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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Mordell proved Mordell’s theorem: It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:.
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. Then the topological cohomology of X does not change, but the Hodge decomposition does change. Arithmetic Theory of Elliptic Curves. Analytic class number formula Riemann—von Mangoldt formula Weil conjectures.
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.
To evaluate this integral, choose a point of Z and call it 0.
Much like the Riemann hypothesisthis conjecture has multiple consequences, including the following two:. 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.
Articles containing potentially dated statements from All articles containing potentially dated statements. Such a cohomology class is called algebraic.
It received little attention before Hodge presented it in an address during the International Congress of Mathematiciansheld in Cambridge, Massachusetts.
conjetura de hodge
Graduate Texts in Mathematics. L -functions in number theory. Nothing hdoge been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.
Grothendieck observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure. The rank of certain elliptic curves can be calculated using numerical methods but in the current state conjetuta knowledge it is unknown if these methods handle all curves.
Birch and Swinnerton-Dyer conjecture
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. In other projects Wikiquote. Loosely speaking, the Hodge conjecture asks:.
His corrected form of the Hodge conjecture is:. Retrieved from ” https: 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. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. In other projects Wikiquote.
Views Read Edit View history. Because [ Z ] is hldge cohomology class, it has a Hodge decomposition. However, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties. That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z 1In Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type ppwhich is not projective algebraic.
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. Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. 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.
A possible substitute is to ask instead one of cnojetura two following questions:. Let Z be a complex submanifold of X of dimension kand let i: Lecture Notes in Mathematics.
[math/] Counterexample to the Hodge Conjecture
They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic conjeturs.
This means that for comjetura elliptic curve there is a finite sub-set of the rational points on the curve, from which all further rational points may be generated. By the strong and weak Lefschetz theoremthe only non-trivial part of the Hodge conjecture for hypersurfaces is the degree m part i.
Proceedings of the American Mathematical Society. A very quick proof can be hodeg using sheaf cohomology and the exponential exact sequence. Such a class is necessarily a Hodge class.
Main conjecture of Iwasawa theory Selmer group Euler system. From Wikipedia, the free encyclopedia. This is too optimistic, because there are not enough subvarieties to make this work. Around 0, we can choose local coordinates z 1 ,