Download Linking Discrete & Continuous in Algebraic Geometry: Weil Conjectures & Etale Cohomology and more Study notes Mathematics in PDF only on Docsity!
The Weil Conjectures and Etale Cohomology:´
A Hand-Waving Introduction
Andrew Potter
May 5, 2009
1 A Weil-gue recollection
Recall (from way back in lecture 15) the Weil Conjectures, a series of now-proved theorems about the zeta function of a variety over a finite field. Let X be a nonsingular, d-dimensional projective variety over the finite field Fq of q elements. Let Nk be the number of points on X over the field of qk elements. The zeta function of X is defined as
ζ(X, s) = exp
k=
Nk k
(q−s)k
Often we will want to make the substitution t = q−s. When we wish to consider the zeta function of X as a function of t, the definition becomes
Z(X, t) = exp
k=
Nk k
tk
With this set-up, we can now state two of the Weil Conjectures:
- Rationality: Z(X, t) is a rational function of t. More specifically, Z(X, t) has the following form:
Z(X, t) =
P 1 (t)P 3 (t)... P 2 d− 1 (t) P 0 (t)P 2 (t)... P 2 d(t)
where each Pi is a polynomial with integer coefficients, P 0 (t) = 1 − t, P 2 d(t) = 1 − qdt, and for 1 ≤ i ≤ 2 d − 1,
Pi(t) =
∏^ βi
j=
(1 − αi,j t),
where the αi,j are algebraic integers, and βi ∈ N for all i and j.
- Riemann Hypothesis: |αi,j | = qi/^2 for all i and j.
The other two assertions say that the zeta function satisfies a functional equation, just like all good zeta functions should, and that the βi are the topo- logical Betti numbers.
What is so astounding about the Weil Conjectures is that they provide a link between the discrete (the number of points on a variety over a finite field) and the continuous (topological notions such as Betti numbers). One of the great triumphs of algebraic geometry (or at least for those with an arithmetic bent) of the 20th century was proving these conjectures. In the rest of this talk, I hope to illustrate how they were proved by way of constructing a suitable cohomology theory. Since “cohomology” is one of those irritating words in maths which is casually slipped into conversation without ever explaining what it means, I’ll also attempt to give a crash course in (co)homology.
2 Rationality via Lefschetz
Although rationality was proved first using p-adic methods by Dwork in 1959, it was Grothendieck in 1964 who discovered the correct cohomology theory which proved everything except the Riemann hypothesis (which was proved by Deligne in 1973). This section will attempt to give an idea of why such a cohomology theory leads to rationality as an immediate corollary. To start with, we state an elemental result whose proof can be found in, for example, Ireland & Rosen’s A Classical Introduction to Modern Number Theory, page 155, Proposition 11.1.1.
Lemma. The zeta function is rational if and only if there exist complex numbers αi and βj such that
Nk =
j
βkj −
i
αki.
So it remains to find such αi and βj. What Weil himself noticed was that it was conceivable that such an expres- sion might be obtained from a formula from topology attributed to Lefschetz, called the Lefschetz Fixed-Point Formula.
Theorem. Lefschetz Fixed-Point Formula Let Y be a topological space and f : Y −→ Y a continuous mapping from Y to itself. Let Λf denote the number of fixed points of f , i.e. points y ∈ Y such that f (y) = y, counted with appropriate multiplicities. Then
Λf =
i
(−1)i^ Tr(f |Hi(Y, Q)).
What this theorem says is that the fixed points can be obtained from the trace of the induced linear map f |Hi(Y, Q) acting on the cohomology Q-vector spaces Hi(Y, Q) for each i. Now Weil’s insight was to realise that the points X(Fqk ) on a variety X over Fqk are precisely the fixed points of the qk-th power Frobenius map. That is, let X¯ denote X considered over an algebraic closure of Fqk , and let φqk : X¯ −→ X¯ be the Frobenius map which takes each coordinate of X¯ to its qk-th power. Then the fixed points of φqk are precisely the points of X(Fqk ). So, suppose that X had some sort of “nice” topology on it, or some suitably complicated mathematical structure comparable to topology, and that we could
Often the modules C^0 , C^1 , C^2 ,... are free modules over some ring, generated by some elements, and so we could construct them as free modules over any ring R. The ring R is often called the coefficient ring. To distinguish between cohomology groups with different coefficient rings, we write them as Hi(X, R). Functoriality. The operators Hn(∗) and Hn(∗) turn out to be functors (covariant and contravariant respectively), so that a morphism X −→ Y induces a morphism on the homology groups Hn(X) −→ Hn(Y ) or on the cohomology groups Hn(Y ) −→ Hn(X). Now if we look back at the Lefschetz fixed-point formula, the terms involved are far less mysterious. The function f : X −→ X is a continuous map, so for each i, f induces a Q-linear map f |Hi(X, Q) between the vector space Hi(X, Q) and itself, thus it has a trace.
4 Etale Cohomology and´ `-adic Cohomology
Now that we know what we are looking for, here is a summary of the key points in the search for the “correct” cohomology theory which proves the Weil conjectures. No Nice Topology! The main problem was that for X a variety over a finite field, no one could come up with a topology on X which was nice enough to use existing cohomology theories. Enter the ´Etale Category. Grothendieck attacked this problem by re- placing the notion of “topology”, which can be thought of as the category of open sets on X, with the ´etale category on X, Et(´ X). Etale Cohomology.^ ´ With this generalisation, it is possible to construct a cohomology theory, called ´etale cohomology, which has certain “nice” properties when the coefficient ring is Z/nZ for n coprime to p, the characteristic of the finite field. -adic Cohomology. However, in order to find “the” cohomology theory, one requires to consider the-adic cohomology, constructed from ´etale cohomol- ogy: Let be a prime not equal to p. For each ´etale cohomology group Hi(X, Z/kZ), define the -adic cohomology group Hi(X, Z) to be the inverse limit: Hi(X, Z) = lim ← Hi(X, Z/kZ).
Finally, we eventually reach the correct cohomology groups that will allow us to use the Lefschetz fixed-point formula, and hence prove three of the four Weil conjectures: Hi(X, Q) = Hi(X, Z) ⊗ Q`. Exercise: Now prove the Riemann hypothesis.
