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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However, for large primes it is computationally intensive.
Around 0, we can choose local coordinates z 1The cohomology class of a divisor turns out to equal to its first Chern class. This is now known to be false. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points.
Totaro reinterpreted their result in the framework of cobordism and found many examples of such classes. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.
conjetura de hodge
An algebraic cycle on X is a formal combination of subvarieties of X ; that is, it is something of the form:. In fact, it predates the conjecture and provided some of Hodge’s motivation. Such a class is necessarily a Hodge class. A Survey of the Hodge Conjecture, Example 7. Initially this was based on somewhat tenuous trends in graphical plots; this induced a measure of skepticism in J. 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.
From Wikipedia, the free encyclopedia. His corrected form of the Hodge conjecture is:.
By the Hard Lefschetz theoremone can prove:. By Chow’s theorema projective complex manifold is also a smooth projective algebraic variety, that is, it is dde zero set of a collection of homogeneous polynomials.
Lecture Notes in Mathematics. This conjecture was first proved by Deuring for elliptic curves with complex multiplication.
In mathematicsthe Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. The coefficients are usually taken to be integral or rational. Dde page was last edited on 30 Decemberat If the degree d is 2, i. Views Read Edit View history. By uodge strong and weak Lefschetz theoremthe only non-trivial part of the Hodge conjecture for hypersurfaces is the degree m part i.
Mathematics > Algebraic Geometry
This is too optimistic, because there are not enough subvarieties to make this work. Birch and Swinnerton-Dyer conjecture at Wikipedia’s sister projects.
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. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order.
Such a cohomology class is called algebraic. Birch, Bryan ; Swinnerton-Dyer, Peter It is a special case of a Hasse—Weil L-function. 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. Talk at the BSD 50th anniversary conference, May Retrieved from ” https: 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.
Since X is a compact oriented manifold, X has a fundamental class. Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture. The assumption in the Hodge conjecture that X be algebraic projective complex manifold cannot be weakened.
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. Hodge made an additional, stronger conjecture than the integral Hodge conjecture.
Hodge Conjecture | Clay Mathematics Institute
Suppose that we vary the complex structure of X over a simply connected base. Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hovge decomposition:. Main conjecture of Iwasawa theory Selmer group Euler system. Dokchitser, Tim ; Dokchitser, Vladimir Introduction to Elliptic Curves and Modular Forms.
That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z 1However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties. Retrieved from ” https: