Weil Conjectures. Note Quote.

Solving Diophantine equations, that is giving integer solutions to polynomials, is often unapproachably difficult. Weil describes one strategy in a letter to his sister, the philosopher Simone Weil: Look for solutions in richer fields than the rationals, perhaps fields of rational functions over the complex numbers. But these are quite different from the integers:

We would be badly blocked if there were no bridge between the two. And voilà god carries the day against the devil: this bridge exists; it is the theory of algebraic function fields over a finite field of constants.

A solution modulo 5 to a polynomial P(X,Y,..Z) is a list of integers X,Y,..Z making the value P(X,Y,..Z) divisible by 5, or in other words equal to 0 modulo 5. For example, X² + Y² − 3 has no integer solutions. That is clear since X and Y would both have to be 0 or ±1, to keep their squares below 3, and no combination of those works. But it has solutions modulo 5 since, among others, 3² + 3² − 3 = 15 is divisible by 5. Solutions modulo a given prime p are easier to find than integer solutions and they amount to the same thing as solutions in the finite field of integers modulo p.

To see if a list of polynomial equations P_i(X, Y, ..Z) = 0 have a solution modulo p we need only check p different values for each variable. Even if p is impractically large, equations are more manageable modulo p. Going farther, we might look at equations modulo p, but allow some irrationals, and ask how the number of solutions grows as we allow irrationals of higher and higher degree—roots of quadratic polynomials, roots of cubic polynomials, and so on. This is looking for solutions in all finite fields, as in Weil’s letter.

The key technical points about finite fields are: For each prime number p, the field of integers modulo p form a field, written F_p. For each natural number r > 0 there is (up to isomorphism) just one field with p^r elements, written as F_p^r or as F_q with q = p^r. This comes from F_p by adjoining the roots of a degree r polynomial. These are all the finite fields. Trivially, then, for any natural number s > 0 there is just one field with q^s elements, namely F_p^(r+s) which we may write F_q^s. The union for all r of the F_p^r is the algebraic closure F^–_p. By Galois theory, roots for polynomials in F_p^r, are fixed points for the r-th iterate of the Frobenius morphism, that is for the map taking each x ∈ F^–_p to x^pr.

Take any good n-dimensional algebraic space (any smooth projective variety of dimension n) defined by integer polynomials on a finite field F_q. For each s ∈ N, let N_s be the number of points defined on the extension field F_(q^s). Define the zeta function Z(t) as an exponential using a formal variable t:

Z(t) = exp(∑_s=1^∞N_st_s/s)

The first Weil conjecture says Z(t) is a rational function:

Z(t) = P(t)/Q(t)

for integer polynomials P(t) and Q(t). This is a strong constraint on the numbers of solutions N_s. It means there are complex algebraic numbers a₁ . . . a_i and b₁ . . . b_j such that

N_s =(a^s₁ +…+ a^s_i) − (b^s₁ +…+ b^s_j)

And each algebraic conjugate of an a (resp. b) also an a (resp. b).

The second conjecture is a functional equation:

Z(1/qⁿt) = ± q^nE/2t^EZ(t)

This says the operation x → qⁿ/x permutes the a’s (resp. the b’s).The third is a Riemann Hypothesis

Z(t) = (P₁(t)P₃(t) · · · P_2n−1(t))/(P₀(t)P₂(t) · · · P_2n(t))

where each P_k is an integer polynomial with all roots of absolute value q^−k/2. That means each a has absolute value q^k for some 0 ≤ k ≤ n. Each b has absolute value q^(2k−1)/2 for some 0 ≤ k ≤ n.

Over it all is the conjectured link to topology. Let B₀, B₁, . . . B_2n be the Betti numbers of the complex manifold defined by the same polynomials. That is, each B_k gives the number of k-dimensional holes or handles on the continuous space of complex number solutions to the equations. And recall an algebraically n-dimensional complex manifold is topologically 2n-dimensional. Then each P_k has degree B_k. And E is the Euler number of the manifold, the alternating sum

∑_k=0²ⁿ (−1)^kB_k

On its face the topology of a continuous manifold is worlds apart from arithmetic over finite fields. But the topology of this manifold tells how many a’s and b’s there are with each absolute value. This implies useful numerical approximations to the numbers of roots N_s. Special cases of these conjectures, with aspects of the topology, were proved before Weil, and he proved more. All dealt with curves (1-dimensional) or hypersurfaces (defined by a single polynomial).

Weil presented the topology as motivating the conjectures for higher dimensional varieties. He especially pointed out how the whole series of conjectures would follow quickly if we could treat the spaces of finite field solutions as topological manifolds. The topological strategy was powerfully seductive but seriously remote from existing tools. Weil’s arithmetic spaces were not even precisely defined. To all appearances they would be finite or (over the algebraic closures of the finite fields) countable and so everywhere discontinuous. Topological manifold methods could hardly apply.