## Weil I §6: A rationality theorem

Last update: 06 July 2012

## §6.1

Let ${ℙ}_{0}$ be a projective space of dimension $\ge 1$ on a nonsingular projective variety, ${A}_{0}\subseteq {ℙ}_{0}$ a linear subspace of codimension two, ${D}_{0}\subseteq {\stackrel{\vee }{ℙ}}_{0}$ the dual line, $\stackrel{—}{{𝔽}_{q}}$ an algebraic closure of ${𝔽}_{q}$ and $ℙ,X,A,D$ on $\stackrel{—}{{𝔽}_{q}}$ obtained from ${ℙ}_{0},{X}_{0},{A}_{0},{D}_{0}$ by extension of scalars. The diagram [De; 5.1.1] of [De; 5.6] provides an analogue of the diagram over ${𝔽}_{q}:$ $X0 X0˜ D0 π0 f0 (W6 6.1.1)$

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 $E\subseteq {H}^{n}\left({X}_{u},{ℚ}_{l}\right)$ is stable under ${\pi }_{1}\left(U,u\right),$ thus defines over $U$ a local system $ℰ$ of ${R}^{n}{f}_{*}{ℚ}_{l}.$ This last object is defined over ${𝔽}_{q}:$ ${R}^{i}{f}_{*}{ℚ}_{l}$ is the reciprocal image of the ${ℚ}_{l}-$sheaf ${R}^{i}{{f}_{0}}_{*}{ℚ}_{l}$ on ${D}_{0}$ and, on is the reciprocal image of a local system $ℰ0 ⊆ Rnf0*ℚl.$

The cup product is an alternating form $ψ: Rnf0*ℚl⊗Rnf0*ℚl → ℚl(-n).$ Denoting the orthogonal complement to ${ℰ}_{0}$ relative to $\psi$ by ${ℰ}_{0}^{\perp },$ in ${R}^{n}{{f}_{0}}_{*}{ℚ}_{l}{|}_{{U}_{0}}$ we see that $\psi$ induces a perfect duality $ψ: ℰ0 ℰ0∩ℰ0⊥ ⊗ ℰ0 ℰ0∩ℰ0⊥ → ℚl(-n).$

For every $x\in |{U}_{0}|,$ the polynomial $\mathrm{det}\left(1-{F}_{x}^{*}t,\frac{{ℰ}_{0}}{{ℰ}_{0}\cap {ℰ}_{0}^{\perp }}\right)$ 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 ${H}^{1}\left(D,{j}_{*}\frac{ℰ}{ℰ\cap {ℰ}^{\perp }}\right)$ are algebraic numbers such that all their complex conjugates satisfy $q n+12 -12 ≤|α|≤ q n+12 +12 .$

By [De; 5.10] and Theorem 1.1 [De; 6.2], the hypotheses of [De; 3.2] are in fact verified for $\left({U}_{0},\frac{{ℰ}_{0}}{{ℰ}_{0}\cap {ℰ}_{0}^{\perp }},\psi \right)$ for $\beta =n,$ and we apply [De; 3.9].

Let ${𝒢}_{0}$ be a locally constant ${ℚ}_{l}-$sheaf on ${U}_{0}$ such that its reciprocal image $𝒢$ on $U$ is a constant sheaf. Then there exist units ${\alpha }_{i}$ in $\stackrel{—}{{ℚ}_{l}}$ such that for every $x\in |{U}_{0}|,$ we have $det( 1-Fx*t ,𝒢0 ) = ∏i( 1-αi deg(x) t ).$

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 $\mathrm{Gal}\left(\stackrel{—}{{𝔽}_{q}}/{𝔽}_{q}\right)$ and we take $det(1-Ft, G0) = ∏i (1-αit).$

The Lemma 1.3 [De; 6.4] applies to ${R}^{i}{{f}_{0}}_{*}{ℚ}_{l}$ $\left(i\ne n\right),$ to ${R}^{n}{{f}_{0}}_{*}{ℚ}_{l}/{ℰ}_{0}$ and to ${ℰ}_{0}\cap {ℰ}_{0}^{\perp }.$

For $x\in |{U}_{0}|,$ 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 ${H}^{i}\left({X}_{\stackrel{—}{x}},{ℚ}_{l}\right)$ is the fibre of ${R}^{i}{f}_{*}{ℚ}_{l}$ at $\stackrel{—}{x}.$ The formula [De; 1.5.4] for the variety ${X}_{x}$ over $k\left(x\right)$ is thus written $Z(Xx,t) = ∏idet( 1-Fx*t , Rif0*ℚl ) (-1) i+1 ,$ and $Z\left({X}_{x},t\right)$ is the product of $Zf = det( 1-Fx*t , Rnf0*ℚl/ℰ0 ) ⋅ det( 1-Fx*t , ℰ0∩ℰ0⊥ ) ⋅ ∏i≠n det( 1-Fx*t , Rif0*ℚl ) (-1)i+1$ with $Zm = det( 1-Fx*t , ℰ0 ℰ0∩ℰ0⊥ ).$

Putting and applying Lemma 1.3 [De; 6.4] to the factors ${Z}^{f}$ and ${Z}^{m}$ we find that there exist $l-$adic units and in $\stackrel{—}{{ℚ}_{l}}$ such that for every $x\in |{U}_{0}|$ $Z(Xx,t) = ∏i( 1-αi degx t ) ∏j( 1-βj degx t ) ⋅ det( 1-Fx*t ,ℱ0 )$ and, in particular, the second factor is in $ℚ\left(t\right).$ 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 $\prod _{i}\left(1-{\alpha }_{i}t\right)$ and $\prod _{j}\left(1-{\beta }_{j}t\right)$ have rational coefficients, i.e. that the family of ${\alpha }_{i}$ (resp. the family of ${\beta }_{j}$) is defined over $ℚ.$ We will deduce the following propositions.

