現代数学の系譜工学物理雑談古典ガロア理論も読む78

[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78 (1002ﾚｽ)
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613: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/08(金) 16:44:18.12 ID:nN7QsxvT

>>608
＞モノドロミーの歴史（だれがいつ？）を調べていたのだが

追加
https://arxiv.org/abs/1507.00711
Monodromy and normal forms
Fabrizio Catanese (Universitaet Bayreuth)
(Submitted on 2 Jul 2015)
https://arxiv.org/pdf/1507.00711.pdf
MONODROMY AND NORMAL FORMS
FABRIZIO CATANESE
Abstract. We discuss the history of the monodromy theorem,
starting from Weierstras, and the concept of monodromy group.
From this viewpoint we compare then the Weierstras, the Legendre and other normal forms for elliptic curves, explaining their
geometric meaning and distinguishing them by their stabilizer in
PSL(2, Z) and their monodromy. Then we focus on the birth of
the concept of the Jacobian variety, and the geometrization of the
theory of Abelian functions and integrals. We end illustrating the
methods of complex analysis in the simplest issue, the difference
equation f(z) = g(z + 1) ? g(z) on C.

Introduction
In Jules Verne’s novel of 1874, ‘Le Tour du monde en quatre-vingts
jours’ , Phileas Fogg is led to his remarkable adventure by a bet made
in his Club: is it possible to make a tour of the world in 80 days?
Idle questions and bets can be very stimulating, but very difficult
to answer when they deal with the history of mathematics, and one
asks how certain ideas, which have been a common knowledge for long
time, did indeed evolve and mature through a long period of time, and
through the contributions of many people.
In short, there are three idle questions which occupy my attention
since some time:

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/613

614: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/08(金) 16:46:01.10 ID:nN7QsxvT

>>613

つづき

(1) When was the statement of the monodromy theorem first fully
formulated (resp. : proven)?
(2) When did the normal form for elliptic curves
y^2 = x(x ? 1)(x ? λ),
which is by nowadays’ tradition called by many (erroneously?)
‘the Legendre normal form’ first appear?
(3) The old ‘Jacobi inversion theorem’ is today geometrically formulated through the geometry of the ‘Jacobian variety J(C)’
of an algebraic curve C of genus g: when did this formulation
clearly show up (and so clearly that, ever since, everybody was
talking only in terms of the Jacobian variety)?

The above questions not only deal with themes of research which
were central to Weierstras’ work on complex function theory, but indeed they single out philosophically the importance in mathematics of
clean formulations and rigorous arguments.
Ath his point it seems appropriate to cite Caratheodory, who wrote
so in the preface of his two volumes on ‘Funktionentheorie’ ([Car50]):
‘ The genius of B. Riemann (1826-1865) intervened not only to bring
the Cauchy theory to a certain completion, but also to create the foundations for the geometric theory of functions. At almost the same time,
K. Weierstras(1815-1897) took up again the above-mentioned idea of
Lagrange’s 1
, on the basis of which he was able to arithmetize Function
Theory and to develop a system that in point of rigor and beauty cannot
be excelled. The Weierstras tradition was carried on in an especially
pure form by A. Pringsheim (1850-1941), whose book (1925-1932) is
extremely instructive.’

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/614

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