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181(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 09:48:56.07 ID:2Y0qBKwb(3/9) AAS
>>180

つづき

In this paper I consider a different approach to this problem of understanding the fluidity of ring structures and in particular to the problem of quantifying the fluidity of the additive structures on the set OΔK ∪ {0} for a p-adic field K.
I began thinking of this problem in Kyoto (Spring 2018) and my preoccupation with it became more or less permanent on my return from Kyoto.

The idea, which I elaborate here, occurred to me in a recent lecture by Michael Hopkins at the Arizona Winter School (2019).
In one of his lectures, Hopkins narrated an anecdote about Daniel Quillen’s discovery of the role of formal groups in topological cohomology theories:
in particular Quillen’s assertion (to Hopkins) that “as addition rule for Chern classes fails to hold,
it must therefore fail in worst possible way?namely by means of a formal group”
(I am paraphrasing both Hopkins and Quillen here).

つづく

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182: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 09:49:20.20 ID:2Y0qBKwb(4/9) AAS
>>181
つづき

As was also pointed out to me by Taylor Dupuy, Mochizuki recognized a long time ago (see for
instance [15, Section 4]) that arithmetic applications of anabelian geometry lead naturally to the
deep and difficult problem of understanding the line bundles and (Arakelov) degrees (or Arakelov
Chern classes) in the presence of anabelian variation of ring structures and he resolved this problem
by means of his theory of Frobenioids and realified Frobenioids [18] and Arakelov-Hodge theoretic
evaluation methods culminating in [10, 11, 12, 13].

This paper would not be possible without Shinichi Mochizuki’s bold, audacious, deep and profoundly original ideas which continue to be a source of inspiration for me?in particular his truly remarkable and astounding discovery that there are arithmetic properties of non-zero elements of a fixed p-adic field which are independent of the ring structure of this field.
I am deeply indebted to him for many conversations on many topics surrounding his ideas and for his continued support and encouragement.
(引用終り)
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