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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The coefficients are usually taken to be integral or rational. This conjecture was first proved by Deuring for elliptic curves with complex multiplication.
In Zucker 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. Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:.
Say 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. Although Mordell’s theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. 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.
Hodge conjecture – Wikidata
However, for large primes it is computationally intensive. By the strong and weak Lefschetz theoremthe only non-trivial xe of the Hodge conjecture for hypersurfaces is the degree m part i. L -functions in number theory.
Around 0, we can choose local coordinates z 1Initially this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J. Finding rational points on a general elliptic curve is a difficult problem.
In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension. Grothendieck observed that this cannot be true, conjetira with rational coefficients, because the right-hand side is not always a Hodge structure.
His corrected form of the Hodge conjecture is:.
Birch and Swinnerton-Dyer conjecture
Hodge conjecture at Wikipedia’s sister projects. Conjeura, Bryan ; Swinnerton-Dyer, Peter Views Read Edit View history.
Hodge made an additional, stronger conjecture than the integral Hodge conjecture.
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 denoted by N p for a large number of primes p on elliptic curves whose rank was known. A projective complex manifold is a complex manifold which can be embedded in complex projective space. A Survey of the Hodge Conjecture, Example 7. Retrieved from ” https: By Chow’s theorema projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogeneous polynomials.
Journal of the American Mathematical Society. For example, the cohomology class of the above cycle would be:. As of [update]only special cases of the conjecture have been proved.
Such a cohomology class is called 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.
The first result on the Hodge conjecture is due to Lefschetz Bhargava, Manjul ; Shankar, Arul Mordell proved Mordell’s theorem: Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. In other projects Wikiquote. If the degree d is 2, i. This page was last edited on 13 Novemberat The cohomology class of a divisor turns out to equal to its first Chern class. Weil generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic fieldthen Hdg 2 X is not generated by products of divisor classes.
Hodge Conjecture — from Wolfram MathWorld
hovge Such a class is necessarily a Hodge class. Let X be a compact complex manifold of complex dimension n. Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. Dokchitser, Tim ; Dokchitser, Vladimir Graduate Texts in Mathematics. In other projects Wikiquote. Proceedings of the American Mathematical Society.