Number theorists and algebraic geometers study algebraic varieties. For us variety is the common zero set of a collection of polynomials having coefficients either in the rational numbers Q or in a finite field like Z/p (the integers modulo p for p a prime). One can put natural geometric conditions (such as transversal crossing, or no folding) on a variety V so that there are no singularities--the variety is smooth. See Figure 1.

Figure 1

If p is a polynomial over a field k, then one may adjoin the roots of p (which certainly may not lie in k) to the field and form thereby a new (larger) field m. The set of algebraic mappings of m which fix k form a group, and this is the Galois group. This is a beautiful idea of Evariste Galois (1811--1832). This group acts in a natural way on the coefficients of the polynomial. As a result, the group also acts on certain geometric invariants (the cohomology groups H^j(V)) of the variety V corresponding to p.

Now let V and V' be two algebraic varieties having the same geometric dimension. Suppose that H^j(V) and H^j(V') (these are the previously mentioned geometric invariants of V, V') are isomorphic, that is they are algebraically equivalent (in such a fashion as to respect the action of the Galois group). The fundamental question, relating algebra to geometry, is whether this algebraic equivalence is induced by a geometric equivalence of V and V'. That this is true is called the Tate conjecture. [An analogous, but simpler, question could be asked in classical Euclidean geometry. Suppose that two geometric figures in the plane are acted on by the same groups of rotations. Must they then be the same figure? The answer is "no," because the group could be the rotation group with four elements--the right-angle rotations--one figure could be a square while the other figure could be a four-pointed star.]

In fact the Tate conjecture may be formulated over more general fields, such as the finitely generated field obtained by adjoining finitely many transcendental numbers to Q (a number is transcendental if it is not the root of a polynomial equation with rational coefficients).

It is anticipated that the Tate conjecture, formulated in suitable generality, is actually stronger than the famous Hodge conjecture. We describe this celebrated conjecture as follows. Mathematicians have discovered powerful ways to investigate the shapes of complicated objects. We ask to what extent we can approximate the shape of a given object by gluing together simple geometric inits of increasing dimension. This technique has turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of geometric objects that arise in various branches of mathematics. Unfortunately, the geometric origins of the procedure became obscured in the process. Complications arose that necessitated adding pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of geometric objects called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles. The Hodge conjecture is one of the Clay Millenium Prize Problems that carries a purse of $1 million.

The Tate conjecture dates from the mid-1960s. The Hodge conjecture is perhaps twenty years older than that. Some years after Tate formulated his problem, Fields Medalist Pierre Deligne proved a version of the conjecture for varieties which are tori sitting in projective space. Among his other achievements, Deligne proved that the Tate conjecture implies the Hodge conjecture for abelian varieties.

There is a second approach to the Tate conjecture which may be described as follows. Consider the celebrated Riemann zeta function

ζ(s) = ∑_n 1/n^s .

It is a basic fact that ζ has a simple pole at s = 1. The Euler product formula says that

ζ(s) = ∏_p 1/(1 - p^-s)

hence

log ζ(s) = ∑_p ∑_{m ≥ 1} (1/[m p^ms]) = ∑_p 1/p^s + ( ∑_{m ≥ 2} ∑_p 1/[m p^ms]) .

Thus one sees that the pole can exist at s = 1 if and only if there are infinitely many primes (particularly ∑_p 1/p = +∞).

It is an important tenet of modern number theory that any time a zeta function (or a closely related L-function first introduced by Dirichlet (1805--1859)) has a zero or a pole, then there is some interesting phenomenon occurring. Poles and zeros are not accidental, but exist for important reasons; this was an important thesis of Hecke (1887-1947). If V is a variety over the field Q or Z/p then there is a standard method for associating a ζ function or an L-function to it. If the function has a pole, then there is a naturally occuring subvariety of V that is causing that pole. The order of the pole should in some sense be the number of such special subvarieties. These ideas are related to the conjecture of Birch/Swinnerton-Dyer--another Clay $1 million Millenium Prize Problem about number theory.

The Tate conjecture is a key idea in modern number theory, and is the driving force behind much modern research in the subject. The AIM workshop explored connections of the conjecture to other parts of mathematics, and particularly the two Millenium Prize problems.