現代数学の系譜工学物理雑談古典ガロア理論も読む79

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751: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 10:01:47.32 ID:uR3g5aDb

>>750

（参考）
https://en.wikipedia.org/wiki/Frey_curve
Frey curve
(抜粋)
Frey curve or Frey?Hellegouarch curve is the elliptic curve
y^2=x(x-a^l)(x+b^l)
associated with a (hypothetical) solution of Fermat's equation
a^l+b^l=c^l.
The curve is named after Gerhard Frey.

History
Yves Hellegouarch (1975) came up with the idea of associating solutions  (a,b,c)}(a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve. If ? is an odd prime and a, b, and c are positive integers such that
a^l+b^l=c^l,
then a corresponding Frey curve is an algebraic curve given by the equation
y^2=x(x-a^l)(x+b^l)
or, equivalently
y^2=x(x-a^l)(x-c^l).
This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q.

(Gerhard Frey 1982) called attention to the unusual properties of the same curve as Hellegouarch, which became called a Frey curve.
This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular.
The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama?Shimura?Weil conjecture implies Fermat's Last Theorem. However, his argument was not complete.
In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture.
In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama?Shimura?Weil conjecture implies Fermat's Last Theorem.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/751

752: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 10:28:53.79 ID:uR3g5aDb

>>751

（参考：これ分り易いかも）
https://inference-review.com/article/fukugen
Fukugen
Ivan Fesenko
Published on September 28, 2016 in Volume 2, Issue 3.
On Shinichi Mochizuki’s Inter-universal Teichmuller Theory
(抜粋)
THE SUNLIGHT is strong in Kyoto, even in winter. In December of 2014, I visited Shinichi Mochizuki at the Research Institute for Mathematical Sciences to discuss his inter-universal Teichmuller theory (IUT).1 A distinguished mathematician and a leading figure in anabelian geometry,
Mochizuki first made his papers about IUT available at the end of August, 2012. Their study has proved challenging.

A term that is frequently used in mathematical discussions about anabelian geometry and IUT is fukugen, which may be translated as restoration or as reconstruction, and which, like so many words in a foreign language, cannot be truly translated.

It must be used without translation. But isn’t this true of mathematics itself?

IUT contributes to a new view of the numbers. This may sound as if Mochizuki had announced, rather than executed, a program in pure mathematics. But IUT yields proofs of several outstanding problems in number theory: the strong Szpiro conjecture for elliptic curves,
Vojta’s conjecture for hyperbolic curves, and the Frey conjecture for elliptic curves.

And it settles the famous Oesterle?Masser or abc conjecture.2

The abc conjecture is easy to state and difficult to prove. Prime numbers are defined in terms of multiplication in the ring of integers.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/752

760: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 18:15:21.09 ID:uR3g5aDb

>>751
＜Frey curve＞
https://en.wikipedia.org/wiki/Abc_conjecture#cite_ref-1
abc conjecture
(抜粋)
The precise statement is given below. The abc conjecture originated as the outcome of attempts by Oesterle and Masser to understand the Szpiro conjecture about elliptic curves.[1]

Citations
[1]
https://www.maths.nottingham.ac.uk/plp/pmzibf/notesoniut.pdf
Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF), European Journal of Mathematics, 1 (3): 405?440, doi:10.1007/s40879-015-0066-0.
(抜粋)
P4
1.3. Conjectural inequalities for the same property.
(a) the effective Mordell conjecture ? a conjectural extension of the Faltings?Mordell theorem which involves an effective bound on the height of rational points of the curve C over the number field K in the Faltings theorem in terms of data associated to C and K,
(b) the Szpiro conjecture, see below,
(c) the Masser?Oesterle conjecture, a.k.a. the abc conjecture (whose statement over Q is well known^6 , and which has an extension to arbitrary algebraic number fields, see Conj. 14.4.12 of [6]),
(d) the Frey conjecture, see Conj. F.3.2(b) of [15],
(f) arithmetic Bogomolov?Miyaoka?Yau conjectures (there are several versions).

The Szpiro conjecture was stated several years before^7 the work of Faltings, who learned much about the subject related to his proof from Szpiro.
Using the Frey curve^8, it is not difficult to show that (c) and (d) are equivalent and that they imply (b), see e.g. see sect. F3 of [15] and references therein.
Using Belyi maps as in 1.1, one can show the equivalence of (c) and (a).
For the equivalence of (c) and (e) see e.g. Th. 14.4.16 of [6]
and [47]. For implications (e) ⇒ (f) see [48].

Footnote
^8 y^2 = x(x+a)(x?b) where a,b,a+b are non-zero coprime integers

http://rio2016.5ch.net/test/read.cgi/math/1573769803/760

762: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 19:14:54.39 ID:uR3g5aDb

>>760 追加

＜スピロ予想−Frey curve−楕円曲線 E−ABC予想＞
https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%94%E3%83%AD%E4%BA%88%E6%83%B3
スピロ予想
(抜粋)
スピロ予想 (Szpiro's conjecture) は、楕円曲線の導手と判別式との間の関係について述べた予想であり、ABC予想と深い関係にある。この予想の名前は、1980年代にこれを定式化した Lucien Szpiro に由来する。

ABC予想との関係
スピロ予想より強い以下の主張がABC予想と同値である[2]。
任意の ε > 0 に対し、定数 C (ε) が存在して、有理数体 Q 上定義された全ての楕円曲線 E に対して、
max{｜c_4｜^3,｜c_6｜^2}<= C(ε)cdot f^{6+ε}}
が成り立つ。
ここに、c4, c6 は楕円曲線 E のよく知られた不変量である。
一般に 1728Δ = (c_4)^3 - (c_6)^2 であるから、上記の主張から通常のスピロ予想は簡単に導かれる。
通常のスピロ予想は、少し弱いヴァージョンのABC予想と同値である[3]。

