Remarks on the Tate Conjecture for beginners

The pair of conjectures of Tate, called T1 and T2 below, are of great importance. They provide analogues of two other famous conjectures, namely the Hodge Conjecture, which is a prediction in Complex Geometry, and the Birch and Swinnerton-Dyer Conjecture, which is a centrepiece of Number theory. This dual aspect is what makes the Tate conjecture central and also very difficult to solve in general. After stating the conjectures, I will discuss them for the special case of a product of an elliptic curve E with itself. First the geometric aspect. Let X be a smooth projective variety over the complex numbers C. The simplest example of X is the projective space P , whose points are represented by non-zero vectors x = (x0, x1, . . . , xN) in C where x is identified with λx for any scalar λ; it is also the image of the (real) (2N − 1)-dimensional sphere S2N−1 under the antipodal map. More generally, X is cut out in P by a finite number of homogeneous polynomials fi(x), 1 ≤ i ≤ r. In other words, X is the set of simultaneous solutions, up to scaling, of these polynomials. We also want it to be smooth, i.e., without any corners or folds (which can be checked by a Jacobian condition). Then, under the complex topology, X becomes a compact complex manifold, equipped with a Kḧler metric, of certain dimension d. When the polynomials form an independent system, d would be N − r. When r = 1 we get a hypersurface (prime divisor), and this is already interesting for N = 2, as X would be a complex algebraic curve. (It is a real surface!) Of fundamental importance are the (singular) cohomology groups H(X,C) of dimension bj, which vanish if j 6∈ [0, 2d], with b0 = b2d = 1. There is a duality relating H to H2d−j, and moreover, there is a Hodge decomposition H(X,C) = ⊕p=0H(X), where H(X) is generated by differential forms ω which are locally of the form ξdzI ∧ dzJ with |I| = p, |J | = q; in other words this differential form