Chapter 2 The Weil conjectures and examples
Original lecture date: October 2, 2019.
In this lecture, we give the full statement of Weil’s conjecture together with some small examples.
Remark 2.0.2.
“projective”“proper”Theorem 2.1.2Section 2.1 Formulation of the conjectures
Definition 2.1.1.
Let \(k=\FF_q\) be a finite field of \(q\) elements, and \(X/k\) be a quasi-projective algebraic variety (or more generally, any \(k\)-scheme of finite type; we will add hypotheses later in the statement). For any integer \(r\geq 1\text{,}\) let \(k_r = \FF_{q^r}\) be one field extension of \(k\) with degree \(r\) (which is unique up to noncanonical isomorphism). Write the zeta function \(\zeta_X(s)\) as \(Z(X, q^{-s})\) for
\begin{equation*}
Z(X,T) \colonequals \exp\left(\sum_{r=1}^{\infty}\frac{T^r}{r}\# X\left(k_r\right)\right) \in \ZZ\llbracket T \rrbracket.
\end{equation*}
The containment \(Z(X,T) \in \QQ\llbracket T \rrbracket\) is more obvious here, but the prior description of \(\zeta_X(s)\) as an infinite product (Definition 1.3.3) shows that \(Z(X,T) \in \ZZ \llbracket T \rrbracket\text{.}\)
Theorem 2.1.2. Weil conjectures.
The series \(Z(X,T)\) has the following properties.
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Rationality.The series \(Z(X,T)\) represents a rational function of \(T\text{.}\) We will often make a minor misuse of language and say that \(Z(X,T)\) is a rational function of \(T\text{.}\)
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Functional equation.Suppose in addition that \(X\) is of pure dimension \(n\text{.}\) Then\begin{equation*} Z\left(X,\frac{1}{q^n T}\right) = \pm q^{nE/2}T^E Z(X,T) \end{equation*}for some integer \(E\text{.}\)
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Analogue of the Riemann hypothesis.Suppose in addition that \(X\) is smooth and projective (see Remark 2.0.2 )of (pure) dimension \(n\text{.}\) Then there is a unique factorization\begin{equation*} Z(X,T) = \frac{P_1(T)\cdots P_{2n-1}(T)}{P_0(T)\cdots P_{2n}(T)} \end{equation*}in which \(P_i(T)\) factors over \(\CC\) as \(\prod_j (1-\alpha_{ij} T)\) where \(|\alpha_{ij}| = q^{i/2}\) for each \(j\) (see Remark 2.1.3). In particular, the integer \(E\) from (2) equals\begin{equation*} E=\sum(-1)^i \deg(P_i). \end{equation*}Moreover, if \(X\) is geometrically irreducible, then \(P_0(T) = 1-T\) and \(P_{2n}(T) = 1-q^n T\text{.}\)
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Betti numbers.Suppose in addition that there exist a number field \(K\text{,}\) a finite set \(S\) of prime ideals of \(\calO_K\text{,}\) a maximal ideal \(\frakp\) of \(\calO_K\) not contained in \(S\) with residue field isomorphic to \(k\text{,}\) and a smooth projective scheme \(\frakX\) over \(\calO_{K,S}\) (the localization of \(\calO_K\) at the primes in \(S\)) such that \(X\) is isomorphic to the base extension \(\frakX \times_{\calO_{K,S}} k\text{.}\) (Informally, \(X\) is the “reduction of \(\frakX\) modulo \(\frakp\text{.}\)”) Then for any embedding \(K\rightarrow \CC\text{,}\) the \(i\)-th Betti number of the topological space \({( \frakX \times_{\calO_{K,S}} \CC)}^{\analytic}\) equals \(\deg(P_i)\text{.}\)
Remark 2.1.3.
In the Riemann hypothesis clause of Theorem 2.1.2, each factor \(P_i(T)\) is itself a polynomial over \(\ZZ\) with constant term 1. Consequently, each \(\alpha_{ij}\) is an algebraic integer whose conjugates in \(\CC\) all have absolute value \(q^{1/2}\text{;}\) such integers are often called Weil numbers.
Remark 2.0.6.
In the Betti numbers clause of Theorem 2.1.2, the choice of an embedding \(K \hookrightarrow \CC\) is there for more than just formal reasons: while the Betti numbers of \({( \frakX \times_{\calO_{K,S}} \CC)}^{\analytic}\) are independent of the choice, the structure of this space can vary significantly as one varies the embedding. For example, a famous example of Serre [112] shows that the fundamental group of the space can vary, so in particular the homotopy type can vary. See [40] for further discussion of this phenomenon, and [99], [107] for additional examples coming from Shimura varieties. (See also the introduction to [107] for references to even more examples.)
Remark 2.0.7.
In the Betti numbers clause of Theorem 2.1.2, the liftability hypothesis is also there for more than just formal reasons: while there are a few classes of varieties for which it always holds, such as curves, abelian varieties, and K3 surfaces (see [32] for this case) there do exist examples of nonliftable varieties. See [111] for the first known example and [124] for a much more robust example (a surface which cannot even be dominated by any liftable smooth proper variety). See also Remark 15.1.12 and Remark 12.1.10.
Section 2.2 Examples and remarks
Example 2.2.1.
