Putting the Hodge and Tate Conjectures to the Test

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.