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Putting the Hodge and Tate Conjectures to the Test

  • Henri Darmon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

Abstract

The Hodge conjecture asserts that the presence of algebraic cycles on a (smooth, projective) variety over the complex numbers can be detected in its Betti cohomology equipped with the Hodge structure arising from its relation with complex deRham cohomology. The Tate conjecture makes a similar assertion with ℓ-adic cohomology replacing Betti cohomology. One of the difficulties with these conjectures is that the predictions that they make are often hard to test numerically, even in specific concrete instances. Unlike closely related parts of number theory (a case in point being the Birch and Swinnerton-Dyer conjecture) the study of algebraic cycles has therefore not been as strongly affected by the growth of the experimental and computational community as it perhaps could be. In this lecture, I will describe some numerical experiments that are designed to “test” the Hodge and Tate conjectures for certain varieties (of arbitrarily large dimension) which arise from elliptic curves with complex multiplication and theta series of CM Hecke characters.

Keywords

Complex Number Number Theory Complex Multiplication Large Dimension Elliptic Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Henri Darmon
    • 1
  1. 1.Department of MathematicsMcGill UniversityMontrealCanada

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