Weil I §6: A rationality theorem

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 06 July 2012

§6.1

Let ${\mathbb{P}}_0$ be a projective space of dimension $\ge 1$ on ${𝔽}_q, X_0\subseteq {\mathbb{P}}_0$ a nonsingular projective variety, $A_0\subseteq {\mathbb{P}}_0$ a linear subspace of codimension two, $D_0\subseteq {\stackrel{\vee }{\mathbb{P}}}_0$ the dual line, $\stackrel{—}{{𝔽}_q}$ an algebraic closure of ${𝔽}_q$ and $\mathbb{P},X,A,D$ on $\stackrel{—}{{𝔽}_q}$ obtained from ${\mathbb{P}}_0,X_0,A_0,D_0$ by extension of scalars. The diagram [De; 5.1.1] of [De; 5.6] provides an analogue of the diagram over ${𝔽}_q:$ $$\begin{array}{ll}\begin{array}{c} X0 X0˜ D0 π0 f0 \end{array}& (W6\; 6.1.1)\end{array}$$

Suppose $X$ is connected of even dimension $n+1=2m+2,$ and the pencil ${\left(X_t\right)}_{t\in D}$ of hyperplane sections of $X$ defined by $D$ is a Lefschetz pencil. The set $S$ of $t\in D$ such that $X_t$ is singular is defined over ${𝔽}_q,$ i.e. providing $S_0\subseteq D_0.$ We set $U_0=D_0-S_0$ and $U=D-S.$

Let $u\in U.$ The vanishing part of the cohomology $E\subseteq H^n(X_u,{\mathbb{Q}}_l)$ is stable under ${\pi }_1(U,u),$ thus defines over $U$ a local system $ℰ$ of $R^n{f}_{*}{\mathbb{Q}}_l.$ This last object is defined over ${𝔽}_q:$ $R^i{f}_{*}{\mathbb{Q}}_l$ is the reciprocal image of the ${\mathbb{Q}}_l-$sheaf $R^i{f_0}_{*}{\mathbb{Q}}_l$ on $D_0$ and, on $U, ℰ$ is the reciprocal image of a local system $${ℰ}_0\subseteq R^n{f_0}_{*}{\mathbb{Q}}_l.$$

The cup product is an alternating form $$\psi :R^n{f_0}_{*}{\mathbb{Q}}_l\otimes R^n{f_0}_{*}{\mathbb{Q}}_l\to {\mathbb{Q}}_l(-n).$$ Denoting the orthogonal complement to ${ℰ}_0$ relative to $\psi$ by ${ℰ}_0^{\perp },$ in $R^n{f_0}_{*}{\mathbb{Q}}_l{|}_{U_0}$ we see that $\psi$ induces a perfect duality $$\psi :\frac{{ℰ}_0}{{ℰ}_0\cap {ℰ}_0^{\perp }}\otimes \frac{{ℰ}_0}{{ℰ}_0\cap {ℰ}_0^{\perp }}\to {\mathbb{Q}}_l(-n).$$

For every $x\in \left|U_0\right|,$ the polynomial $\det (1-F_x^{*}t,\frac{{ℰ}_0}{{ℰ}_0\cap {ℰ}_0^{\perp }})$ has rational coefficients.

Let $j_0$ be the inclusion of $U_0$ in $D_0,$ and $j$ that of $U$ in $D.$ The eigenvalues of $F^{*}$ acting on $H^1(D,j_{*}\frac{ℰ}{ℰ\cap {ℰ}^{\perp }})$ are algebraic numbers such that all their complex conjugates satisfy $$q^{\frac{n+1}{2}-\frac{1}{2}}\le \left|\alpha \right|\le q^{\frac{n+1}{2}+\frac{1}{2}}.$$

By [De; 5.10] and Theorem 1.1 [De; 6.2], the hypotheses of [De; 3.2] are in fact verified for $(U_0,\frac{{ℰ}_0}{{ℰ}_0\cap {ℰ}_0^{\perp }},\psi )$ for $\beta =n,$ and we apply [De; 3.9].

