It would cost several time whether true or not true.
On http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/
Terence Tao on September 26, 2012 said:
It looks like Vesselin has located a serious “red flag” in Mochizuki’s argument, in that the main Diophantine inequality claimed in IUTT-IV appears to have a robust family of counterexamples (assuming the truth some plausible conjectures, including abc): http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279
This doesn’t tell us exactly where the source of the error is coming from, though, or how fixable it would be. But it would be difficult to be optimistic about the proof until this issue is somehow resolved.
I quote from above site and Vesselin Dimitrov's arguement appears on MathOverFlow.
It looks like Vesselin has located a serious “red flag” in Mochizuki’s argument, in that the main Diophantine inequality claimed in IUTT-IV appears to have a robust family of counterexamples (assuming the truth some plausible conjectures, including abc): http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279
This doesn’t tell us exactly where the source of the error is coming from, though, or how fixable it would be. But it would be difficult to be optimistic about the proof until this issue is somehow resolved.
We discuss a minor error in the theory of [IUTchIII], [IUTchIV] concerning the precise content of the “\epsilon-portion” of the ABC Conjecture. This error is easily repaired and, moreover, has no effect on the conclusion constituted by the ABC Conjecture.
I find the below sentence on someone's BLOG and I'll read it.
For higher genus curves $g \geq 2$ (which is the typical situation when $F$ has degree $4$ or higher), we are quite far from a satisfactory theory, in large part because there is little group structure (or group-like structure) to exploit here. The celebrated proof by Faltings of the Mordell conjecture shows that the number of rational points is necessarily finite in this case, but as the proof proceeds by comparing rational points to each other, rather than to an external reference point, there is no effective bound known on the height of these points (though one can get a weak but effective bound on the number of points in terms of the rank of the Jacobian, which is in turn controlled by a "weak Mordell-Weil conjecture"). It would be a substantial breakthrough to get this effective bound. In particular there are the effective Mordell conjectures, one (rather naive) version of which can be stated as follows: given any curve $C$ of genus $g \geq 2$, one should have $h(x_1,x_2) \ll_g h(C)$ for all rational points $(x_1,x_2)$ in $C$, where the height $h(x_1,x_2)$ of the rational point is the maximum of the logarithms of the numerators and denominators of $x_1,x_2$, and the height $h(C)$ of the curve $C$ is (say) $1$ plus the logarithm of the largest coefficient in the defining polynomial $F$. This conjecture is already so strong that it implies the abc conjecture. It is in turn implied by an even stronger bound which was motivated from the function field model (i.e. complex surfaces rather than arithmetic ones) as an analogy. More precisely, since the well-known Bogomolov-Miyaoka-Yau inequality $c_1(X)^2 \leq 3 c_2(X)$ for complex surfaces controls the self-intersection of the first Chern class of the canonical complex line bundle of $X$ by the second Chern class; one can conjecture an arithmetic analogue of this inequality in which the self-intersection of a "relative canonical bundle" from Arakelov theory (a substitute for the canonical line bundle, which is not available due to the lack of differential structure in $\hbox{Spec}({\Bbb Z}))$ would be controlled by the logarithm of some sort of discriminant (plus some other correction terms), times a factor depending on the genus; I didn't fully understand the statement here, but apparently it implies the effective Mordell conjecture.
本記事の趣旨は、次の２点のことを申し上げたいのが趣旨です．
返信削除１、『ABC予想とは、』という話ですが、数論の大事件、快挙として『Fermat最終定理』と『Mordell予想』の証明がありました．ABC予想は、これら双方を従えるような重要な予想であることです．（初等的に２大予想の別証明をあたえるであろう、大予想であること）
２、数論の基本的な問題に、代数方程式の整数解の数を求めるという問題があります．歴史的にはディオファントス方程式の議論があるわけです．このような基本的問題の重要な目標となっています．
ということを申し上げたかったのです．大変大きなアクセスを頂いております．
It would cost several time whether true or not true.
返信削除On http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/
Terence Tao on September 26, 2012 said:
It looks like Vesselin has located a serious “red flag” in Mochizuki’s argument, in that the main Diophantine inequality claimed in IUTT-IV appears to have a robust family of counterexamples (assuming the truth some plausible conjectures, including abc): http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279
This doesn’t tell us exactly where the source of the error is coming from, though, or how fixable it would be. But it would be difficult to be optimistic about the proof until this issue is somehow resolved.
I quote from above site and Vesselin Dimitrov's arguement appears on MathOverFlow.
MathOverFlowのDimitrovさんの論調は、『、、、私の間違いかもしれないが、、、望月先生も、ABC予想を証明たと主張しているのだろうか？、、、』というトーン、、、
返信削除もちろん、DimitrovさんはABC予想が誤っているという反例ではなく、証明として成立していないのではという証拠として挙げられています．論文の中の定理１．１０の反例があると主張しています．
Terence Taoさんは、このDimitrovさんの主張を支持する内容になってます。
すでに英語圏では、Mochizuki denialとかいうブログサイトまで出てきていている．これも行き過ぎと思うが、、、worldpressのサイトだ．Wikipedia（英語版）が批判の対象のようである．（これは日本語版Wikipediaも単純に右ならえしている）
「ABC予想が証明された」というwikipediaの記事だけでない．
削除さらに、wikipedia（英語版）のFrobenioidまで、批判の対象となっているようだ．他にも色々と古い資料まで、、、
Tさんが『佐藤幹夫先生、久保田富雄先生、藤原正彦先生に学ぶBLOG』: http://study-underskf.blogspot.jp/2012/10/201210214.html
返信削除に『数理物理セミナ』のことを記載頂いています．
また、本記事のことも同時に記載いただいております．
6 Oct. 2012
Terence Taoさん、ほぼ否定的になっている．Dimitrovさんを支持は確定的のようだ．
返信削除It looks like Vesselin has located a serious “red flag” in Mochizuki’s argument, in that the main Diophantine inequality claimed in IUTT-IV appears to have a robust family of counterexamples (assuming the truth some plausible conjectures, including abc): http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279
This doesn’t tell us exactly where the source of the error is coming from, though, or how fixable it would be. But it would be difficult to be optimistic about the proof until this issue is somehow resolved.