If \(X\) is set-theoretically the disjoint union of an open subscheme \(Y\) and a closed subscheme \(S\text{,}\) then \(X(k_r)\) is likewise the disjoint union of \(Y(k_r)\) and \(S(k_r)\text{,}\) so formally
\begin{equation*}
Z(X,T)=Z(Y,T)\cdot Z(S,T).
\end{equation*}
Let us apply this to the decomposition \(\PP^n= \AAA^n\sqcup\PP^{n-1}\text{.}\) We obtain:
\begin{equation*}
Z(\PP^n,T)=Z(\AAA^n,T)\cdot Z(\PP^{n-1},T)=\frac{1}{1-q^nT}\cdot Z(\PP^{n-1},T).
\end{equation*}
In particular, as we have seen before,
\begin{equation*}
Z(\PP^1,T) = \frac{1}{(1-T)(1-qT)}
\end{equation*}
and similarly for \(\PP^n\) (see Exercise 19.2.2).
Example 2.2.2.
For \(X = C\) an elliptic curve, it can be shown by (relatively) elementary methods that
\begin{equation*}
Z(C,T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}
\end{equation*}
where \(a\) is an integer depending on \(C\text{.}\) It was shown by Hasse that moreover \(|a| \leq 2q^{1/2}\text{;}\) see [116], Chapter V for an efficient proof.
Remark 2.2.3.
Let’s see in detail what the Weil conjectures say for \(\PP^1\) and \(C\text{.}\)
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Rationality is obviously true in both cases.
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The functional equation for \(\PP^1\text{:}\)\begin{equation*} Z(\PP^1,\frac{1}{qT})=\frac{qT^2}{(1-T)(1-qT)}\quad E=2. \end{equation*}The functional equation for \(C\text{:}\)\begin{equation*} Z(C,\frac{1}{qT})=\frac{1-aT+qT^2}{(1-T)(1-qT)}\quad E=0. \end{equation*}
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The factorization for \(\PP^1\) is obvious, and the analogue of the Riemann hypothesis carries no new information. The factorization for \(C\) gives something nontrivial:\begin{equation*} P_i (T)=\ \begin{cases} 1-T & i=0 \\ 1-aT+qT^2 & i=1 \\ 1-q T & i=2. \\ \end{cases} \end{equation*}The analogue of the Riemann hypothesis asserts that the roots of \(P_1(T)\) lie on the circle \(|T|=q^{-1/2}\text{;}\) given the shape of the factorization, this is equivalent to the Hasse bound.
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The Betti numbers of a topological \(\PP^1\) are \(1,0,1\text{.}\) The Betti numbers of a topological elliptic curve are \(1,2,1\text{.}\)
Remark 2.2.4.
The factorization assertion was largely inspired by the example of Fermat hypersurfaces considered in Section 1.4. In that example, the numbers \(\alpha_{ij}\) are the products of Gauss sums appearing in Weil’s formula.
Remark 2.2.5.
The Betti number statement is a proxy for a stronger statement that Weil was not in a position to formulate precisely: what we wanted is to have \(P_i(T)=\det(1-FT,V_i)\) where \(V_i\) is some “naturally occurring” vector space over a field and \(F:V_i\rightarrow V_i\) is some endomorphism of the vector space. This perspective gives rise to the notion of Weil cohomology around which this course is centered.
But before we get there, note that the Betti number statement has a fair bit of power on its own. One important example computed by Weil in [132] is that of Grassmannian varieties, whose points correspond to subspaces of a fixed vector space. It is elementary to compute the number of points on a Grassmannian over a finite field (see Exercise 19.2.4); according to the Weil conjectures, this should then predict the Betti numbers of a Grassmannian over \(\CC\text{.}\) These had been computed previously by Ehresmann using totally different methods.
Now that the Weil conjectures are a theorem, one can go further with this logic: in some cases, the first known computation of the Betti numbers of a topological space have used the Weil conjecture. A famous example is the Hilbert schemes of points on a smooth projective surface, by Göttsche [51].
Section 2.3 From conjectures to theorems
We conclude this lecture with a very brief summary of how the Weil conjectures became a theorem. We will spend much of the course partially unpacking this summary.
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The rationality was first proved in 1958 by Dwork [39] using an interpretation of \(Z(X,T)\) in terms of \(p\)-adic analysis (where \(p\) is the characteristic of the finite field).
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During the 1960s, Grothendieck [58], [59] led a heroic effort to develop modern foundations of algebraic geometry, including a theory of étale cohomology that was meant to simulate the role of topological (singular) cohomology for complex algebraic varieties. This led to a new proof of rationality (via a form of the Lefschetz trace formula as per Remark 2.2.5), together with the first proofs of the functional equation (arising from Poincaré duality) and the Betti number condition (arising from a comparison theorem with singular cohomology).
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Grothendieck proposed an approach to the analogue of the Riemann hypothesis via the so-called “Standard Conjectures” [60], but this approach never bore fruit.
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An important simplification of “Weil II” was discovered by Laumon [84], inspired by the stationary phase approximation from classical analysis.
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Subsequently, Dwork’s methods were adapted to give a parallel cohomology theory, again based on \(p\)-adic analysis, in which the entire étale-cohomological proof of the Weil conjectures can be emulated. For example, a \(p\)-adic adaption of Laumon’s argument was given by Kedlaya [73].