脚注
3^D. Goldfeld, Modular forms, elliptic curves, and the ABC-conjecture.
http://www.math.columbia.edu/~goldfeld/
DORIAN GOLDFELD Professor Mathematics Columbia University New York
http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
MODULAR FORMS, ELLIPTIC CURVES AND THE ABC?CONJECTURE
Dorian Goldfeld? Dedicated to Alan Baker on the
occasion of his sixtieth birthday （1999）
(抜粋)
P7
Consider the Frey?Hellegouarch curve
EA,B : y^2 = x(x - A)(x + B).
A minimal model for EA,B has discriminant (ABC)^2・2^-s and conductor N・2^-t for
certain absolutely bounded integers s, t, (see Frey [F1]).
Plugging this data into Szpiro’s conjecture immediately shows the equivalence.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/762

764: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 22:50:26.55 ID:uR3g5aDb

>>762

追加
https://en.wikipedia.org/wiki/Dorian_M._Goldfeld
(抜粋)
Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in analytic number theory and automorphic forms at Columbia University.

Professional career
Goldfeld received his B.S. degree in 1967 from Columbia University. His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia.
He has held positions at the University of California at Berkeley (Miller Fellow, 1969?1971), Hebrew University (1971?1972), Tel Aviv University (1972?1973), Institute for Advanced Study (1973?1974), in Italy (1974?1976), at MIT (1976?1982), University of Texas at Austin (1983?1985) and Harvard (1982?1985).
Since 1985, he has been a professor at Columbia University.[1]

His work on the Birch and Swinnerton-Dyer conjecture includes the proof of an estimate for a partial Euler product associated to an elliptic curve,[12] bounds for the order of the Tate?Shafarevich group[13]

Awards and honors
In 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the prizes in Number Theory, for his solution of Gauss' class number problem for imaginary quadratic fields.
In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/764

766: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 23:08:10.15 ID:uR3g5aDb

>>765
>>765

この日本文は不正確
正確には、下記英文
なお、下記(a, b, c, n)は (a, b, c, p)が正確かもね、y^2 = x (x - a^p)(x + b^p)だからね
https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
Fermat's Last Theorem
(抜粋)
Contents
2.5 Connection with elliptic curves

Ribet's theorem for Frey curves
Main articles: Frey curve and Ribet's theorem
In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture.
If Fermat's equation had any solution (a, b, c) for exponent p > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-Hellegouarch[note 3])
y^2 = x (x - a^p)(x + b^p)
would have such unusual properties that it was unlikely to be modular.[122]
This would conflict with the modularity theorem, which asserted that all elliptic curves are modular.
As such, Frey observed that a proof of the Taniyama?Shimura?Weil conjecture might also simultaneously prove Fermat's Last Theorem.[123]
By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama?Shimura?Weil conjecture.

In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama?Shimura?Weil conjecture.
Therefore if the latter were true, the former could not be disproven, and would also have to be true.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1573769803/766

767: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 23:08:38.66 ID:uR3g5aDb

>>766
つづき

Following this strategy, a proof of Fermat's Last Theorem required two steps.
First, it was necessary to prove the modularity theorem ? or at least to prove it for the types of elliptical curves that included Frey's equation (known as semistable elliptic curves).
This was widely believed inaccessible to proof by contemporary mathematicians.[121]:203?205, 223, 226
Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular.
Frey showed that this was plausible but did not go as far as giving a full proof.
The missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was identified by Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by Ken Ribet.[124]
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1573769803/767

768: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 23:25:31.39 ID:uR3g5aDb

>>767 追加

英文の方が圧倒的に詳しいね
まあ、和文→英文の順に読めば良い（＾＾

（英文）
https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
Wiles's proof of Fermat's Last Theorem
(抜粋)
Contents
1 Precursors to Wiles' proof
1.1 Fermat's Last Theorem and progress prior to 1980
1.2 The Taniyama?Shimura?Weil conjecture
1.3 Frey's curve
1.4 Ribet's theorem
1.5 Situation prior to Wiles' proof
2 Andrew Wiles
3 Announcement and subsequent developments
3.1 Announcement and final proof (1993?1995)
3.2 Subsequent developments
4 Summary of Wiles' proof
5 Mathematical detail of Wiles proof
5.1 Overview
5.2 General approach and strategy
5.3 3-5 trick
5.4 Structure of Wiles's proof
5.5 Overviews available in the literature
6 References

（和文）
https://ja.wikipedia.org/wiki/%E3%83%AF%E3%82%A4%E3%83%AB%E3%82%BA%E3%81%AB%E3%82%88%E3%82%8B%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9E%E3%83%BC%E3%81%AE%E6%9C%80%E7%B5%82%E5%AE%9A%E7%90%86%E3%81%AE%E8%A8%BC%E6%98%8E
ワイルズによるフェルマーの最終定理の証明
(抜粋)
目次
1 ワイルズの証明以前の進展
1.1 フェルマーの最終定理
1.2 ワイルズ以前の特定の指数に関する部分的な解
1.3 谷山・志村・ヴェイユ予想
1.4 フライ曲線
1.5 フライ曲線を用いたフェルマーの最終定理への挑戦
1.6 リベットの定理
2 アンドリュー・ワイルズ
3 証明の発表とその後の発展
3.1 証明の発表と最終的な証明 (1993?1995)
3.2 その後の発展
4 脚注
5 参考文献
6 外部リンク
6.1 ワイルズの証明の解説

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