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Travaux de Laumon. (Works of Laumon). (English) Zbl 0698.14014

Sémin. Bourbaki, 40ème Année, Vol. 1987/88, Exp. No. 691, Astérisque 161-162, 105-132 (1988).
[For the entire collection see Zbl 0659.00006.]
This is an overview of G. Laumon’s work in which he uses Deligne’s geometric Fourier transform to study \(\ell\)-adic étale sheaves on the affine line \({\mathbb{A}}^ 1\) over a finite field \({\mathbb{F}}_ q\). The main results are:
(i) There exists a local Fourier transform on representations of Gal(\(\overline{{\mathbb{F}}_ q((t))}/{\mathbb{F}}_ q((t)));\)
(ii) a product formula for the constant in the functional equation of the global L-function;
(iii) a geometric realization of the Artin representation.

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry
11S37 Langlands-Weil conjectures, nonabelian class field theory
14-03 History of algebraic geometry
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
01A60 History of mathematics in the 20th century

Citations:

Zbl 0659.00006
Full Text: Numdam EuDML