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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 29 (p.181) The Situation over ℤ: Questions
Convolution and Equidistribution

Nicholas M. Katz

Princeton University Press

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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