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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Suppose that we vary the complex structure of X over a simply connected base. 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.

From Wikipedia, the free encyclopedia. Let X be a compact complex manifold of complex dimension n.

L -functions in number theory. Bhargava, Manjul ; Shankar, Arul 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. Views Read Edit View history.

## Mathematics > Algebraic Geometry

In other projects Wikiquote. Birch and Swinnerton-Dyer conjecture at Wikipedia’s sister projects. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. Analytic class number formula Riemannâ€”von Mangoldt formula Weil conjectures.

That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z 1Around 0, we can choose local coordinates z 1Another way of phrasing the Hodge conjecture involves the idea of conketura algebraic cycle. If the degree d is 2, i.

Let Z be a complex submanifold of X of dimension kand let i: His corrected form of the Hodge conjecture is:. The number of independent basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve. As of hidgeonly special cases of the conjecture have been proved.

Arithmetic Theory of Elliptic Curves. They show that the rational Hodge conjecture is equivalent to an fonjetura Hodge conjecture for this modified motivic cohomology.

## Hodge Conjecture

A projective complex manifold is a complex manifold which can be embedded in complex projective space. Views Read Edit View history. In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension. Retrieved from ” https: Grothendieck observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure.

Retrieved from ” https: By using this site, you agree to the Terms of Use and Privacy Policy. 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.

Such a class is necessarily a Hodge class. The cohomology class of a divisor turns out to equal to its first Chern class. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between and to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. 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.

CS1 French-language sources fr. Initially this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J. The coefficients are usually taken to be integral or rational. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components.

If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order.

### Hodge Conjecture | Clay Mathematics Institute

Hodhe rational points on a general elliptic curve is a difficult problem. Hodge made an additional, stronger conjecture than the integral Hodge conjecture. 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. Mordell proved Mordell’s theorem: