(抜粋) In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves. (引用終り)
政治的に場合分けすると
１．もし、望月IUTが正しいとすると、 ”the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.” の３つは、今後、「おれが解決したぞ」という人が出てきても、二番煎じの扱いにしかならない
(引用開始) １．もし、望月IUTが正しいとすると、 ”the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.” の３つは、今後、「おれが解決したぞ」という人が出てきても、二番煎じの扱いにしかならない (引用終り)
もし、IUTを経由しない、従来の数学手法の改良で、 ”the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.” を証明できて、それがずっと簡明ならば、かなり高評価なのでしょうね 政治的には
などの円周率 π や自然対数の底 e の大抵の和、積、べき乗は、有理数であるのか無理数であるのか超越的であるのか否かは証明されていない[注 4]。
とあるけど？ 証明できたなら論文になるよ。 でもこの人の数学内容は全然信用できないけどね。 0322１３２人目の素数さん2019/12/07(土) 16:23:51.68ID:qvPzzpXn>>317 以前、IUT とかの数論幾何は標数0の実数体Rには無力ではないか、 という旨の内容のレスを IUT スレに書いたことがある。 そうしたら、IUT スレでその通りという返事が返って来た。 0323現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 16:24:59.86ID:H2e5WMAT>>306 補足 http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV:LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki October 2019 (抜粋) P3 Theorem A. (Diophantine Inequalities)
Thus, Theorem A asserts an inequality concerning the canonical height [i.e.,“htωX(D)”], the logarithmic different [i.e., “log-diffX”], and the logarithmic conductor [i.e., “log-condD”] of points of the curve UX valued in number fields whose extension degree over Q is ? d . In particular, the so-called Vojta Conjecture forhyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A.
P54 Corollary 2.3. (Diophantine Inequalities)
P57 Remark 2.3.3. Corollary 2.3 may be thought of as an effective version of the Mordell Conjecture. From this point of view, it is perhaps of interest to compare the “essential ingredients” that are applied in the proof of Corollary 2.3 [i.e., in effect, that are applied in the present series of papers!] with the “essential ingredients” applied in [Falt]. The following discussion benefited substantially from numerous e-mail and skype exchanges with Ivan Fesenko during the summer of 2015. (引用終り)
Vojta Conjecture forhyperbolic curves、ABC Conjecture、Szpiro Conjecture for elliptic curves、an effective version of the Mordell Conjecture 全部IUTの射程内だという 本当なら、面白いじゃない？ｗ（＾＾ 0324１３２人目の素数さん2019/12/07(土) 16:31:09.21ID:qvPzzpXn>>320 あの問いは、元々私が証明したと主張した代数的な命題から派生してスレ主に出題した問いだ。
https://en.wikipedia.org/wiki/Weil_conjectures Weil conjectures (抜粋) Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:
Use of Lefschetz pencils ・The theory of monodromy of Lefschetz pencils, introduced for complex varieties (and ordinary cohomology) by Lefschetz (1924), and extended by Grothendieck (1972) and Deligne & Katz (1973) to l-adic cohomology, relates the cohomology of V to that of its fibers. The relation depends on the space Ex of vanishing cycles, the subspace of the cohomology Hd?1(Vx) of a non-singular fiber Vx, spanned by classes that vanish on singular fibers. ・The Leray spectral sequence relates the middle cohomology group of V to the cohomology of the fiber and base. c(U,E), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces Ex of vanishing cycles.
The key estimate The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks. This is done by studying the zeta functions of the even powers Ek of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even k powers of E was inspired by the paper Rankin (1939), who used a similar idea with k=2 for bounding the Ramanujan tau function. Langlands (1970, section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization. つづく 0337現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 19:39:38.79ID:H2e5WMAT>>336
つづき
Completion of the proof The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
Deligne's second proof Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallee Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.
Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallee Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.
https://en.wikipedia.org/wiki/Lefschetz_pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
It has been shown that Lefschetz pencils exist in characteristic zero. They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds. It has also been shown that Lefschetz pencils exist in characteristic p for the etale topology.
Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them. (引用終り) 以上 0339現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 19:40:56.78ID:H2e5WMAT>>336
(>>306) http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: ¨ LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki October 2019 Abstract. (抜粋) n the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.
