Counting local systems with principal unipotent local monodromy
成果类型：
Article
署名作者：
Deligne, Pierre; Flicker, Yuval Z.
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2013.178.3.3
发表日期：
2013
页码：
921-982
关键词：
摘要：
Let X-1 be a curve of genus g, projective and smooth over F-q. Let S-1 subset of X-1 be a reduced divisor consisting of N-1 closed points of X-1. Let (X, S) be obtained from (X-1, S-1) by extension of scalars to an algebraic closure F of F-q. Fix a prime l not dividing q. The pullback by the Frobenius endomorphism Fr of X induces a permutation Fr* of the set of isomorphism classes of rank n irreducible (Q(l)) over bar -local systems on X - S. It maps to itself the subset of those classes for which the local monodromy at each s is an element of S is unipotent, with a single Jordan block. Let T(X-1, S-1, n, m) be the number of fixed points of Fr*(m) acting on this subset. Under the assumption that N-1 >= 2, we show that T(X-1, S-1, n, m) is given by a formula reminiscent of a Lefschetz fixed point formula: the function bar right arrow T(X-1, S-1, n, m) is of the form Sigma n(i)gamma(m)(i) for suitable integers n(i), and eigenvalues gamma(i). We use Lafforgue to reduce the computation of T(X-1, S-1, n, m) to counting automorphic representations of GL(n), and the assumption N-1 >= 2 to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use.