Weil’s cohomological metaconjecture

Chapter 3 Weil’s cohomological metaconjecture

Original lecture date: October 7, 2019.

In the previous lecture, we stated the Weil conjectures for an algebraic variety (or a scheme of finite type) \(X\) over a finite field \(\FF_q\text{,}\) which imply that the zeta function

\begin{equation*} Z(X,T)=\exp\left(\sum\limits_{n=1}^{\infty}\frac{T^n}{n}\# X(\FF_{q^n}) \right) \end{equation*}

has properties that we identified as follows:

(rationality)

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(functional equation)

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(Riemann Hypothesis)

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(Betti numbers)

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As we pointed out in Remark 2.2.5, Weil went further and suggested an approach to these conjectures inspired by algebraic topology. In this lecture, we explain this approach.

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Readings 3.0.1.

We continue to follow [63], Appendix C.

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Section 3.1 The metaconjecture

Definition 3.1.1.

Let \(R\) be a commutative algebra over \(\FF_{q}\text{;}\) then the map \(x\rightarrow x^q\) is an \(\FF_{q}\)-homomorphism from \(R\) to itself. For any scheme \(X\) over \(\FF_{q}\text{,}\) this construction induces a morphism \(F\colon X\rightarrow X\) of \(\FF_{q}\)-schemes, called the absolute Frobenius of X (more precisely, of \(X\) over \(\FF_q\)). One easily sees that we have an action of \(F\) on the set \(X(\overline{\FF_{q}})\text{,}\) whose set of fixed points is exactly \(X(\FF_{q})\text{.}\) Also if we consider the action of \(F^n\) on \(X(\overline{\FF_{q}})\text{,}\) then the fixed points would be \(X(\FF_{q^n})\text{.}\)

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Remark 3.1.2.

The inspiration for what follows is the general principle that the problem of counting fixed points of a self-map on a space should have something to do with computing traces of some associated linear map. A simple example of this principle is the following: if \(\sigma\) is a permutation of \(\{1,\dots,n\}\text{,}\) then the number of fixed points of \(\sigma\) is equal to the trace of the permutation matrix assocated to \(\sigma\text{.}\)

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A vastly more sophisticated example is the Lefschetz trace formula. Let \(T:S\rightarrow S\) be a continuous self-map of a topological space. Under suitable conditions, the quantity

\begin{equation*} \sum_{i}(-1)^i \Trace(T,H^{i}(S)) \end{equation*}

gives a weighted count of the fixed points of \(T\text{;}\) in particular, the nonvanishing of this quantity can be used to establish the existence of a fixed point of \(T\) (as in the Brouwer fixed point theorem).

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With the above considerations Weil proposed the following. (This statement can and should be refined somewhat, see below.)

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Metaconjecture 3.1.3. Weil.

For some field \(K\) of characteristic \(0\text{,}\) there is a series of contravariant “cohomology” functors

\begin{equation*} H^{i}:\{\text{algebraic varieties over} \ \FF_{q} \}\rightarrow \{\text{finite dimensional vector spaces over} \ K \} \end{equation*}

satisfying the following formula: for \(i=0,\dots,2d = 2\dim(X)\text{,}\) satisfying the formula

\begin{equation*} \#X(\FF_{q^n})=\sum\limits^{2d}_{i=0}(-1)^i \Trace(F^n\vert H^{i}(X) ) \end{equation*}

for every positive integer \(n\text{,}\) where \(F^n: H^{i}(X) \to H^{i}(X)\) denotes (by abuse of notation) the linear transformation induced by the morphism \(F^n: X \to X\text{.}\) (One can also formulate a similar metaconjecture in terms of a sequence of covariant “homology” functors \(H_{i}\text{.}\))

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Section 3.2 Remarks about the metaconjecture

Remark 3.2.1.

Let us see what the metaconjecture says, or could say with some refinement, about the Weil conjectures.

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Firstly, it immediately implies rationality because

\begin{equation*} Z(X,T)=\prod\det(1-FT, H^{i}(X))^{(-1)^{i+1}}. \end{equation*}

Note that here, we use crucially that \(K\) is of characteristic 0; otherwise, we would only get this relation modulo the characteristic of \(K\text{.}\)

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Secondly, the functional equation would hold if the functors \(H^i(X)\) satisfied “Poincaré Duality”, in the sense of admitting a perfect, \(F\)-equivariant pairing

\begin{equation*} H^i(X)\times H^{2d-i}(X)\rightarrow K(-d) \end{equation*}

where \(K(-d)\) denotes the field \(K\) with the “twisted” \(F\)-action, sending \(1\) to \(q^d\text{.}\)

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Thirdly, the Betti number statement would follow from an equality of dimensions between our \(H^i(X)\) and the usual singular cohomology groups of the analytification.

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It is not clear where the Riemann hypothesis would come from in this framework. We will discuss this later.

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Let us note that we haven’t talked much about the field of coefficients \(K\) which plays an important role in our cohomology theory here (except to note that it must be of characteristic \(0\)). The following example shows that we cannot hope to take \(K = \QQ\text{.}\)

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Example 3.2.2.

Suppose the metaconjecture holds for some \(K\text{.}\) Let \(X/\FF_q\) be a supersingular elliptic curve; we then have an action of \(\End(X)\) on \(H^1(X)\text{.}\) As we have seen in the previous lecture, \(H^1(X)\) is of dimension 2. However, if the endomorphisms of \(X_{\overline{\FF_q}}\) are all defined over \(\FF_q\text{,}\) then \(\End(X)\) is a \(\ZZ\)-module of rank 4 contained in a (nonsplit) quaternion algebra over \(\QQ\text{,}\) and a quaternion algebra over \(\QQ\) cannot act on a 2-dimensional \(\QQ\)-vector space unless it splits (i.e., is isomorphic to the matrix ring \(\mathrm{M}_2(\QQ)\)). Thus we cannot have \(K = \QQ\text{.}\)

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In this example, the quaterion algebra in question remains nonsplit after tensoring over \(\QQ\) with either \(\RR\) or \(\QQ_p\) (where \(p\) is the characteristic of \(\FF_q\)). Consequently, the same argument rules out the possibility of satisfying the metaconjecture with \(K=\RR\) or \(K = \QQ_p\) (but it does not rule out extensions of these fields).

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Remark 3.2.3.

There are essentially two known approaches to constructing a Weil cohomology theory over a finite field of characteristic \(p\text{.}\)

For \(K = \QQ_{\ell}\) where \(\ell \neq p\) is prime (which is not precluded by Example 3.2.2), the construction of étale cohomology by Grothendieck et al. will satisfy the metaconjecture.

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For \(K=\overline{\QQ}_{p}\text{,}\) the construction of rigid cohomology developed by Berthelot et al. will satisfy the metaconjecture. (Note that we cannot take \(K = \QQ_p\) because of Example 3.2.2.)

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