Let ${𝒢}_0$ be a locally constant ${\mathbb{Q}}_l-$sheaf on $U_0$ such that its reciprocal image $𝒢$ on $U$ is a constant sheaf. Then there exist units ${\alpha }_i$ in $\stackrel{—}{{\mathbb{Q}}_l}$ such that for every $x\in \left|U_0\right|,$ we have $$\det (1-F_x^{*}t,{𝒢}_0)=\prod _i(1-{\alpha }_i^{\mathrm{deg}\left(x\right)}t).$$

This lemma expresses ${𝒢}_0$ as the reciprocal image of a sheaf on $\mathrm{Spec}\left({𝔽}_q\right)$ knowing its direct image on $\mathrm{Spec}\left({𝔽}_q\right).$ This last is identified with an $l-$adic representation $G_0$ of $\mathrm{Gal}(\stackrel{—}{{𝔽}_q}/{𝔽}_q)$ and we take $$\det (1-Ft,G_0)=\prod _i(1-{\alpha }_{i}t).$$

The Lemma 1.3 [De; 6.4] applies to $R^i{f_0}_{*}{\mathbb{Q}}_l$ $(i\ne n),$ to $R^n{f_0}_{*}{\mathbb{Q}}_l/{ℰ}_0$ and to ${ℰ}_0\cap {ℰ}_0^{\perp }.$

For $x\in \left|U_0\right|,$ the fibre $X_x=f_0^{-1}\left(x\right)$ is a variety over the finite field $k\left(x\right).$ If $\stackrel{—}{x}$ is a point of $U$ over $x,$ $X_{\stackrel{—}{x}}$ is deduced from $X_{\stackrel{—}{x}}$ by extension of scalars of $k\left(x\right)$ to its algebraic closure $k\left(\stackrel{—}{x}\right)=\stackrel{—}{{𝔽}_q},$ and $H^i(X_{\stackrel{—}{x}},{\mathbb{Q}}_l)$ is the fibre of $R^i{f}_{*}{\mathbb{Q}}_l$ at $\stackrel{—}{x}.$ The formula [De; 1.5.4] for the variety $X_x$ over $k\left(x\right)$ is thus written $$Z(X_x,t)=\prod _i\det {(1-F_x^{*}t,R^i{f_0}_{*}{\mathbb{Q}}_l)}^{{(-1)}^{i+1}},$$ and $Z(X_x,t)$ is the product of $$Z^f=\det (1-F_x^{*}t,R^n{f_0}_{*}{\mathbb{Q}}_l/{ℰ}_0)\cdot \det (1-F_x^{*}t,{ℰ}_0\cap {ℰ}_0^{\perp })\cdot \prod _{i\ne n}\det {(1-F_x^{*}t,R^i{f_0}_{*}{\mathbb{Q}}_l)}^{{(-1)}^{i+1}}$$ with $$Z^m=\det (1-F_x^{*}t,\frac{{ℰ}_0}{{ℰ}_0\cap {ℰ}_0^{\perp }}).$$

Putting ${ℱ}_0=\frac{{ℰ}_0}{{ℰ}_0\cap {ℰ}_0^{\perp }}, ℱ=\frac{ℰ}{ℰ\cap {ℰ}^{\perp }}$ and applying Lemma 1.3 [De; 6.4] to the factors $Z^f$ and $Z^m$ we find that there exist $l-$adic units ${\alpha }_i (1\le i\le N)$ and ${\beta }_j (1\le j\le M)$ in $\stackrel{—}{{\mathbb{Q}}_l}$ such that for every $x\in \left|U_0\right|$ $$Z(X_x,t)=\frac{\prod _i(1-{\alpha }_i^{\mathrm{deg}x}t)}{\prod _j(1-{\beta }_j^{\mathrm{deg}x}t)}\cdot \det (1-F_x^{*}t,{ℱ}_0)$$ and, in particular, the second factor is in $\mathbb{Q}\left(t\right).$ If ${\alpha }_i$ coincides with ${\beta }_j,$ it is reasonable to simultaneously remove this ${\alpha }_i$ from the list of $\alpha$ and this ${\beta }_j$ from the list of the $\beta .$ Thus we may assume, and we will assume, that ${\alpha }_i\ne {\beta }_j$ for all $i$ and all $j.$

§6.5

It suffices to prove that the polynomials $\prod _i(1-{\alpha }_{i}t)$ and $\prod _j(1-{\beta }_{j}t)$ have rational coefficients, i.e. that the family of ${\alpha }_i$ (resp. the family of ${\beta }_j$) is defined over $\mathbb{Q}.$ We will deduce the following propositions.

