Deligne I Section 3
Last updates: 3 June 2011
The fundamental bound
The result of this paragraph was catalysed by a lecture of Rankin [Ra].
Let be a curve over
the complement in of a
finite number of closed points, the induced curve over
a closed point of ,
twisted constant -sheaf
its reciprocal image over .
Let . We say that
if for every
proper values of act on
(1.13) are algebraic numbers all
of whose complex conjugates have absolute value
is of weights .
Make the following hypotheses:
is endowed with a nondegenerate
alternating bilinear form
in is an open subgroup
of the symplectic group .
polynomial has rational coefficients.
On may assume, and we will assume that is affine and that
be an even integer and let
tensor power of
is a formal series with positive rational coefficients.
The hypothesis (iii) assures that, for every ,
. The number
is thus positive rational, and one applies (1.5.3).
The local factors
are formal series with positive rational coefficients.
The formal series
is without constant term; by (3.3), its coefficients are ;
the coefficients of its exponential are also positive.
a sequence of formal series with constant term one, and with positive real coefficients.
Assume that the order of
goes to infinity with , and one puts
Then the radius of absolute convergence of the is
at least equal to that of .
Under the hypotheses of (3.5), if
are the Taylor series expansions of meromorphic functions then
In fact these numbers are the radii of absolute convergence.
(3.7) For each partition of
in subsets of two elements
Let be a closed point of .
The hypothesis (ii) guarantees that the coinvariants of
are the coinvariants in
of the full symplectic group (
is Zariski-dense in ). Let be
the set of partitions . Following H. Weyl
(The classical groups, Princeton University Press, chap. VI § 1), for
the for define an isomorphism
be the number of elements of
By (2.10), the formula above gives