1. Weil's conjectures
For $ \operatorname{Re}(z)>1 $, write
\[\zeta(z) = \sum_{n=1}^{\infty} \frac{1}{n^z}\]for the Riemann zeta function. Its Dirichlet series and Euler product converge in this half-plane; the logarithmic-derivative and Euler-product identities below hold there, while the functional equation refers to its meromorphic continuation.
- The logarithmic derivative is \( \frac{d}{dz} \ln(\zeta(z)) = -\sum_{n=1}^{\infty} \Lambda(n) n^{-z} \), where \( \Lambda(n) \) is the von Mangoldt function, equal to \( \ln p \) when \( n=p^m \) for a prime \(p\) and an integer \(m\geq 1\), and \(0\) otherwise.
- It satisfies the functional equation \[ \zeta(z) = 2^{z} \pi^{z-1} \sin(\pi z/2) \Gamma(1-z) \zeta(1-z). \]
- It has an Euler product expansion \[ \zeta(z) = \prod_{p \text{ prime}} \frac{1}{1-p^{-z}}. \]
Let \(V\) be a smooth, projective, geometrically connected variety of pure dimension \(n\) over the finite field \(k=\FF_q\). For instance, a homogeneous polynomial in \(n+2\) variables whose projective zero locus is smooth and geometrically connected defines such a variety as a hypersurface in \(\PP_{\FF_q}^{n+1}\). Write \(N_d=\#V(\FF_{q^d})\).
In his 1949 paper [5], submitted in 1948, Weil formulated conjectures about the zeta function
\[Z(V,U)=\exp\left(\sum_{d=1}^{\infty}N_d\frac{U^d}{d}\right).\]He was guided by the case of curves and by computable higher-dimensional examples such as Grassmannians. These conjectures are now theorems.
Weil Conjectures. With \(V\) and \(Z\) as above, the following are true:
- The logarithmic derivative of \( Z(V,U) \) is the generating function for the \( N_d \), meaning \[ \sum_{d=1}^{\infty} N_d U^{d-1} = \frac{d}{dU} \ln(Z(V,U)). \] Furthermore, \( Z(V,U) \) is a rational function.
- \( Z(V,U) \) satisfies the functional equation \[ Z(V,(q^n U)^{-1}) = \pm q^{n \chi/2} U^{\chi} Z(V,U), \] where \( \chi \) is the Euler characteristic of our variety (see Remark 1.1).
- We have \[ Z(V,U) = \frac{P_1(U) P_3(U) \cdots P_{2n-1}(U)}{P_0(U) P_2(U) \cdots P_{2n}(U)}, \] where \( P_i(U)\in\mathbb Z[U] \), \( P_0(U)=1-U \), \(P_{2n}(U)=1-q^nU\), and \[P_i(U)=\prod_{j=1}^{B_i}(1-\alpha_{i,j}U).\] The \( \alpha_{i,j} \) are algebraic integers, and every complex embedding \( \iota \) satisfies \( |\iota(\alpha_{i,j})|=q^{i/2} \).
- The integers \( B_i=\dim_{\QQ_\ell}H^i_{\mathrm{\acute et}}(V_{\overline{\FF}_q},\QQ_\ell) \) are the Betti numbers of \(V\); they are independent of \(\ell\neq\operatorname{char}\FF_q\) and satisfy \( \chi=\sum_i(-1)^iB_i \).
Remark 1.1. For a smooth projective variety, the Euler characteristic \(\chi\) equals the self-intersection number \(I(\Delta,\Delta)\) of the diagonal in \(V\times V\). More generally, when the intersection is proper, \(I(\operatorname{graph}(f),\Delta)\) counts fixed points with intersection multiplicity. The identity map has a non-isolated fixed locus, so its self-intersection is understood intersection-theoretically rather than as a literal count of fixed points.
Remark 1.2. Write \(M\) for a smooth compact oriented manifold. Then we can again define \(\chi(M) = I(\Delta, \Delta)\). This quantity is related to the Betti numbers \(b_i = \dim_{\QQ} H_i(M, \QQ)\) by Corollary 2.1.