Let and be two families of $l-$adic units in $\stackrel{—}{{ℚ}_{l}}.$ 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 $|{U}_{0}|$ of density 0, then, if $x\in |{U}_{0}|$ satisfies $k\nmid \mathrm{deg}\left(x\right)$ (for every $k\in K$) and $x\notin L,$ the denominator of $det( 1-Fx*t , ℱ0 ) ∏i( 1-γi deg(x) t ) ∏j( 1-δj deg(x) t ) , (W6 6.6.1)$ written in irreducible form, is $\prod _{j}\left(1-{\delta }_{j}^{\mathrm{deg}\left(x\right)}t\right).$

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 $x\in |{U}_{0}|.$

Let $K$ be a finite set of integers $\ne 1$ and and 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 and

Let and be two families of $l-$adic units in $\stackrel{—}{{ℚ}_{l}},$ $R\left(t\right)=\prod _{i}\left(1-{\gamma }_{i}t\right)$ and $S\left(t\right)=\prod _{j}\left(1-{\delta }_{j}t\right).$ We assume that for every $x\in |{U}_{0}|$ $\prod _{i}\left(1-{\delta }_{i}^{\mathrm{deg}\left(x\right)}t\right)$ divides $∏i ( 1-γideg(x)t ) ⋅ det(1-Fx*t,ℱ0).$ 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 $∏i ( 1-γideg(x)t ) ⋅ det(1-Fx*t,ℱ0).$ It is the lcm of the polynomials $S\left(t\right)=\prod _{j}\left(1-{\delta }_{j}t\right)$ 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 Since these are in $ℚ\left(t\right),$ the family of ${\beta }_{j}$ is defined over $ℚ.$

The polynomials $\prod _{i}\left(1-{\alpha }_{i}^{\mathrm{deg}\left(x\right)}t\right)\cdot \mathrm{det}\left(1-{F}_{x}^{*}t,{ℱ}_{0}\right)$ are thus in $ℚ\left[t\right].$ 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 $ℚ.$

## §6.10

Preliminaries: Let $u\in U$ and ${ℱ}_{u}$ be the fibre of $ℱ$ at $u.$ The arithmetic fundamental group ${\pi }_{1}\left({U}_{0},u\right),$ extension of $\stackrel{^}{ℤ}=\mathrm{Gal}\left(\stackrel{—}{{𝔽}_{q}}/{𝔽}_{q}\right)$ (generator: $\phi$) by the geometric fundamental group ${\pi }_{1}\left(U,u\right),$ acts on ${ℱ}_{u}$ by symplectic similitudes: $ρ: π1(U0,u) → CSp(ℱu,ψ).$ We denote by $\mu \left(g\right)$ the multiplier of a symplectic similitude $g.$ Let $H ⊆ ℤ^×CSp(ℱu,ψ)$ be the subgroup defined by the equation $q-n = μ(g)$ ($q$ being an $l-$adic unit, ${q}^{n}\in {ℚ}_{l}^{*}$ is defined for all $n\in \stackrel{^}{ℤ}$). The fact that $\psi$ has values in ${ℚ}_{l}\left(-n\right)$ can be expressed by saying that the map from ${\pi }_{1}$ into $\stackrel{^}{ℤ}×\mathrm{CSp},$ with coordinates the canonical projection on $\stackrel{^}{ℤ}$ and $\rho ,$ factors ${\rho }_{1}:{\pi }_{1}\left({U}_{0},u\right)\to H.$

The image ${H}_{1}$ of ${\rho }_{1}$ is open in $H.$

In fact, ${\pi }_{1}\left({U}_{0},u\right)$ projects onto $\stackrel{^}{ℤ},$ and the image of ${\pi }_{1}\left(U,u\right)=\mathrm{Ker}\left({\pi }_{1}\left({U}_{0},u\right)\to \stackrel{^}{ℤ}\right)$ in $\mathrm{Sp}\left({ℱ}_{u},\psi \right)=\mathrm{Ker}\left(H\to \stackrel{^}{ℤ}\right)$ is open [De; 5.10].

For $\delta \in \stackrel{—}{{ℚ}_{l}}$ an $l-$adic unit, the set $Z$ of $\left(n,g\right)\in {H}_{1}$ such that ${\delta }^{n}$ is an eigenvalue of $g$ is closed of measure 0.

It is clear that $Z$ is closed. For each $n\in \stackrel{^}{ℤ},$ let ${\mathrm{CSp}}_{n}$ be the set of $g\in \mathrm{CSp}\left({ℱ}_{u},\psi \right)$ 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], ${H}_{1}\cap \left(\left\{n\right\}×{Z}_{n}\right)$ is thus of measure 0 in the inverse image of $n$ in ${H}_{1},$ and we apply Fubini to the projection ${H}_{1}\to \stackrel{^}{ℤ}.$

## §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 $x\in |{U}_{0}|$ 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.