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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Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:.
A projective complex manifold is a complex manifold which can be embedded in complex projective space. See Hodge theory for more details. Suppose that we vary the complex structure of X over a simply connected base. His corrected form of the Hodge conjecture is:.
Hidge page was last edited on 30 Decemberat However, for large primes it is computationally intensive. The cohomology class of a divisor turns out to equal to its first Chern class. Totaro reinterpreted their result in the framework of cobordism and found many examples of such classes.
If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension.
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. Around 0, we can choose local coordinates z 1For example, the cohomology class of the above cycle would be:.
As of [update]only special cases of the conjecture have been proved. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. A possible substitute is to ask instead one of the two following questions:. 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. Bhargava, Manjul ; Shankar, Arul 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.
A Survey of the Hodge Conjecture, Example 7. Arithmetic Theory of Elliptic Curves.
Birch and Swinnerton-Dyer conjecture – Wikipedia
Let Z be a complex submanifold of X of dimension kand let i: However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.
They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic cohomology. 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.
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.
This is also false. This is too optimistic, because there are not enough subvarieties to make this work. Because [ Z ] is a cohomology class, it has a Hodge decomposition. 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.
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. 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.
Birch, Bryan ; Swinnerton-Dyer, Peter Loosely speaking, the Hodge conjecture asks:. This conjecture was first nodge by Deuring for elliptic curves with complex multiplication. Talk at the BSD 50th anniversary conference, May Views Read Edit View history.
Hodge Conjecture | Clay Mathematics Institute
It is conjecturally given by. In other projects Wikiquote. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. Journal of the American Mathematical Society.
conjetura de hodge
This is now known to be false. The Millennium prize problems. This is an example of the cycle class map of de Rham conjteura, see Weil cohomology. Dokchitser, Tim ; Dokchitser, Vladimir Main conjecture of Iwasawa theory Selmer group Euler system.
In other projects Wikiquote. It was subsequently shown to be true for all elliptic curves over Qas a consequence of the modularity theorem. Hodgd containing potentially dated statements from All articles containing potentially dated statements.