Example 1.3. Let \(\Gr(m,n)\) be the Grassmannian of \(m\)-dimensional subspaces of an \(n\)-dimensional vector space. Set \(k=\FF_p\) and \(V=\Gr(m,n)_{\FF_p}\). Writing \(Q=p^d\), a well-known formula gives
\[N_d=\#\Gr(m,n)(\FF_Q) =\prod_{j=0}^{m-1}\frac{Q^n-Q^j}{Q^m-Q^j}.\]Take \(V=\Gr(1,2)_{\FF_p}=\PP_{\FF_p}^1\). Solving for \(Z(U)\) in (i),
\[\begin{aligned} Z(U) & = \exp \left( \sum_{d=1}^{\infty} (p^d+1) \frac{U^d}{d} \right) \\ & = \exp \left( \sum_{d=1}^{\infty} \frac{(pU)^d}{d} + \sum_{d=1}^{\infty} \frac{U^d}{d} \right) \\ & = \exp ( -\ln(1-pU) - \ln(1-U) ) \\ & = \frac{1}{(1-U)(1-pU)}, \end{aligned}\]since
\[\sum_{n=1}^{\infty} \frac{x^n}{n} = -\ln(1-x)\]for \(x\) near \(0\).
The displayed factorization verifies (iii), with \(B_0=1\), \(B_1=0\), and \(B_2=1\). Thus (iv) gives \(\chi=1-0+1=2\), agreeing with the Euler characteristic of the Riemann sphere in the complex case.
Finally, we verify (ii) by computing
\[\begin{aligned} Z((pU)^{-1}) & = \frac{1}{(1-(pU)^{-1})(1-p(pU)^{-1})} \\ & = pU^2 \frac{1}{(pU-1)(U-1)} = pU^2 Z(U). \end{aligned}\]We conclude that the Weil Conjectures hold for \(\PP_{\FF_p}^1\).
2. Lefschetz theory
Fix an algebraic closure \(\overline{\FF}_p\) of \(\FF_p\). An element \(a\in\overline{\FF}_p\) belongs to the subfield \(\FF_q\) with \(q=p^n\) elements precisely when it is fixed by the \(q\)-power Frobenius \(F_q(a)=a^q\). This is an extension of Fermat's little theorem: every nonzero \(a\in\FF_p\) satisfies \(a^{p-1}=1\), and \(\FF_q\) is the set of roots of \(X^q-X\).
Hence, with our previous setup, we can compute \(N_d\) by counting the fixed geometric points of \(F_q^d\): on coordinates, \(F_q^d(x)=x^{q^d}\). In particular, the point-counting problem becomes a fixed-point problem.
It was well known when Weil published his conjectures that algebraic topology can be used to prove the existence of and count fixed points. The following is a simple example:
Brouwer fixed point theorem. Let \(f \colon D^2 \to D^2\) be a continuous map from the disk to itself. Then \(f\) has a fixed point.
Proof. Suppose \(f\) has no fixed points. Let \(h \colon D^2 \to S^1\) be the map which sends \(x\) to the point on \(S^1\) which intersects the ray starting at \(f(x)\) and passing through \(x\). Since \(f\) has no fixed points this map is well-defined and continuous. Writing \(\iota \colon S^1 \to D^2\) for the inclusion, we see \(h \circ \iota = \text{Id}_{S^1}\). But \(\pi_1(S^1)=\ZZ\) and \(\pi_1(D^2)=0\) imply that \(\pi_1(h\circ\iota)\) factors through zero, contradicting that it is the identity on \(\ZZ\). \(\blacksquare\)
Our main use of algebraic topology will be Lefschetz theory, as it allows us to write zeta functions using trace formulas. Let \(M\) be a compact oriented manifold and \(f\colon M\to M\) smooth. We call the oriented intersection
\[L(f) = I(\text{graph}(f),\Delta)\]the global Lefschetz number of \(f\). If the graph is transverse to the diagonal, its local contribution at a fixed point \(x\) is
\[L_x(f)=\operatorname{sgn}\det(I-Df_x);\]in general one uses the local fixed-point index. The global Lefschetz number is related to singular homology by the Lefschetz-Hopf theorem.
