## 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 ${ℙ}^{1}$ of a finite number of closed points, $U$ the induced curve over ${\stackrel{‾}{𝔽}}_{q}$, $u$ a closed point of $U$, ${ℱ}_{0}$ a twisted constant ${ℚ}_{\ell }$-sheaf over ${U}_{0}$ and $ℱ$ its reciprocal image over $U$.

Let $\beta \in ℚ$. We say that ${ℱ}_{0}$ has weight $\beta$ if for every $x\in |{U}_{0}|$, 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, ${ℚ}_{\ell }\left(r\right)$ is of weights $-2r$.

(3.2) Make the following hypotheses:
(i)   ${ℱ}_{0}$ is endowed with a nondegenerate alternating bilinear form
 $\psi :{ℱ}_{0}\otimes {ℱ}_{0}\to {ℚ}_{\ell }\left(-\beta \right)\phantom{\rule{2em}{0ex}}\left(\beta \in ℤ\right).$
(ii)   The image ${\pi }_{1}\left(U,u\right)$ in $\mathrm{GL}\left({ℱ}_{u}\right)$ is an open subgroup of the symplectic group $\mathrm{Sp}\left({ℱ}_{u},{\psi }_{u}\right)$.
(iii)   For every $x\in |{U}_{0}|$, the polynomial $\mathrm{det}\left(1-{F}_{x}t,{ℱ}_{0}\right)$ has rational coefficients.
Then $ℱ$ has weight $\beta$.

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

(3.3) Let $2k$ be an even integer and let $\stackrel{2k}{⨂}{ℱ}_{0}$ be the $\left(2k\right)$th tensor power of ${ℱ}_{0}$. For $x\in |{U}_{0}|$, the logarithmic derivative
 $t\frac{d}{dt}\mathrm{log}\left(\mathrm{det}\left(1-{F}_{x}{t}^{\mathrm{deg}\left(x\right)},\stackrel{2k}{⨂}{ℱ}_{0}{\right)}^{-1}\right)$
is a formal series with positive rational coefficients.

The hypothesis (iii) assures that, for every $n$, $\mathrm{Tr}\left({F}_{x}^{n},{ℱ}_{0}\right)\in ℚ$. The number

 $\mathrm{Tr}\left({F}_{x}^{n},\stackrel{2k}{⨂}{ℱ}_{0}\right)={\mathrm{Tr}\left({F}_{x}^{n},{ℱ}_{0}\right)}^{2k}$
is thus positive rational, and one applies (1.5.3).

(3.4) The local factors $\mathrm{det}\left(1-{F}_{x}{t}^{\mathrm{deg}\left(x\right)},\stackrel{2k}{⨂}{ℱ}_{0}{\right)}^{-1}$ are formal series with positive rational coefficients.

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

(3.5) Let ${f}_{i}=\sum _{n}{a}_{i,n}{t}^{n}$ a sequence of formal series with constant term one, and with positive real coefficients. Assume that the order of ${f}_{i}-1$ goes to infinity with $i$, and one puts $f=\prod _{i}{f}_{i}$. Then the radius of absolute convergence of the ${f}_{i}$ is at least equal to that of $f$.
(3.6) Under the hypotheses of (3.5), if $f$ and the ${f}_{i}$ are the Taylor series expansions of meromorphic functions then
 $\mathrm{inf}\left\{|z|\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}f\left(z\right)=\infty \right\}\le \mathrm{inf}\left\{|z|\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}{f}_{i}\left(z\right)=\infty \right\}$

In fact these numbers are the radii of absolute convergence.

(3.7) For each partition $P$ of $\left[1,2k\right]$ in subsets of two elements $\left\{{i}_{\alpha },{j}_{\alpha }\right\}$ $\left({i}_{\alpha }<{j}_{\alpha }\right)$, one defines

 $\begin{array}{cccc}{\psi }_{P}:& \stackrel{2k}{⨂}{ℱ}_{0}& ⟶& {ℚ}_{\ell }\left(-k\beta \right)\\ & {x}_{1}\otimes \cdots \otimes {x}_{2k}& ⟼& \prod _{\alpha }\psi \left({x}_{{i}_{\alpha }},{x}_{{j}_{\alpha }}\right).\end{array}$

Let $x$ be a closed point of $X$. The hypothesis (ii) guarantees that the coinvariants of ${\pi }_{1}\left(U,u\right)$ 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

