Deligne I Section 3

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 3 June 2011

The fundamental bound

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

(3.1) Let U0 be a curve over 𝔽q, the complement in 1 of a finite number of closed points, U the induced curve over 𝔽q, u a closed point of U, 0 a twisted constant -sheaf over U0 and its reciprocal image over U.

Let β. We say that 0 has weight β if for every x |U0|, the proper values of Fx act on 0 (1.13) are algebraic numbers all of whose complex conjugates have absolute value qxβ/2. For example, (r) is of weights -2r.

(3.2) Make the following hypotheses:
(i)   0 is endowed with a nondegenerate alternating bilinear form
ψ: 0 0 (-β) (β) .
(ii)   The image π1(U,u) in GL(u) is an open subgroup of the symplectic group Sp(u ,ψu).
(iii)   For every x |U0|, the polynomial det(1-Fxt ,0) has rational coefficients.
Then has weight β.

On may assume, and we will assume that U is affine and that 0.

(3.3) Let 2k be an even integer and let 2k 0 be the (2k)th tensor power of 0. For x |U0|, the logarithmic derivative
t ddt log( det( 1-Fx tdeg(x), 2k 0 ) -1 )
is a formal series with positive rational coefficients.

The hypothesis (iii) assures that, for every n, Tr (Fxn , 0) . The number

Tr (Fxn , 2k 0 ) = Tr (Fxn , 0) 2k
is thus positive rational, and one applies (1.5.3).

(3.4) The local factors det( 1-Fx tdeg(x), 2k 0 ) -1 are formal series with positive rational coefficients.

The formal series log det( 1-Fx tdeg(x), 2k 0 ) -1 is without constant term; by (3.3), its coefficients are 0; the coefficients of its exponential are also positive.

(3.5) Let fi = n ai,n tn a sequence of formal series with constant term one, and with positive real coefficients. Assume that the order of fi-1 goes to infinity with i, and one puts f= i fi. Then the radius of absolute convergence of the fi is at least equal to that of f.
(3.6) Under the hypotheses of (3.5), if f and the fi are the Taylor series expansions of meromorphic functions then
inf{ |z| | f(z)= } inf{ |z| | fi(z)= }

In fact these numbers are the radii of absolute convergence.

(3.7) For each partition P of [1,2k] in subsets of two elements {iα ,jα} (iα< jα), one defines

ψP: 2k 0 (-kβ) x1 x2k α ψ( xiα , xjα ) .

Let x be a closed point of X. The hypothesis (ii) guarantees that the coinvariants of π1(U,u) in 2k u are the coinvariants in 2k u of the full symplectic group (π1 is Zariski-dense in Sp). Let 𝒫 be the set of partitions P. Following H. Weyl (The classical groups, Princeton University Press, chap. VI § 1), for appropriate 𝒫𝒫 the ψP for P 𝒫 define an isomorphism

( 2k u ) π1 = ( 2k u ) Sp (-kβ) 𝒫 .
Let N be the number of elements of 𝒫. By (2.10), the formula above gives
Hc2 (U, 2k ) (-kβ-1) N .
Since Hc0 (U, 2k ) =0 , the formula (1.14.3) reduces to
Z(U0, 2k 0,t) = det(1-F*t , H1 (U, 2k ) ) (1 -qkβ+1t) N .
This function Z is thus the Taylor expansion of a rational function which has no pole except at t=1/ qkβ+1. We will use only the fact that the poles are of absolute value has no pole except at 1/ qkβ+1 in . This can be deduced from general arguments on reductive groups. If α is an eigenvalue of Fx on 0, then α2k is an eigenvalue of Fx on 2k 0 . We will also denote any complex conjugate of α by α. The inverse power 1/ α2k/deg(x) is a pole of det(1-Fx tdeg(x), 2k ) -1 . By (3.4) and (3.6), it follows that
| 1/q kβ+1 | | 1/α 2k/deg(x) |,
so that
|α| qxβ2 +12k.
Letting k tend to infinity, one finds that
|α| qxβ/2.
On the other hand, the existence of ψ guarantees that qxβ α-1 is also an eigenvalue, whence the inequality
| qxβ α-1 | qxβ/2.
so that
qxβ/2 |α| .
This completes the proof.

(3.8) Let α be an eigenvalue of F* acting on Hx*(U, ). Then α is an algebraic number, and all its complex conjugates satisfy
|α| qβ+12 +12.

The formula (1.14.3) for 0 reduces to

Z(U0,0, t) = det(1-F*t, Hx1(U,) ).
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/α is a root. This already proves that α is algebraic. To complete the proof, it suffices to check that the infinite product that defines Z(U0,0, t) converges absolutely (thus is nonzero) for |t| q-β2 -1.

Let N be the rank of , and put

det(1-Fxt, ) = i=1N (1- αi,xt) .
By (3.2), |αi,x| =qx3/2. The convergence of the infinite product is a result of the series
i,x |αi,x tdet(x) |,
For |t| q-β2 -1-x (ε>0), we have
i,x |αi,x tdet(x) | = Nx qx-1-x .
On the affine line, there are qn points with values in 𝔽qn, hence at most qn closed points of degree n. Thus we have
x qx-1-x x qn qn(-1-x) = x q-nx < ,
which completes the proof.

(3.9) Let j0 be the inclusion of U0 in 1, and α an eigenvalue of F* acting on H1( 1,j* ). Then α is an algebraic number, and all its complex conjugates satisfy
qβ+12 -12 |α| qβ+12 +12.

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

0 j! j* j*/ j! 0
(j! = extension by 0) is written
Hc1(U,) H1(1, j*) 0 .
Thus the eigenvalue α appears already in Hc1(U,) , and on account of (3.8):
|α| qβ+12 +12.
Poincaré duality (2.12) guarantees that qβ+1 α-1 is also an eigenvalue, whence the inequality
| qβ+1 α-1 | qβ+12 +12.
and the corollary.

Notes and References

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


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

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