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 L -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. If the degree d is 2, i. Analytic class number formula Riemann—von Mangoldt formula Weil conjectures. 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.

They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic cohomology. This is too optimistic, because there are not enough subvarieties to make this work. Views Read Edit View history. In mathematicsthe Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. This is an example of the cycle class map of de Rham cohomology, see Weil cohomology.

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. It was subsequently shown to be true for all elliptic curves over Qas a consequence of the modularity theorem. 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.


By the Hard Lefschetz theoremone can prove:. hoddge

The coefficients are usually taken to be integral or rational. Dokchitser, Tim ; Dokchitser, Vladimir Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle.

By using this site, you agree to the Terms of Use and Privacy Policy. Finding the points on an elliptic curve modulo a hogde prime p is conceptually straightforward, as there are only a finite number of possibilities to check. Initially this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J.

Birch and Swinnerton-Dyer conjecture – Wikipedia

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. The Millennium prize problems. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations.

If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. Finding rational points on a general elliptic curve is a difficult problem. In the latter case, the Hodge conjecture is only known in special cases.


We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components. Arithmetic Theory of Elliptic Curves.

Hodge Conjecture

Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is cpnjetura 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. A projective complex manifold is a complex manifold which can be embedded in complex projective space.

A Survey of conjetur Hodge Conjecture, Example 7.

Mathematics > Algebraic Geometry

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2 nso its cohomology groups lie in degrees zero through 2 n. Proceedings of the American Mathematical Society. For example, the cohomology class of the above cycle would be:.

Such a cohomology class is called algebraic. Hodge made an additional, stronger conjecture than the integral Hodge conjecture. Graduate Texts in Mathematics. The Lefschetz theorem on 1,1 -classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:.

The rank of certain elliptic curves can be calculated using numerical methods but in the current state of knowledge it is unknown if these methods handle all curves. Talk at the BSD 50th anniversary conference, May