Lefschetz-Hopf theorem. We have the equality
\[L(f) = \sum_{i=0}^{\infty} (-1)^i \text{Tr}(f_* \mid H_i(M, \QQ)).\]Corollary 2.1. Let \(b_i = \dim_{\QQ} H_i(M, \QQ)\) be the \(i\)th Betti number of our manifold. Then \(\chi(M) = \sum_i (-1)^i b_i\).
Proof. As in Remark 1.2 let \(f\) be the identity. Then \(f_*\) is the identity, and hence
\[\text{Tr}(f_* \mid H_i(M, \QQ)) = \dim_{\QQ} H_i(M, \QQ).\]Our result follows. \(\blacksquare\)
Suppose now that the fixed points of \(f\) are isolated, and let \(L_x(f)\) denote the local fixed-point index. Then
\[\sum_{f(x) = x} L_x(f) = L(f),\]and combining this with the Lefschetz-Hopf Theorem we arrive at the following.
Corollary 2.2. We have the equality
\[\sum_{f(x)=x} L_x(f) = \sum_{i=0}^{\infty} (-1)^i \text{Tr}(f_* \mid H_i(M, \QQ)).\]3. Étale cohomology
In order to develop a Lefschetz theory for algebraic varieties we need a cohomology theory analogous to singular cohomology. For a locally contractible topological space, sheaf cohomology with coefficients in a constant sheaf recovers singular cohomology, but the Zariski topology of a scheme is too coarse for this purpose. Grothendieck's solution was to use the étale site: its covering families consist of étale maps, and étale cohomology is the derived functor cohomology of sheaves on this site. The following results can be found in SGA 4 1/2.
Let \(X_0\) be a separated scheme of finite type over \(\FF_q\), so it is covered by spectra of finitely generated \(\FF_q\)-algebras, and put
\[X=X_0\times_{\FF_q}\operatorname{Spec}\overline{\FF}_q.\]Write \(A_0\) for a constructible \(\QQ_\ell\)-sheaf on \(X_0\), where \(\ell\neq\operatorname{char}\FF_q\), and \(A\) for its pullback to \(X\). We use geometric Frobenius \(F\) on geometric points, normalized so that the fixed points of \(F^d\) are exactly \(X_0(\FF_{q^d})\).
An analogue of Corollary 2.2 is the following:
Theorem 3.1. For every integer \(n\geq1\),
\[\sum_{F^n(x) = x} \text{Tr}(F^n, A_x) = \sum_{i=0}^{\infty}(-1)^i \text{Tr}(F^n, H_c^i(X, A)).\]Likewise we can prove the following product expansion of our zeta function using trace formulas. Set
\[\zeta_{X_0}(z) = \prod_{x \in |X_0|} (1-\text{Card}(k(x))^{-z})^{-1},\]where \(\lvert X_0\rvert\) is the set of closed points of \(X_0\). If we set \(\deg(x)=[k(x):\FF_q]\) and \(U=q^{-z}\), we can write
\[\zeta_{X_0}(z) = Z_{X_0}(q^{-z}),\]where
\[Z_{X_0}(U) = \prod_{x \in |X_0|} (1-U^{\deg(x)})^{-1}.\]Again \(Z_{X_0}(U)\) converges for \(U\) small.
Theorem 3.2. We have
\[Z_{X_0}(U) = \prod_{i=0}^{2\dim X_0} \det(1-FU\mid H_c^i(X,\QQ_\ell))^{(-1)^{i+1}}.\]Since these cohomology groups vanish for \(i>2\dim X_0\), the zeta function is a rational function.
4. References
- Pierre Deligne. "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS, Volume 43 (1974), pp. 273-307. Link
- Pierre Deligne, Sga 4 1/2: Cohomologie étale, Springer, 1977.
- Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, 1974.
- Robin Hartshorne, Algebraic geometry, Springer, 1977.
- André Weil. "Numbers of solutions of equations in finite fields". Bulletin of the American Mathematical Society, 55(5) pp. 497-508, May 1949.