Deligne I Section 3

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

Last updates: 3 June 2011

The fundamental bound

The result of this paragraph was catalysed by a lecture of Rankin [Ra].

(3.1) Let $U_0$ be a curve over ${𝔽}_q$, the complement in ${\mathbb{P}}^1$ of a finite number of closed points, $U$ the induced curve over ${\overline{𝔽}}_q$, $u$ a closed point of $U$, ${ℱ}_0$ a twisted constant ${\mathbb{Q}}_{\ell }$-sheaf over $U_0$ and $ℱ$ its reciprocal image over $U$.

Let $\beta \in \mathbb{Q}$. We say that ${ℱ}_0$ has weight $\beta$ if for every $x\in \left|U_0\right|$, the proper values of $F_x$ act on ${ℱ}_0$ (1.13) are algebraic numbers all of whose complex conjugates have absolute value $q_x^{\beta /2}$. For example, ${\mathbb{Q}}_{\ell }\left(r\right)$ is of weights $-2r$.

On may assume, and we will assume that $U$ is affine and that $ℱ\ne 0$.

The hypothesis (iii) assures that, for every $n$, $\mathrm{Tr}(F_x^n,{ℱ}_0)\in \mathbb{Q}$. The number

$\mathrm{Tr}(F_x^n,\stackrel{2k}{⨂}{ℱ}_0)={\mathrm{Tr}(F_x^n,{ℱ}_0)}^{2k}$

is thus positive rational, and one applies (1.5.3).

The formal series $\log \det (1-F_x{t}^{\mathrm{deg}\left(x\right)},\stackrel{2k}{⨂}{ℱ}_0{)}^{-1}$ is without constant term; by (3.3), its coefficients are $\ge 0$; the coefficients of its exponential are also positive.

In fact these numbers are the radii of absolute convergence.

(3.7) For each partition $P$ of $[1,2k]$ in subsets of two elements $\{i_{\alpha },j_{\alpha }\}$ $(i_{\alpha }<j_{\alpha })$, one defines

$\begin{array}{cccc}{\psi }_P:& \stackrel{2k}{⨂}{ℱ}_0& ⟶& {\mathbb{Q}}_{\ell }(-k\beta )\\ & x_1\otimes \cdots \otimes x_{2k}& ⟼& \prod _{\alpha }\psi (x_{i_{\alpha }},x_{j_{\alpha }}).\end{array}$

Let $x$ be a closed point of $X$. The hypothesis (ii) guarantees that the coinvariants of ${\pi }_1(U,u)$ in $\stackrel{2k}{⨂}{ℱ}_u$ are the coinvariants in $\stackrel{2k}{⨂}{ℱ}_u$ of the full symplectic group (${\pi }_1$ is Zariski-dense in $\mathrm{Sp}$). Let $𝒫$ be the set of partitions $P$. Following H. Weyl (The classical groups, Princeton University Press, chap. VI § 1), for appropriate $𝒫\prime \subseteq 𝒫$ the ${\psi }_P$ for $P\in 𝒫\prime$ define an isomorphism

${(\stackrel{2k}{⨂}{ℱ}_u)}_{{\pi }_1}={(\stackrel{2k}{⨂}{ℱ}_u)}_{\mathrm{Sp}}\stackrel{\sim }{\to }{{\mathbb{Q}}_{\ell }(-k\beta )}^{𝒫\prime }.$

Let $N$ be the number of elements of $𝒫\prime$. By (2.10), the formula above gives

$H_c^2(U,\stackrel{2k}{⨂}ℱ)\simeq {{\mathbb{Q}}_{\ell }(-k\beta -1)}^N.$

Since $$ H_c^0(U,\stackrel{2k}{⨂}ℱ)=0$,\; the\; formula\; (1.14.3)\; reduces\; to$

$Z(U_0,\stackrel{2k}{⨂}{ℱ}_0,t)={\displaystyle \frac{\det (1-F^{*}t,H^1(U,\stackrel{2k}{⨂}ℱ))}{{(1-q^{k\beta +1}t)}^N}}.$

