last check: 2007-10-21) For Abel's theorem on algebraic curves, see Jacobian variety. Let a = {ai: i ≥ 0} be any sequence of real or complex numbers and let be the power series with coefficients a. Galton-Watson processes, Jacobian variety, Niels Henrik Abel, Tauberian theorems, algebraic curves, divergent series, generating function, generating functions, one-sided limit, power series, probability-generating functions, radius of convergence, real analysis, sequences www.experiencefestival.com.cob-web.org:8888 /a/Abels_theorem/id/409774   (453 words)

Mathematics of Computation A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields. Efficient reduction on the Jacobian variety of Picard curves. Efficient algorithms for the Riemann-Roch problem and for addition in the jacobian of a curve. www.ams.org /mcom/2002-71-237/S0025-5718-00-01297-7/home.html   (403 words)

University of New South Wales - Faculty of Science - News - HONOURS TALK In the study of compact Riemann surfaces, the Jacobian variety plays an important role. The Torelli theorem states that given a Jacobian variety, as well as an additional piece of data called the principal polarisation, the compact Rieman surface is determined up to isomorphism. This assures us that in studying a Riemann surface using via its Jacobian variety, no information is lost. www.maths.unsw.edu.au /news/2004/honours2004.html   (188 words)

Abelian Surfaces over Finite Fields as Jacobians, Daniel Maisner, Enric Hart Then, we analyze numerically what surfaces are k-isogenous to the Jacobian of a smooth projective curve of genus 2 defined over k. For instance, we show that every absolutely simple abelian surface is k-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials {\small $t^4+(1-2q)t^2+q^2$} (for all q) and {\small $t^4+(2-2q)t^2+q^2$} (for q odd) are never the characteristic polynomial of Frobenius of a Jacobian. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1057777425   (244 words)

Mathlets: Jacobians   (Site not responding. Last check: 2007-10-21) Shows the effect of a change of variables in two dimensions on area units, using the Jacobian to approximate the transformed areas. =" can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates. With the point (u,v), shown in the left graph by the position of the cs.jsu.edu /mcis/faculty/leathrum/Mathlets/jacobian.html   (317 words)

School of Mathematics :: In this talk we give a family of positive polynomials of degree 8 in y that are not a sum of three squares in R(x,y). This problem can be reformulated as the search for an antineutral point of the jacobian variety associated to some hyperelliptic curve C. Following a method invented by Cassels, Ellison and Pfister, we check the nonexistence of an antineutral torsion point of the jacobian associated to C, and we use a 2-descent to show the triviality of the R(x)-Mordell-Weil rank of this jacobian variety. www.mth.uea.ac.uk /maths/pure-sem05-06.htm   (1376 words)

Graduate Colloquia Spring 2001 Abstract: Jacobians are a very classical and fascinating topic in the study of algebraic curves. You can associate to any Riemann Surface its Jacobian, which is a complex torus; the amazing thing is that given a Jacobian and a little extra information you can pin down exactly the curve you started with. It has been used to study the phenomenon of collective synchronization in a variety of biological systems (neural oscillators, flashing fireflies, circadian pacemaker cells, cardiac pacemaker cells) and physical systems (laser arrays, superconducting Josephson junction arrays). www.math.utah.edu /gsac/colloq_past/colloq_spring01.html   (1263 words)

Efficient algorithms for the Jacobian variety of hyperelliptic curves y² = x^p - x + 1 over a finite field of odd ...   (Site not responding. Last check: 2007-10-21) Efficient algorithms for the Jacobian variety of hyperelliptic curves y² = x^p - x + 1 over a finite field of odd characteristic (Extended Abstract) (ResearchIndex) (via CobWeb/3.1 planetlab2.tamu.edu) Efficient algorithms for the Jacobian variety of hyperelliptic curves y² = x^p x + 1 over a finite field of odd characteristic (Extended Abstract) (1999) We develop efficient algorithms for the Jacobian of the hyperelliptic curve defined by the equation y 2 = x p \Gamma x + 1 over a finite field F p n of odd... citeseer.ist.psu.edu.cob-web.org:8888 /duursma99efficient.html   (633 words)

Citebase - A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic   (Site not responding. Last check: 2007-10-21) Citebase - A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic is not algebraically equivalent to zero in the Jacobian variety J(C). citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0508433   (124 words)

Counting Points on the Jacobian Variety of a Hyperelliptic Curve defined by y² = x^5 + ax over a Prime Field ...   (Site not responding. Last check: 2007-10-21) Abstract: Counting the order of the Jacobian group of a hyperelliptic curve over a finite field is very important for constructing a hyperelliptic curve cryptosystem (HECC), but known algorithms to compute the order of a Jacobian group over a given large prime field need very long running times. In this note, we propose a practical polynomialtime algorithm to compute the order of the Jacobian group for a hyperelliptic curve of type y + ax over a given large prime field F p, e.g. 2 the Jacobian Varieties of Hyperelliptic Curves over Fields o.. citeseer.ist.psu.edu.cob-web.org:8888 /558804.html   (428 words)

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