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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 Situation over ℤ: Questions

The Situation over ℤ: Questions

Chapter:
Chapter 29 (p.181) The Situation over ℤ: Questions
Source:
Convolution and Equidistribution
Author(s):

Nicholas M. Katz

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153308.003.0030

This chapter considers that there is another sense in which we might ask about “situations over ℤ,” namely we might try to mimic the setting of a theorem of Pink [Ka-ESDE, 8.18.2] about how “usual” (geometric) monodromy groups vary in a family. For each geometric point s in a normal noetherian connected scheme S, we have the closed subgroup Γ‎(s) ⊂ GL(n,ℚℓ¯) which is the image of π‎₁(Xₛ; xₛ) in the representation corresponding to Ƒₛ. The assertion is that these groups Γ‎(s) are, up to GL(n)-conjugacy, constant on a dense open set of S, and that they decrease under specialization. This chapter treats the following question: suppose a normal noetherian connected scheme S which is of finite type over ℤ [1/𝓁], an object N in the derived category Dcb((𝔾ₘ)ₛ;ℚℓ¯), and an integer n ≤ 1.

Keywords:   number theory, noetherian connected scheme, monodromy groups

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