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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His corrected form of the Hodge conjecture is:. CS1 French-language sources fr. It was subsequently shown to be true for all elliptic curves over Qas a consequence of the modularity theorem. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite hoge.
Lecture Notes in Mathematics.
La conjetura de hodge pdf files
Grothendieck observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure. Finding rational points on a general elliptic curve is a difficult problem.
It is conjecturally given by. 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 particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension. Mordell proved Mordell’s theorem: Arithmetic Theory of Elliptic Curves. This conjecture was first proved nodge Deuring for elliptic curves with complex multiplication. Hodge made an additional, stronger conjecture than the integral Hodge conjecture. By Chow’s theorema projective complex manifold is also a smooth projective algebraic variety, that is, it is cpnjetura zero set of a collection of homogeneous polynomials.
The Millennium prize problems. In 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.
We define the cohomology class of an algebraic cycle to be the sum of conjeura cohomology classes of its components. This is too optimistic, because there are not enough subvarieties to make this work. Bhargava, Manjul ; Shankar, Arul Graduate Texts in Mathematics. In other projects Wikiquote.
Proceedings of the American Mathematical Society. In other projects Wikiquote. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points.
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.
The cohomology class of a divisor turns out to equal to its first Chern class. By the strong and weak Lefschetz theoremthe only non-trivial part of the Hodge conjecture for hypersurfaces is the degree m part i. Much like the Riemann hypothesisthis conjecture has multiple consequences, including the following two:.
This page was last edited on 13 Novemberat Journal of the American Mathematical Society.
Articles containing potentially dated statements from All articles containing potentially dated statements. 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. Talk at the BSD 50th anniversary conference, May Hodge conjecture at Wikipedia’s sister projects.
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Hodge Conjecture | Clay Mathematics Institute
Such a cohomology class is called algebraic. 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. For example, the cohomology cknjetura of the above cycle would be:. They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic cohomology.
This page was last edited on 30 Decembercnojetura