[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む79 (1002ﾚｽ)
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181(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/26(火) 23:13:42.54 ID:oYs7jyeH(12/12) AAS
>>124
>楕円曲線のホッジ・アラケロフ理論は、アラケロフ理論（英語版）(Arakelov theory)のフレームワークで考える p-進ホッジ理論（英語版）(p-adic Hodge thory)の楕円曲線についての類似理論である。
(参考)
https://en.wikipedia.org/wiki/P-adic_Hodge_theory
p-adic Hodge theory
(抜粋)
The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge?Tate representation.
Hodge?Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory.
Further developments were inspired by properties of p-adic Galois representations arising from the etale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
182(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/27(水) 07:49:43.49 ID:qnEhNItW(1/3) AAS
>>181
つづき
Contents
1 General classification of p-adic representations
2 Period rings and comparison isomorphisms in arithmetic geometry
General classification of p-adic representations
Let K be a local field with residue field k of characteristic p. In this article, a p-adic representation of K (or of GK, the absolute Galois group of K) will be a continuous representation ρ : GK→ GL(V), where V is a finite-dimensional vector space over Qp.
The collection of all p-adic representations of K form an abelian category denoted \mathrm {Rep} _{\mathbf {Q} _{p}}(K)}{\mathrm {Rep}}_{{{\mathbf {Q}}_{p}}}(K) in this article.
p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:[2]
{Rep} _{\mathrm {cris} }(K)\subsetneq {Rep} _{st}(K)\subsetneq {Rep} _{dR}(K)\subsetneq {Rep} _{HT}(K)\subsetneq {Rep} _{\mathbf {Q} _{p}}(K)}
where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge?Tate representations, and all p-adic representations.
In addition, two other categories of representations can be introduced, the potentially crystalline representations Reppcris(K) and the potentially semistable representations Reppst(K).
The latter strictly contains the former which in turn generally strictly contains Repcris(K); additionally, Reppst(K) generally strictly contains Repst(K), and is contained in RepdR(K) (with equality when the residue field of K is finite, a statement called the p-adic monodromy theorem).
つづく
184: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2019/11/27(水) 07:57:19.21 ID:qnEhNItW(3/3) AAS
>>181 補足
p-adic Hodge theory
キーワードを拾うと
・The collection of all p-adic representations of K form an abelian category
・and also provides faithful functors to categories of linear algebraic objects that are easier to study.
・where each collection is a full subcategory properly contained in the next.
category、faithful functors、full subcategory properly
てのは、p-adic Hodge theory 由来なのかな？
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