The -adic etale cohomology of algebraic varieties is much richer than their classical cohomology in the sense that it admits the action of Galois groups. In the 1960's, motivated by its relation to poles of zeta functions and a geometric analogue of the BSD conjecture, Tate was led to formulate his famous conjecture saying roughly that the -adic cohomology classes fixed by the Galois action should arise from algebraic cycles. Besides making the Tate conjecture precise, we will spend most of the time discussing Tate's original motivation, various evidence supporting it and a few surprising implications.
Let be a field and let be a smooth geometrically irreducible projective variety over of dimension . We denote by the base change of to the algebraic closure . The Galois group then acts on via the second factor.
Let be a prime different from . Recall from John's talk that there is cycle class map (of -modules) which associates to every algebraic cycle an -adic etale cohomology class.
Notice the inclusions induces surjections By the theorem of the base, is a finitely generated abelian group. The kernel of the first surjection is a divisible group (points of the Picard variety) and the kernel of the second surjection is the torsion subgroup of . So we have , a free abelian group of rank equal to the Picard number . In particular, does not depend on the choice of . It also follows that the map (2) is at least injective when .
With this example in mind, we are now ready to state the famous Tate conjecture.
Why should one believe the Tate conjecture? One should because it is a conjecture of Tate (proof by authority, QED). We are going to discuss two of Tate's major original motivations: one is the relation between and the geometric analogue of the remarkable conjecture of Birch and Swinnerton-Dyer concerning the rational points of elliptic curves; one has to do with the following conjecture (now a theorem) on homomorphisms between abelian varieties.
The truth of allows us to prove the following
Even better, Tate proved that
Together with the previous proposition, we see that is true for any that is dominated by products of varieties for which are known to be true.
The Hodge conjecture is known for codimension 1 by Lefschetz 1-1 theorem. In contrast, is still not known in general (and as you expected, even less is known when ). We list some known case about K3 surfaces (the theme of this seminar).
Due to the work of many authors, is also known to be true for Hilbert modular surfaces, quaternionic Shimura surfaces, Picard modular surfaces, Siegel modular threefolds... See the references in .
There are many implications of the Tate conjecture. One surprising implication among them is the BSD conjecture for elliptic curves over global function fields. As we mentioned, this is one of Tate's original motivations for formulating .
Assume now that is a finite field of characteristic . Recall that the zeta function of is defined to be the generating function of , By the Lefschetz trace formula, this is in fact a rational function in , where here is the characteristic polynomial of the geometric Frobenius on . By Deligne's proof of the Weil conjecture, this is a polynomial in independent of the choice of . The inverse roots of have all absolute values . For example, and .
It follows that the poles of the zeta function lie on the lines . In particular, the order of pole at is exactly the multiplicity of occurring as the eigenvalue of on , or the multiplicity of occurring as the eigenvalue of on . If acts semisimply, this is exactly the dimension of . In fact, by carefully doing linear algebra Tate showed that
This may ring some bells if you have seen the remarkable conjecture of Birch and Swinnerton-Dyer, which predicts that the order of zero of the -function at of an elliptic curve over a global field (more generally, any finitely generated field) is equal to the rank of the group of -rational points . When is the function field of a smooth projective curve defined over , by resolving singularities one can always find a regular projective elliptic surface such that the generic fiber of is . If is smooth, the connection is extremely simple: on the one hand, unraveling the definition one finds that Since has simple poles at , it follows that at , On the other hand, where is the class of fibers and is the class of the zero section. Hence Therefore The left hand side is the BSD conjecture for and by the previous theorem the right hand side is equivalent to the Tate conjecture for ! In general has singular fibers and the theorem of Shioda-Tate says that where is the number of different components of the fiber over the closed point of . When taking into account of the modification to from bad places, the change of the order of zero at miraculously matches up with . Hence we know that
In view of Remark 4, one naturally wonders if the finiteness of the Brauer group of the elliptic surface has anything to do the finiteness of the Shafarevich-Tate group . Amazingly, Grothendieck  proved that
Finally it is worth mentioning another interesting implication of the Tate conjecture for K3 surfaces recently proved in .
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