English Spanish online dictionary Term Bank, translate words and terms with different pronunciation options. Products of it with itself give candidates for counterexamples to the Hodge conjecture which may be of interest. We also study the Kuga-Satake. The Hodge conjecture asserts that, for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually rational.

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This page was last edited on 13 Novemberat Hodge conjecture at Wikipedia’s sister projects.

Conjeyura that a cohomology class on X is of co-level c coniveau c if it is the pushforward of a cohomology class on a c -codimensional subvariety of X. L -functions in number theory.

In Dr showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type ppwhich is not projective algebraic. Voisin proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes.

Arithmetic Theory of Elliptic Curves. Nothing has been proved for curves with rank cojjetura than 1, although there is extensive numerical evidence for the truth of the conjecture.

Birch and Swinnerton-Dyer conjecture

Such a class is necessarily a Hodge class. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components. From Wikipedia, the free encyclopedia. In the latter case, the Hodge conjecture is only known in special cases.

In other projects Wikiquote. Totaro reinterpreted their result in the framework of cobordism and found many examples of such classes. The assumption in the Hodge conjecture that X be algebraic projective complex manifold cannot be weakened.


A possible substitute is to ask instead one of the two following questions:. It received little attention before Hodge presented it in an address during the International Congress of Mathematiciansheld in Cambridge, Massachusetts. To evaluate this integral, choose a point of Z and call it 0. Birch, Bryan ; Swinnerton-Dyer, Peter In fact, it predates the conjecture and provided some of Hodge’s motivation.

With this notation, the Hodge conjecture becomes:.

[math/] Counterexample to the Hodge Conjecture

Then X is an orientable smooth manifold of real dimension connetura nso its cohomology groups lie in degrees zero through 2 n. The cohomology classes of co-level at least c filter the cohomology of Xand it is easy to see that the c th step of the filtration N c H k XZ satisfies. Retrieved from ” https: On the other hand, if the rank of the curve is greater conhetura 0, then the curve has an infinite number of rational points. In the early s Peter Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number of points modulo p conjethra by N p for a large number of primes p on elliptic curves whose rank was known.

A Survey of the Hodge Conjecture, Example 7. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve, from which all further rational points may be generated. Suppose that we vary the complex structure of X over a simply connected base. Since X is a compact oriented manifold, X has a fundamental class.

If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. By using this site, you agree to the Terms of Use and Privacy Policy. The coefficients are usually taken to be integral or rational. Birch and Swinnerton-Dyer conjecture at Wikipedia’s sister projects. Then the topological cohomology of X does not change, but the Hodge decomposition does change.


NB that the reciprocal of the L-function is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes. Lecture Notes in Mathematics.

Term Bank – conjetura de hodge – Spanish English Dictionary

Retrieved from ” https: Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:. In particular, the Hofge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension.

Hodge made an additional, stronger conjecture than the integral Hodge conjecture. It was subsequently shown to be true for all elliptic curves over Qas a consequence of the modularity theorem. An algebraic cycle on X is a formal combination of subvarieties of X ; that is, it is something of the form:.

The number of independent basis points with infinite order is called the rank of the curve, and is dee important invariant property of an elliptic curve. This L -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form.