 ${\left(\stackrel{2k}{⨂}{ℱ}_{u}\right)}_{{\pi }_{1}}={\left(\stackrel{2k}{⨂}{ℱ}_{u}\right)}_{\mathrm{Sp}}\stackrel{\sim }{\to }{{ℚ}_{\ell }\left(-k\beta \right)}^{𝒫\prime }.$
Let $N$ be the number of elements of $𝒫\prime$. By (2.10), the formula above gives
 ${H}_{c}^{2}\left(U,\stackrel{2k}{⨂}ℱ\right)\simeq {{ℚ}_{\ell }\left(-k\beta -1\right)}^{N}.$
Since ${H}_{c}^{0}\left(U,\stackrel{2k}{⨂}ℱ\right)=0, the formula \left(1.14.3\right) reduces to$
 $Z\left({U}_{0},\stackrel{2k}{⨂}{ℱ}_{0},t\right)=\frac{\mathrm{det}\left(1-{F}^{*}t,{H}^{1}\left(U,\stackrel{2k}{⨂}ℱ\right)\right)}{{\left(1-{q}^{k\beta +1}t\right)}^{N}}.$
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 $\mathrm{det}\left(1-{F}_{x}{t}^{\mathrm{deg}\left(x\right)},\stackrel{2k}{⨂}ℱ{\right)}^{-1}$. By (3.4) and (3.6), it follows that
 $|1/{q}^{k\beta +1}|\le |1/{\alpha }^{2k/\mathrm{deg}\left(x\right)}|,$
so that
 $|\alpha |\le {q}_{x}^{\frac{\beta }{2}+\frac{1}{2k}}.$
Letting $k$ tend to infinity, one finds that
 $|\alpha |\le {q}_{x}^{\beta /2}.$
On the other hand, the existence of $\psi$ guarantees that ${q}_{x}^{\beta }{\alpha }^{-1}$ is also an eigenvalue, whence the inequality
 $|{q}_{x}^{\beta }{\alpha }^{-1}|\le {q}_{x}^{\beta /2}.$
so that
 ${q}_{x}^{\beta /2}\le |\alpha |.$
This completes the proof.

(3.8) Let $\alpha$ be an eigenvalue of ${F}^{*}$ acting on ${H}_{x}^{*}\left(U,ℱ\right)$. Then $\alpha$ is an algebraic number, and all its complex conjugates satisfy
 $|\alpha |\le {q}^{\frac{\beta +1}{2}+\frac{1}{2}}.$

The formula (1.14.3) for ${ℱ}_{0}$ reduces to

 $Z\left({U}_{0},{ℱ}_{0},t\right)=\mathrm{det}\left(1-{F}^{*}t,{H}_{x}^{1}\left(U,ℱ\right)\right).$
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 $Z\left({U}_{0},{ℱ}_{0},t\right)$ converges absolutely (thus is nonzero) for $|t|\le {q}^{\frac{-\beta }{2}-1}$.

Let $N$ be the rank of $ℱ$, and put

 $\mathrm{det}\left(1-{F}_{x}t,ℱ\right)=\prod _{i=1}^{N}\left(1-{\alpha }_{i,x}t\right).$
By (3.2), $|{\alpha }_{i,x}|={q}_{x}^{3/2}$. The convergence of the infinite product is a result of the series
 $\sum _{i,x}|{\alpha }_{i,x}{t}^{\mathrm{det}\left(x\right)}|,$
For $|t|\le {q}^{\frac{-\beta }{2}-1-x}$ ($\epsilon >0$), we have
 $\sum _{i,x}|{\alpha }_{i,x}{t}^{\mathrm{det}\left(x\right)}|=N\sum _{x}{q}_{x}^{-1-x}.$
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
 $\sum _{x}{q}_{x}^{-1-x}\le \sum _{x}{q}^{n}{q}^{n\left(-1-x\right)}=\sum _{x}{q}^{-nx}<\infty ,$
which completes the proof.

(3.9) Let ${j}_{0}$ be the inclusion of ${U}_{0}$ in ${ℙ}^{1}$, and $\alpha$ an eigenvalue of ${F}^{*}$ acting on ${H}^{1}\left({ℙ}^{1},{j}_{*}ℱ\right)$. Then $\alpha$ is an algebraic number, and all its complex conjugates satisfy
 ${q}^{\frac{\beta +1}{2}-\frac{1}{2}}\le |\alpha |\le {q}^{\frac{\beta +1}{2}+\frac{1}{2}}.$

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
 ${H}_{c}^{1}\left(U,ℱ\right)⟶{H}^{1}\left({ℙ}^{1},{j}_{*}ℱ\right)⟶0.$
Thus the eigenvalue $\alpha$ appears already in ${H}_{c}^{1}\left(U,ℱ\right)$, and on account of (3.8):
 $|\alpha |\le {q}^{\frac{\beta +1}{2}+\frac{1}{2}}.$
Poincaré duality (2.12) guarantees that ${q}^{\beta +1}{\alpha }^{-1}$ is also an eigenvalue, whence the inequality
 $|{q}^{\beta +1}{\alpha }^{-1}|\le {q}^{\frac{\beta +1}{2}+\frac{1}{2}}.$
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????????.