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Convolution and EquidistributionSato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)$
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Nicholas M. Katz

Print publication date: 2012

Print ISBN-13: 9780691153308

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153308.001.0001

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A First Construction of Autodual Objects

A First Construction of Autodual Objects

Chapter:
(p.53) Chapter 10 A First Construction of Autodual Objects
Source:
Convolution and Equidistribution
Author(s):

Nicholas M. Katz

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153308.003.0011

This chapter presents the construction of autodual objects. It begins with a geometrically irreducible middle extension sheaf Ƒ on G/k which is ι‎-pure of weight zero, and which is not geometrically an Lᵪ. Thus Ƒ(1/2)[1] is a geometrically irreducible object in in Garith. Its dual in in Garith is [x ↦ 1/x]★ ℱ¯ (1/2)[1], for 𝓕 with hook and macron the linear dual middle extension sheaf. Via ι‎, Ƒ and 𝓕 with hook and macron have complex conjugate trace functions; this holds by ι‎-purity on the dense open set where Ƒ is lisse, and then on all of 𝓖 by a result of Gabber [Fuj-Indep, Thm. 3], cf. also [Ka-MMP, proof of 1.8.1 (i)].

Keywords:   number theory, autoduality, autodual objects

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