Let $\left({\gamma }_i\right) (1\le i\le P)$ and $\left({\delta }_j\right) (1\le j\le Q)$ be two families of $l-$adic units in $\stackrel{—}{{\mathbb{Q}}_l}.$ We assume that ${\gamma }_i\ne {\delta }_j.$ If $K$ is a very large finite set of integers $\ne 1,$ and $L$ is a very large subset of $\left|U_0\right|$ of density 0, then, if $x\in \left|U_0\right|$ satisfies $k\nmid \mathrm{deg}\left(x\right)$ (for every $k\in K$) and $x\notin L,$ the denominator of $$\begin{array}{ll}\frac{\det (1-F_x^{*}t,{ℱ}_0)\prod _i(1-{\gamma }_i^{\mathrm{deg}\left(x\right)}t)}{\prod _j(1-{\delta }_j^{\mathrm{deg}\left(x\right)}t)},& (W6\; 6.6.1)\end{array}$$ written in irreducible form, is $\prod _j(1-{\delta }_j^{\mathrm{deg}\left(x\right)}t).$

The proof will be given in [De; 6.10-13]: By Lemma 2.2 [De; 6.7] below, Proposition 2.1 [De; 6.6] furnishes an intrinsic description of the family of ${\delta }_j$ in terms of the rational fractions (W6 6.6.1) for $x\in \left|U_0\right|.$

Let $K$ be a finite set of integers $\ne 1$ and $\left({\delta }_j\right) (1\le j\le Q)$ and $\left({\epsilon }_j\right) (1\le j\le Q)$ two families of elements of a field. If, for $n$ very large, not divisible by any of the $k\in K$ the family of ${\delta }_j^n$ coincides with that of the ${\epsilon }_j^n$ (up to rearrangement), then the family of ${\delta }_j$ coincides with the family of ${\epsilon }_j$ (up to rearrangement).

We proceed by induction on $Q.$ The set of integers $n$ such that ${\delta }_Q^n={\epsilon }_j^n$ is an ideal $\left(n_j\right).$ We prove that there exists $j_0$ such that ${\delta }_Q={\epsilon }_{j_0}.$ If not the $n_j$ will be different from 1, and there exist arbitrarily large integers $n,$ not divisible by any of the $n_j,$ nor by any of the $k\in K.$ We will have ${\delta }_Q^n\ne {\epsilon }_j^n,$ and this will contradict the hypothesis. Thus there exists $j_0$ such that ${\delta }_Q={\epsilon }_{j_0}.$ We conclude by applying the induction hypothesis to the families $\left({\delta }_j\right) (j\ne Q)$ and $\left({\epsilon }_j\right) (j\ne j_0).$

Let $\left({\gamma }_i\right) (1\le i\le P)$ and $\left({\delta }_j\right) (1\le j\le Q)$ be two families of $l-$adic units in $\stackrel{—}{{\mathbb{Q}}_l},$ $R\left(t\right)=\prod _i(1-{\gamma }_{i}t)$ and $S\left(t\right)=\prod _j(1-{\delta }_{j}t).$ We assume that for every $x\in \left|U_0\right|$ $\prod _i(1-{\delta }_i^{\mathrm{deg}\left(x\right)}t)$ divides $$\prod _i(1-{\gamma }_i^{\mathrm{deg}\left(x\right)}t)\cdot \det (1-F_x^{*}t,{ℱ}_0).$$ Then $S\left(t\right)$ divides $R\left(t\right).$

Disregarding the pairs of common elements in the families $\left({\gamma }_i\right)$ and $\left({\delta }_j\right)$ verify the hypothesis of Proposition 2.1 [De; 6.6]. Apply Proposition 2.1 [De; 6.6]. By hypothesis, the rational fractions (W6 6.6.1) are polynomials. Thus no $\delta$ remains, which means that $S\left(t\right)$ divides $R\left(t\right).$

This proposition furnishes an intrinsic characterisation of $R\left(t\right)$ in terms of the family of polynomials $$\prod _i(1-{\gamma }_i^{\mathrm{deg}\left(x\right)}t)\cdot \det (1-F_x^{*}t,{ℱ}_0).$$ It is the lcm of the polynomials $S\left(t\right)=\prod _j(1-{\delta }_{j}t)$ which satisfies the hypothesis of Proposition 2.3 [De; 6.8].

