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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Hodge conjecture at Wikipedia’s sister projects. In fact, it predates the conjecture and provided some of Hodge’s motivation. This is also false.

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. From Wikipedia, the free encyclopedia. This is an example of the cycle class map of de Rham cohomology, see Weil cohomology. As of [update]only special cases of the conjecture have been proved.

Introduction to Elliptic Curves and Modular Forms. 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.

A projective complex manifold is a complex manifold which can be embedded in complex projective space. A possible substitute is to ask instead one of the two following questions:. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order.

## Mathematics > Algebraic Geometry

Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. It is conjecturally given by. By using this site, you agree to the Terms of Use and Privacy Policy. The coefficients are usually taken to be integral or rational. 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.

It was subsequently shown to be true for all elliptic curves over Qas a consequence of the modularity theorem. To evaluate this integral, choose a point of Z conjdtura call it 0.

### [math/] Counterexample to the Hodge Conjecture

They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic cohomology. Such a class is necessarily a Hodge class. CS1 French-language sources fr. Grothendieck observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure.

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. The Millennium prize problems. This page was last edited on 30 Decemberat However, for large primes it is computationally intensive. Bhargava, Manjul ; Shankar, Arul Lecture Notes in Mathematics. Totaro reinterpreted their result in the framework of cobordism and found many examples of such classes. This conjecture was first proved by Deuring for elliptic curves with complex multiplication.

This L -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. L -functions in number theory. Retrieved from ” https: However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.

## Birch and Swinnerton-Dyer conjecture

Much like the Riemann hypothesisthis conjecture has multiple consequences, including the following two:. Weil generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field hldge, then Hdg 2 X is not generated by products of divisor classes.

Talk at the BSD 50th anniversary conference, May