This function $Z$ is thus the Taylor expansion of a rational function which has no pole except at $t=1/q^{k\beta +1}$. We will use only the fact that the poles are of absolute value has no pole except at $1/q^{k\beta +1}$ in $ℂ$. This can be deduced from general arguments on reductive groups. If $\alpha$ is an eigenvalue of $F_x$ on ${ℱ}_0$, then ${\alpha }^{2k}$ is an eigenvalue of $F_x$ on $\stackrel{2k}{⨂}{ℱ}_0$. We will also denote any complex conjugate of $\alpha$ by $\alpha$. The inverse power $1/{\alpha }^{2k/\mathrm{deg}\left(x\right)}$ is a pole of $\det (1-F_x{t}^{\mathrm{deg}\left(x\right)},\stackrel{2k}{⨂}ℱ{)}^{-1}$. By (3.4) and (3.6), it follows that

$\left|1/q^{k\beta +1}\right|\le \left|1/{\alpha }^{2k/\mathrm{deg}\left(x\right)}\right|,$

so that

$\left|\alpha \right|\le q_x^{\frac{\beta }{2}+\frac{1}{2k}}.$

Letting $k$ tend to infinity, one finds that

$\left|\alpha \right|\le q_x^{\beta /2}.$

On the other hand, the existence of $\psi$ guarantees that $q_x^{\beta }{\alpha }^{-1}$ is also an eigenvalue, whence the inequality

$\left|q_x^{\beta }{\alpha }^{-1}\right|\le q_x^{\beta /2}.$

so that

$q_x^{\beta /2}\le \left|\alpha \right|.$

This completes the proof.

The formula (1.14.3) for ${ℱ}_0$ reduces to

$Z(U_0,{ℱ}_0,t)=\det (1-F^{*}t,H_x^1(U,ℱ\left)\right).$

The left hand side is a formal power series with rational coefficients, as seen from its expression as a product and the hypothesis (iii). The right hand side is thus a polynomial with rational coefficients; $1/\alpha$ is a root. This already proves that $\alpha$ is algebraic. To complete the proof, it suffices to check that the infinite product that defines $Z(U_0,{ℱ}_0,t)$ converges absolutely (thus is nonzero) for $\left|t\right|\le q^{\frac{-\beta }{2}-1}$.

Let $N$ be the rank of $ℱ$, and put

$\det (1-F_{x}t,ℱ)=\prod _{i=1}^N(1-{\alpha }_{i,x}t).$

By (3.2), $\left|{\alpha }_{i,x}\right|=q_x^{3/2}$. The convergence of the infinite product is a result of the series

$\sum _{i,x}\left|{\alpha }_{i,x}{t}^{\det \left(x\right)}\right|,$

For $\left|t\right|\le q^{\frac{-\beta }{2}-1-x}$ ($\epsilon >0$), we have

$\sum _{i,x}\left|{\alpha }_{i,x}{t}^{\det \left(x\right)}\right|=N\sum _x{q}_x^{-1-x}.$

On the affine line, there are $q^n$ points with values in ${𝔽}_{q^n}$, hence at most $q^n$ closed points of degree $n$. Thus we have

$\sum _x{q}_x^{-1-x}\le \sum _x{q}^n{q}^{n(-1-x)}=\sum _x{q}^{-nx}<\infty ,$

which completes the proof.

A part of the long exact sequence in cohomology defined by the short exact sequence

$0⟶j_{!}ℱ⟶j_{*}ℱ⟶j_{*}ℱ/j_{!}ℱ⟶0$

($j_{!}$ = extension by 0) is written

$H_c^1(U,ℱ)⟶H^1({\mathbb{P}}^1,j_{*}ℱ)⟶0.$

Thus the eigenvalue $\alpha$ appears already in $H_c^1(U,ℱ)$, and on account of (3.8):

$\left|\alpha \right|\le q^{\frac{\beta +1}{2}+\frac{1}{2}}.$

Poincaré duality (2.12) guarantees that $q^{\beta +1}{\alpha }^{-1}$ is also an eigenvalue, whence the inequality

$\left|q^{\beta +1}{\alpha }^{-1}\right|\le q^{\frac{\beta +1}{2}+\frac{1}{2}}.$

and the corollary.

Notes and References

This is an attempt to translate Section 3 of [DeI].

References

[DeI] P. Deligne, La Conjecture de Weil I, Publ. Math. IHÉS (1974) 273-307. MR????????.

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