§6.9

We prove [De; 6.5] and thus Theorem 1.1 [De; 6.2] (modulo Proposition 2.1 [De; 6.6]). Take $\left({\gamma }_i\right)=\left({\alpha }_i\right)$ and $\left({\delta }_j\right)=\left({\beta }_j\right)$ in Proposition 2.1 [De; 6.6]. We find an intrinsic characterization of the family of ${\beta }_j$ in terms of the family of rational fractions $Z(X_x,t) (x\in \left|U_0\right|).$ Since these are in $\mathbb{Q}\left(t\right),$ the family of ${\beta }_j$ is defined over $\mathbb{Q}.$

The polynomials $\prod _i(1-{\alpha }_i^{\mathrm{deg}\left(x\right)}t)\cdot \det (1-F_x^{*}t,{ℱ}_0)$ are thus in $\mathbb{Q}\left[t\right].$ Proposition 2.3 [De; 6.8] provides an intrinsic description of the family of ${\alpha }_i$ in terms of this family of polynomials. The family of ${\alpha }_i$ is thus defined over $\mathbb{Q}.$

§6.10

Preliminaries: Let $u\in U$ and ${ℱ}_u$ be the fibre of $ℱ$ at $u.$ The arithmetic fundamental group ${\pi }_1(U_0,u),$ extension of $\hat{\mathbb{Z}}=\mathrm{Gal}(\stackrel{—}{{𝔽}_q}/{𝔽}_q)$ (generator: $\phi$) by the geometric fundamental group ${\pi }_1(U,u),$ acts on ${ℱ}_u$ by symplectic similitudes: $$\rho :{\pi }_1(U_0,u)\to \mathrm{CSp}({ℱ}_u,\psi ).$$ We denote by $\mu \left(g\right)$ the multiplier of a symplectic similitude $g.$ Let $$H\subseteq \hat{\mathbb{Z}}\times \mathrm{CSp}({ℱ}_u,\psi )$$ be the subgroup defined by the equation $$q^{-n}=\mu \left(g\right)$$ ($q$ being an $l-$adic unit, $q^n\in {\mathbb{Q}}_l^{*}$ is defined for all $n\in \hat{\mathbb{Z}}$). The fact that $\psi$ has values in ${\mathbb{Q}}_l(-n)$ can be expressed by saying that the map from ${\pi }_1$ into $\hat{\mathbb{Z}}\times \mathrm{CSp},$ with coordinates the canonical projection on $\hat{\mathbb{Z}}$ and $\rho ,$ factors ${\rho }_1:{\pi }_1(U_0,u)\to H.$

The image $H_1$ of ${\rho }_1$ is open in $H.$

In fact, ${\pi }_1(U_0,u)$ projects onto $\hat{\mathbb{Z}},$ and the image of ${\pi }_1(U,u)=\mathrm{Ker}({\pi }_1(U_0,u)\to \hat{\mathbb{Z}})$ in $\mathrm{Sp}({ℱ}_u,\psi )=\mathrm{Ker}(H\to \hat{\mathbb{Z}})$ is open [De; 5.10].

For $\delta \in \stackrel{—}{{\mathbb{Q}}_l}$ an $l-$adic unit, the set $Z$ of $(n,g)\in H_1$ such that ${\delta }^n$ is an eigenvalue of $g$ is closed of measure 0.

It is clear that $Z$ is closed. For each $n\in \hat{\mathbb{Z}},$ let ${\mathrm{CSp}}_n$ be the set of $g\in \mathrm{CSp}({ℱ}_u,\psi )$ such that $\mu \left(g\right)=q^{-n},$ and let $Z_n$ be the set of $g\in {\mathrm{CSp}}_n$ such that ${\delta }^n$ is an eigenvalue of $g.$ Then ${\mathrm{CSp}}_n$ is a homogeneous space for $\mathrm{Sp},$ and we check that $Z_n$ is a proper algebraic subspace, thus of measure 0. By Lemma 4.1 [De; 6.11], $H_1\cap (\left\{n\right\}\times Z_n)$ is thus of measure 0 in the inverse image of $n$ in $H_1,$ and we apply Fubini to the projection $H_1\to \hat{\mathbb{Z}}.$

§6.13

We prove Proposition 2.1 [De; 6.6]. For each $i$ and $j,$ the set of integers $n$ such that ${\gamma }_i^n={\delta }_j^n$ is the set of multiples of a fixed integer $n_{ij}$ (we don't exclude $n_{ij}=0$). By hypothesis, $n_{ij}\ne 1.$

By Lemma 4.2 [De; 6.12] and the density theorem of Čebotarev, the set of $x\in \left|U_0\right|$ such that a ${\delta }_j^{\mathrm{deg}\left(x\right)}$ is an eigenvalue of $F_x^{*}$ acting on ${ℱ}_0$ is of density 0. We take for $K$ the set of $n_{ij}$ and for $L$ the set of $x$ as above.

References

[De] P. Deligne, La conjecture de Weil: I, Publications mathématiques de l'I.H.É.S., tome 43, (1974), p. 273-307.

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