Terence Taoさんの
返信削除http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/
で最後の発言を日本語化しておきます．Dimitrovさんの支持は確定的のようですが、『どこに誤りがあるかは、Dimitrovさんの反例からは分からない』とおっしゃっておられるようです．
--ここからTaoさんの発言
Vesselinさんは望月さんの議論には重大な｢赤信号」を点灯させたように思えます．IUTT-IVの中で主張している主要なDiophantine不等式は、反例の一群があるように思えます．（abc予想を含むいくつかの正しい予想を前提にしているようです．）http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279
これ（Domitrovさんの議論）は、どこから誤りがくるのかとか、あるいはどのくらい柔軟性があるかを、正確には教えてはくれません．しかし、この見解についての何らかの解決策がでるまでは、証明について楽観的な見通しを与えるのは難しいように思います．
--以上がTaoさんの発言
：発言中のURLは、Dimitrovさんの議論の掲載されているMathOverFlowのURLです．
13 Oct. 2012 望月先生が、Dimitrovさん他からの批判に対して、コメントを出している.
返信削除We discuss a minor error in the theory of [IUTchIII], [IUTchIV]
concerning the precise content of the “\epsilon-portion” of the ABC Conjecture. This error is easily repaired and, moreover, has no effect on the conclusion constituted by the ABC Conjecture.
とのこと．1か月のたたずに、、、凄いことです．
URLを書き忘れた．
返信削除URLは、
http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20%28comments%29.pdf
です．
9 Dec. 2012
返信削除I find the below sentence on someone's BLOG and I'll read it.
For higher genus curves $g \geq 2$ (which is the typical situation when $F$ has degree $4$ or higher), we are quite far from a satisfactory theory, in large part because there is little group structure (or group-like structure) to exploit here. The celebrated proof by Faltings of the Mordell conjecture shows that the number of rational points is necessarily finite in this case, but as the proof proceeds by comparing rational points to each other, rather than to an external reference point, there is no effective bound known on the height of these points (though one can get a weak but effective bound on the number of points in terms of the rank of the Jacobian, which is in turn controlled by a "weak Mordell-Weil conjecture"). It would be a substantial breakthrough to get this effective bound. In particular there are the effective Mordell conjectures, one (rather naive) version of which can be stated as follows: given any curve $C$ of genus $g \geq 2$, one should have $h(x_1,x_2) \ll_g h(C)$ for all rational points $(x_1,x_2)$ in $C$, where the height $h(x_1,x_2)$ of the rational point is the maximum of the logarithms of the numerators and denominators of $x_1,x_2$, and the height $h(C)$ of the curve $C$ is (say) $1$ plus the logarithm of the largest coefficient in the defining polynomial $F$. This conjecture is already so strong that it implies the abc conjecture. It is in turn implied by an even stronger bound which was motivated from the function field model (i.e. complex surfaces rather than arithmetic ones) as an analogy. More precisely, since the well-known Bogomolov-Miyaoka-Yau inequality $c_1(X)^2 \leq 3 c_2(X)$ for complex surfaces controls the self-intersection of the first Chern class of the canonical complex line bundle of $X$ by the second Chern class; one can conjecture an arithmetic analogue of this inequality in which the self-intersection of a "relative canonical bundle" from Arakelov theory (a substitute for the canonical line bundle, which is not available due to the lack of differential structure in $\hbox{Spec}({\Bbb Z}))$ would be controlled by the logarithm of some sort of discriminant (plus some other correction terms), times a factor depending on the genus; I didn't fully understand the statement here, but apparently it implies the effective Mordell conjecture.
Distinguished Lecture Series III: Shou-wu Zhang, “Triple L-series and effective Mordell conjecture”：http://bitly.com/UnVDbv
返信削除Distinguished Lecture Series II: Shou-wu Zhang, “Gross-Zagier formula and Birch and Swinnerton-Dyer conjecture ”：http://bitly.com/WU6mww
Distinguished Lecture Series I: Shou-wu Zhang, “Overview of rational points on curves”：http://bitly.com/TORy3S
が、タイトルとURLです．
the AMS blog: http://shar.es/ZG0IQ に、ABC予想のことでている.
返信削除25 Aug 2013
返信削除Jinさんという方が次のようなことをtwitterでおしゃっております。
1. また、望月新一さんの最近の論文では、現在のabc予想を中心とした宇宙際テータ関数についての研究から、宇宙際メリン変換の研究をすることで、リーマン予想に近づけるのでは、というコメントを追加しているそうです！
2．望月新一さん自ら養成している理解者とは、同じRIMSの星裕一郎さん、そしてもう一人はトヨタ研究所の山下剛さん。お二人ともまだ三十代前半の若き数学者。特に山下剛さんは望月新一さんの今までの論文を完全に理解し、abc予想の論文本体も理解しつつあるとのこと！
18 Dec. 2013
返信削除望月先生は、現在の状況を新しく公開しておられます．
http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Kenshou%20Houkoku%202013-12.pdf
10 Feb 2014
返信削除Langの本を読むべきかもしれない．