>>685
追加 佐藤　周友先生ね（＾＾
https://ncatlab.org/nlab/show/Tate+twist
Tate twist Last revised on February 9, 2018
Contents
1. Idea
2. Definition
(抜粋)
1. Idea
Cohomology theories often have two aspects, which one might refer to as geometric and arithmetic. A prototypical example is the l-adic cohomology Hi(X,Zl) of a (sufficiently nice) scheme X over a field k of characteristic p, with l coprime to p.
This is not only an abelian group (the geometric aspect), but a representation of the absolute Galois group of k (the arithmetic aspect).
In the case of the singular cohomology of a complex manifold, the ‘arithmetic’ aspect arises as the Hodge structure on the cohomology groups.
It has been speculated (for example by Manin?) that there should be some kind of ‘Galois group’ whose representations are Hodge structures, (and similarly for mixed Hodge modules vs perverse sheaves, and so on) but this remains mysterious;
it may be that a good theory of algebraic geometry over F1 (the would-be “field with one element”) would provide an explanation.
Tate twists play an important role in cohomology theories with this dual geometric and arithmetic aspect, allowing one to express Poincare duality canonically, that is, without choosing an orientation of one’s geometric object (scheme, complex manifold, …).
3. References
https://arxiv.org/abs/math/0610426
[Submitted on 13 Oct 2006]
p-adic etale Tate twists and arithmetic duality
Kanetomo Sato Graduate School of Mathematics Nagoya University
In this paper, we define, for arithmetic schemes with semistable reduction, p-adic objects playing the roles of Tate twists in etale topology, and establish their fundamental properties.
Comments: 66 papges. to appear in Ann. Sci. Ec. Norm. Sup. (4)
https://arxiv.org/pdf/math/0610426.pdf
https://researchers.chuo-u.ac.jp/Profiles/3/0000248/profile.html?lang=ja
教授 サトウ　カネトモ 佐藤　周友