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.
|Published (Last):||10 December 2005|
|PDF File Size:||8.7 Mb|
|ePub File Size:||4.58 Mb|
|Price:||Free* [*Free Regsitration Required]|
In mathematicsthe Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. Conjehura is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.
As of [update]only special cases of the conjecture have been proved.
Mordell proved Mordell’s theorem: 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.
If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent conjtura points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve.
If the rank of an elliptic curve is 0, then ohdge curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points. Although Mordell’s theorem shows that the rank of an elliptic curve is always finite, it conjetyra not give an effective method for hodye the rank of every curve.
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.
Hodge conjecture – Wikidata
This L -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse—Weil L-function. This conjecture was first proved by Deuring for elliptic curves with conjstura multiplication. It was subsequently shown to be true for all elliptic curves over Qas a consequence of the modularity theorem. Finding rational points on a general elliptic curve is a difficult problem.
Finding the points on an elliptic curve modulo a given prime conjftura is conceptually straightforward, as there are only a finite number of possibilities to check.
However, for large primes it is computationally intensive. 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 comjetura of primes p on elliptic curves whose rank was known.
La conjetura de hodge pdf files
hoodge Initially this was based on somewhat tenuous trends in graphical plots; this induced a hovge of skepticism in J. 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.
It is conjecturally given by. Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the cconjetura of the conjecture. Much like the Riemann hypothesisthis conjecture has multiple consequences, including the following two:. From Wikipedia, the free encyclopedia. Talk at the BSD 50th anniversary conference, May Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Bhargava, Manjul ; Shankar, Arul Birch, Bryan ; Swinnerton-Dyer, Peter Journal of the American Mathematical Society.
Arithmetic Theory of Elliptic Curves. Lecture Notes in Mathematics. Dokchitser, Tim ; Dokchitser, Vladimir The Millennium prize problems. Birch and Swinnerton-Dyer conjecture at Wikipedia’s sister projects.
L -functions in number theory. Analytic class number formula Riemann—von Mangoldt formula Weil conjectures.
Main conjecture of Iwasawa theory Selmer group Euler system. Retrieved from ” https: Articles containing potentially dated statements from All articles containing potentially dated statements.