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Fourier transforms and \(p\)-adic ‘Weil II’. (English) Zbl 1119.14014

This important paper completes the author’s program of proving the Weil conjectures using purely \(p\)-adic techniques. It is a companion paper to the author’s article [Duke Math. J. 134, No. 1, 15–97 (2006; Zbl 1133.14019)]. The main result of the paper under review (theorem 5.3.2, rigid Weil II over a point) is a weak analogue in rigid cohomology of P. Deligne’s ”Weil II” theorem on purity of higher direct images [Publ. Math., Inst. Hautes Étud. Sci. 52, 137–252 (1980; Zbl 0456.14014)]. The proof follows G. Laumon’s Fourier theoretic approach [Publ. Math., Inst. Hautes Étud. Sci. 65, 131–210 (1987; Zbl 0641.14009)], transposed into the setting of arithmetic \(\mathcal{D}\)-modules, and builds on the work of many people. We refer to the paper’s informative introduction for more specific details and comments.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
11G25 Varieties over finite and local fields
12H25 \(p\)-adic differential equations
14G22 Rigid analytic geometry
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