この夏の成果
[Header Image Credit] Carving of Maitreya (future Buddha) and disciples in Feilai Feng Caves in Hangzhou in the Eurasian continent (via Wikipedia), inspired by J.S. Milne's "Basic Theory of Affine Group Schemes"
「夏休みの成果」ではない。Why?
But throughout the summer, I wasn't in a daze, nor was I in a coma:
After some struggles in late May, I found a trailhead for the following historical paper which influenced a lot on Fermat's Last Theorem:
It was
"p-adic Properties of Modular Schemes and Modular Forms"
by Nicholas M. Katz, a historical article in International Summer School on Modular Functions Antwerp 1972.
This article is not easy at all. Although I managed to decipher its important starting point, figuring out that I had to study by looking into various books, lecture notes and papers in order to understand the article by N. Katz, I came to realize that I had to acquire solid understanding of Group Schemes, of which my understanding had been very blurry if not very superficial.
I tried tackling the following paper:
"Finite Flat Groups Schemes" (John Tate).
I have to admit that I was unable to decipher the last section on
Raynaud's Results on Commutative p-Groups Schemes
in the following paper:
After several days of struggle of reviewing the proof, I was convinced that my difficulty stemmed from a lack of deep understanding of a particular tool used in algebraic number theory.
Then I started reading
"Introduction to Finite Group Schemes" (René Schoof),
which I managed to understand and I marked the lecture notes "adequately covered"💪, though there are some propositions that I do not understand with full confidence.
In a sense, the achievement of this scorching summer was that I managed to understand the claims and proofs in the following paper, though I regret to say I cannot claim to have fully grasped every detail completely:
言い換えると、焦げるようなこの夏の成果は以下に示す論文の主張とその証明を、詳細に至る迄完全に理解したとは残念ながら言い難いが、何とか理解したことである：
« Il n'y a pas de variété abélienne sur Z. » (by Jean-Marc Fontaine)
Translating into plain English, it means:
There is no abelian scheme over Z.
簡単な日本語にすると：
整数環 Z 上のアーベル・スキームは存在しない。
This is a striking result on the following conjecture by Igor Rostislavovich Shafarevich (Игорь Ростиславович Шафаревич):
これは以下のシャファレビッチ予想に対する驚くべき結果である：
and is established independently by Fontaine and Abrashkin:
J.-M. Fontaine, Il n’y a pas de variété abélienne sur Z. Compos. Math. 54-2 (1985),121-171.
V. A. Abrashkin, Galois moduli of group schemes of period p over the ring of Witt vectors. Math. USSR-Izv. 31-1 (1998), 1-46.
As always, the more I study, the more keenly I realize how extremely limited my knowledge is. However, studying Group Schemes led to unexpected byproducts.
いつものことだが勉強すればする程自分の知識が極めて限られていることを痛感する。但し、この Group Scheme を勉強したことは期待しえない副産物に繋がった：
PD-structure (puissances divisées)
Le cristal de Dieudonné (Dieudonné crystal)
Cohomologie cristalline (Crystalline cohomology)
Considering that I failed to understand Crystalline cohomology 45 years ago, it is a kind of a tiny, tiny and tiny accomplishment.
45年前、クリスタリン・コホモロジーが全く理解できなかったことを考えると、極めて小さいとは言え成果である。
🔳