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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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Group-Theoretic Facts about Ggeom and Garith

Group-Theoretic Facts about Ggeom and Garith

Chapter:
(p.33) Chapter 6 Group-Theoretic Facts about Ggeom and Garith
Source:
Convolution and Equidistribution
Author(s):

Nicholas M. Katz

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153308.003.0007

This chapter presents proofs of following theorems. Theorem 6.1: Suppose N in Garith is geometrically semisimple. Then Ggeom,N is a normal subgroup of Garith,N. Theorem 6.2: Suppose that N in Garith is arithmetically semisimple and pure of weight zero. If Garith,N is finite, then N is punctual. Indeed, if every Frobenius conjugacy class FrobE,X in Garith,N is quasiunipotent, then N is punctual. Theorem 6.4: Suppose that N in Garith is arithmetically semisimple and pure of weight zero. If Ggeom is finite, then N is punctual. Theorem 6.5: Suppose that N in Garith is arithmetically semisimple and pure of weight zero. Then the group Ggeom,N/G⁰geom,N of connected components of Ggeom,N is cyclic of some prime to p order n.

Keywords:   number theory, theorems, semisimple, Frobenius conjugacy class

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