Weil cohomology in practice

Note that one can also define the cohomology of \(X\) itself, without base change to an algebraically closed field, but in the context of these lectures this is not the construction of interest. An extreme case of this is taking the cohomology of a point \(\Spec k\text{,}\) which coincides with Galois cohomology (group cohomology for the Galois group of the field \(k\)).

Even after unpacking this, the definition of étale cohomology is not really suited for machine computations, as it involves constructions using Grothendieck topologies that are quite difficult to make finitistic. In fact, it is a true but nontrivial theorem that there is an algorithm which, given a scheme \(X\) of finite type over an algebraically closed field \(k\) and a prime \(\ell\) which is nonzero in \(k\text{,}\) computes the groups \(H^i(X, \FF_\ell)\) (see [92]). The strategy is to reduce to the case of curves (this strategy is often called dévissage; see Section 17.1), then apply Example 9.1.3.

In general, if \(X\) has good reduction at \(\frakp\), meaning that it extends in some way to a smooth proper scheme over \((\calO_K)_{\frakp}\text{,}\) then this extension is not guaranteed to be unique up to isomorphism. It is unique if \(\dim(X) = 1\text{,}\) and in some isolated cases of higher dimension (e.g., when \(X\) is an abelian variety); but in general, when \(\dim(X) \geq 2\) there can be multiple lifts which differ by a birational transformation. (Note that the lifts will have dimension \(\geq 3\text{,}\) so constructions like flips become relevant.)

In particular, the reduction modulo \(\frakp\) is not uniquely determined by \(X\text{.}\) However, its zeta function is independent of choices by Proposition 9.2.5 below.

The case where \(\frakp\) divides \(\ell\) behaves differently; in that case, even if \(X\) has good reduction at \(\frakp\text{,}\) the action of \(G_K\) on \(\ell\)-adic étale cohomology will not in general be unramified at \(\frakp\text{.}\) For example, this is already true for \(\PP^1\) with \(i=2\text{.}\)

Based on ideas of Tate, Fontaine managed to define a condition on \(p\)-adic Galois representations which is satisfied by étale cohomology in this context but “usually” fails for other representations; this is called the crystalline condition. Its study is part of the subject of \(p\)-adic Hodge theory, which we will not pursue here.

The case where the action of \(G_K\) is ramified at \(\frakp\text{,}\) but \(\frakp\) does not divide \(\ell\text{,}\) is also different. In this case, there is no well-defined action of \(\Frob_\frakp\) on all of \(H^{i}_{\et}(\frakX_{\overline{K}},\QQ_\ell)\text{,}\) only on the subspace invariant under the action of \(I_{\frakp}\text{.}\) The resulting action can be used to predict “missing” Euler factors needed to complete the \(L\)-function to achieve its functional equation (in addition to factors corresponding to infinite places, as in the case of Dedekind zeta functions).

However, these can be somewhat difficult to compute in practice. For elliptic curves, the relevant computation is Tate’s algorithm, which is somewhat of a nuisance to implement from scratch (but is fortunately implemented in many existing software packages). The corresponding algorithm for curves of genus 2 has been worked out, but is even more complicated. Beyond that, few general results are known, except in a few special cases like cyclic covers of \(\PP^1\) [12] (but see [36] for some recent progress on general curves).

For empirical purposes (i.e., where a rigorous proof is not required), one can sometimes short-circuit this issue by guessing a value for the missing Euler factors and then verifying numerically (by contour integration) that the resulting \(L\)-function appears to satisfy the correct functional equation. A more extreme version of this is [45], in which \(L\)-functions are detected by guessing all of their Euler factors up to some bound.

For general \(X\text{,}\) there is no converse of Proposition 9.2.2. For example, if \(X\) is a curve of genus 2, then \(H^i_{\et}(X_{\overline{K}}, \QQ_\ell)\) is unramified at \(\frakp\) if and only if the same is true with \(X\) replaced by its Jacobian (as this does not change \(H^1_{\et}\)). However, it is possible for the Jacobian of \(X\) to degenerate to a product of two elliptic curves (with the product polarization), which obstructs \(X\) itself from having good reduction (as otherwise taking the reduction would commute with taking the Jacobian, whereas the product of two elliptic curves cannot be a Jacobian).

One way to fix this is to consider “nonabelian étale cohomology”. Roughly speaking, this means that étale cohomology arises from some sort of abelianization process on homotopy groups, which we replace with a less drastic quotienting operation.

One might expect (as Grothendieck did) that Theorem 9.3.1 would be a structural property of étale cohomology that shows up relatively early in its development. Instead, it turns out that the only known proofs depend on the Riemann hypothesis aspect of the Weil conjectures!

Let \(A_1,A_2\) be abelian varieties over a field \(K\text{.}\) Then \(\Hom(A_1, A_2)\) is a finitely generated, torsion-free \(\ZZ\)-module; for \(K = \CC\) this is apparent from the action of endomorphisms on singular homology, and then it follows for any field of characteristic 0. For \(K\) of positive characteristic one can similarly argue using the \(\ell\)-adic Tate module for any prime \(\ell\) which is nonzero in \(K\text{.}\)

A consequence of the previous observation is that for any prime \(\ell\text{,}\) \(\Hom(A_1,A_2) \to \Hom(A_1, A_2) \otimes_{\ZZ} \ZZ_\ell\) is injective, and moreover \(\Hom(A_1, A_2) \neq 0\) if and only if \(\Hom(A_1, A_2) \otimes_{\ZZ} \ZZ_{\ell} \neq 0\text{.}\)