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180: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/19(日) 09:47:59.66 ID:2Y0qBKwb

>>179
つづき

; later he has refined this result and shown that K may be recovered from the topological group GK and one Lubin-Tate character of GK (see [14] and [20]).

On the other hand, given a padic field K1, the Jarden-Ritter Theorem (see [8]) provides a characterization of all p-adic fields
K2 such that one has a topological isomorphism GK2 ' GK1 of their absolute Galois groups and
it is well-known that for every prime p, pairs of fields with this property always exists.
Mochizuki’s anabelian reconstruction yoga (see [22] and its references) provides, starting with
the topological group G ' GK, the reconstruction amphora of G (see [6, 5, 7, 20, 21, 22] and
its references for the contents of the reconstruction amphora; in [9] I introduced this term as a
convenient short-form and memory-aid) which contains all quantities related to K which are reconstructed from the topological group G such as the prime p,
the topological monoids O^*K (the group of units of the ring of integers OK of K) and OΔK the multiplicative monoid of non-zero elements of OK (this notation is due to Mochizuki).

However the ring OK is not contained in the reconstruction amphora of G.
Moreover Mochizuki’s Reconstruction yoga also asserts that if one has an isomorphism of
topological groups
GK1 〜＝ GK2
then an isomorphism of the topological monoids
OΔK1 〜＝ OΔK2
may also be reconstructed from it (see [6, Section 2]).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/180

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

>>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

http://rio2016.5ch.net/test/read.cgi/math/1592654877/181

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