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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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The Main Theorem

The Main Theorem

Chapter:
(p.39) Chapter 7 The Main Theorem
Source:
Convolution and Equidistribution
Author(s):

Nicholas M. Katz

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153308.003.0008

This chapter takes up the proof of the main theorem. Theorem 7.2: Suppose N in Garithι‎-pure of weight zero and arithmetically semisimple, such that the quotient group Garith,N/Ggeom,N is ℤ / nℤ. Fix an integer d mod n. Then as E/k runs over larger and larger extension fields whose degree is d mod n, the conjugacy classes {θ‎E,ρ‎}goodρ‎ become equidistributed in the space Karith# for the measure μd# of total mass one. Equivalently, as E/k runs over larger and larger extension fields whose degree is d mod n, the conjugacy classes {θ‎E,ρ‎}goodρ‎ become equidistributed in the space Karith# for the measure i*μd# of total mass one.

Keywords:   Frobenius conjugacy class, number theory, theorem

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