（>>323） http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV:LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki October 2019 (抜粋) P57 Remark 2.3.3. Corollary 2.3 may be thought of as an effective version of the Mordell Conjecture. 0355現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/11(水) 17:44:13.83ID:W0aIOzhV メモ
・ブルース・クライナーとジョン・ロット, Notes on Perelman's Papers（2006年5月） ペレルマンによる幾何化予想についての証明の細部を解明・補足 ・朱熹平と曹懐東、A Complete Proof of the Poincare and Geometrization Conjectures - application of the Hamilton- Perelman theory of the Ricci flow（2006年7月、改訂版2006年12月） ペレルマン論文で省略されている細部の解明・補足 ・ジョン・モーガンと田剛、Ricci Flow and the Poincare Conjecture（2006年7月） ペレルマン論文をポアンカレ予想に関わる部分のみに絞って詳細に解明・補足
まあ、IUTの人たちは説明責任を果たさないといけないね（＾＾ 0362現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/11(水) 21:19:43.41ID:6dBaZfdC なお、素人のドテ勘だが、IUTは成立しているんだろうと思っている （不成立にしては、IUTに関わる人が、大杉だ） 0363現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/11(水) 23:31:24.62ID:6dBaZfdC メモ https://arxiv.org/abs/1512.04389 https://arxiv.org/pdf/1512.04389.pdf Semi-galois Categories I: The Classical Eilenberg Variety Theory Takeo Uramoto (Submitted on 14 Dec 2015 (v1), last revised 20 Jan 2017 (this version, v4)) 0364現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/12(木) 07:34:32.59ID:aEgA7HUg>>363 追加
メモ http://www.math.is.tohoku.ac.jp/~uramoto/preprints/preprint_christol.pdf Semi-galois Categories II: An arithmetic analogue of Christol’s theorem Takeo Uramoto Graduate School of Information Sciences, Tohoku University February 12, 2018 (抜粋) 5.2 Canonicity of Eilenberg theory for DFAs and its geometric extension Geometric extension of Eilenberg theory
参考文献 [3] https://books.google.co.jp/books?id=_H7PuKhJHZkC&;printsec=frontcover&dq=Szamuely+Galois+groups+and+fundamental+groups&hl=ja&sa=X&ved=0ahUKEwjGv5nFirDmAhXDdXAKHUErAO4Q6AEIKTAA#v=onepage&q=Szamuely%20Galois%20groups%20and%20fundamental%20groups&f=false Galois Groups and Fundamental Groups Cambridge University Press 著者: Tamas Szamuely 2009
[12] https://arxiv.org/abs/0906.3146 https://arxiv.org/pdf/0906.3146.pdf Λ-RINGS AND THE FIELD WITH ONE ELEMENT JAMES BORGER 2009 Abstract. The theory of Λ-rings, in the sense of Grothendieck’s Riemann? Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.
Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumeration formulas in linear algebra over finite fields Fq as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geometry over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, following Tits [60], to call this mythical base F1, the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F1, that Spec Z ?F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function fields.1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes.
The most obvious way of doing this is to consider weaker algebraic structures than rings (commutative, as always), such as commutative monoids, and to try using them as the affine building blocks for a more rigid theory of algebraic geometry. This has been pursued in a number of papers, which I will cite below. Another natural approach is motived by the following question, first articulated by Soul´e [57]: Which rings over Z can be defined over F1? Less set-theoretically, on a ring over Z, what should descent data to F1 be? The main goal of this paper is to show that a reasonable answer to this question is a Λ-ring structure, in the sense of Grothendieck’s Riemann?Roch theory [31]. More precisely, we show that a Λ-ring structure on a ring can be thought of as descent data to a deeper base in the precise sense that it gives rise to a map from the big ´etale topos of Spec Z to a Λ-equivariant version of the big ´etale topos of Spec Z, and that this deeper base has many properties expected of the field with one element. Not only does the resulting algebraic geometry fit into the supple formalism of topos theory, it is also arithmetically rich?unlike the category of sets, say, which is the deepest topos of all. For instance, it is closely related to global class field theory, complex multiplication, and crystalline cohomology. (引用終り) 以上
