https://arxiv.org/pdf/2004.13108.pdf PROBABILISTIC SZPIRO, BABY SZPIRO, AND EXPLICIT SZPIRO FROM MOCHIZUKI’S COROLLARY 3.12 TAYLOR DUPUY AND ANTON HILADO Date: April 30, 2020. P14 Remark 3.8.3. (1) The assertion of [SS17, pg 10] is that (3.3) is the only relation between the q-pilot and Θ-pilot degrees. The assertion of [Moc18, C14] is that [SS17, pg 10] is not what occurs in [Moc15a]. The reasoning of [SS17, pg 10] is something like what follows: P15 (2) We would like to point out that the diagram on page 10 of [SS17] is very similar to the diagram on §8.4 part 7, page 76 of the unpublished manuscript [Tan18] which Scholze and Stix were reading while preparing [SS17]. References [SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017. 1, 1, 1e, 2, 7.5.3 ( http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html ) [Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2 つづく
http://www.kurims.kyoto-u.ac.jp/~motizuki/Tan%20---%20Introduction%20to%20inter-universal%20Teichmuller%20theory%20(slides).pdf Introduction to Inter-universal Teichm¨uller theory Fucheng Tan RIMS, Kyoto University 2018 To my limited experiences, the following seem to be an option for people who wish to get to know IUT without spending too much time on all the details. ・ Regard the anabelian results and the general theory of Frobenioids as blackbox. ・ Proceed to read Sections 1, 2 of [EtTh], which is the basis of IUT. ・ Read [IUT-I] and [IUT-II] (briefly), so as to know the basic definitions. ・ Read [IUT-III] carefully. To make sense of the various definitions/constructions in the second half of [IUT-III], one needs all the previous definitions/results. ・ The results in [IUT-IV] were in fact discovered first. Section 1 of [IUT-IV] allows one to see the construction in [IUT-III] in a rather concrete way, hence can be read together with [IUT-III], or even before. S. Mochizuki, The ´etale theta function and its Frobenioid-theoretic manifestations. S. Mochizuki, Inter-universal Teichm¨uller Theory I, II, III, IV.
http://www.kurims.kyoto-u.ac.jp/daigakuin/Tan.pdf 教員名: 譚 福成（Tan, Fucheng） P-adic Hodge theory plays an essential role in Mochizuki's proof of Grothendieck's Anabelian Conjecture. Recently, I have been studying anabeian geometry and Mochizuki's Inter-universal Teichmuller theory, which is in certain sense a global simulation of p-adic comparison theorem.
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf Research Institute for Mathematical Sciences - Kyoto University, Japan PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元 Online Seminar - Algebraic & Arithmetic Geometry Laboratoire Paul Painleve - Universite de Lille, France Version 1 ? ε - 09/10/2020
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/IUT-references.html Promenade in Inter-Universal Teichmuller Theory Org.: Collas (RIMS); Debes, Fresse (Lille). The Programme of the seminar contains a selection of ~30 references with respect to (1) Diophantine Geometry, (2) IUT Geometry, and (3) Anabelian Geometry. We indicate some links towards the key opuses as well as some complementary notes and proceedings.
https://en.wikipedia.org/wiki/Belyi%27s_theorem#Belyi_functions Belyi's theorem In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only.
This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.
Contents 1 Quotients of the upper half-plane 2 Belyi functions 3 Applications
Quotients of the upper half-plane It follows that the Riemann surface in question can be taken to be H/Γ with H the upper half-plane and Γ of finite index in the modular group, compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular curve.
Belyi functions A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C) ramified only over three points, which after a Mobius transformation may be taken to be {\displaystyle \{0,1,\infty \}}\{0,1,\infty \}. Belyi functions may be described combinatorially by dessins d'enfants.
Belyi functions and dessins d'enfants ? but not Belyi's theorem ? date at least to the work of Felix Klein; he used them in his article (Klein 1879) to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).[1]
Applications Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings". A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite. The faces of the embedding must be topological disks. The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.
Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.
For a more detailed treatment of this subject, see Schneps (1994) or Lando & Zvonkin (2004).
Contents 1 History 1.1 19th century 1.2 20th century 2 Riemann surfaces and Belyi pairs
History 19th century Early proto-forms of dessins d'enfants appeared as early as 1856 in the icosian calculus of William Rowan Hamilton;[1] in modern terms, these are Hamiltonian paths on the icosahedral graph.
Recognizable modern dessins d'enfants and Belyi functions were used by Felix Klein (1879). Klein called these diagrams Linienzuge (German, plural of Linienzug "line-track", also used as a term for polygon); he used a white circle for the preimage of 0 and a '+' for the preimage of 1, rather than a black circle for 0 and white circle for 1 as in modern notation.[2] He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with monodromy group PSL(2,11), following earlier constructions of a 7-fold cover with monodromy PSL(2,7) connected to the Klein quartic in (Klein 1878?1879a, 1878?1879b). These were all related to his investigations of the geometry of the quintic equation and the group A5 ? PSL(2,5), collected in his famous 1884/88 Lectures on the Icosahedron. The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of trinity.
20th century Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in 1984 in his Esquisse d'un Programme.[3] Zapponi (2003) quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants:
This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact. This is surely because of the very familiar, non-technical nature of the objects considered, of which any child’s drawing scrawled on a bit of paper (at least if the drawing is made without lifting the pencil) gives a perfectly explicit example. To such a dessin we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke.
Part of the theory had already been developed independently by Jones & Singerman (1978) some time before Grothendieck. They outline the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators, but do not consider the Galois action. Their notion of a map corresponds to a particular instance of a dessin d'enfant. Later work by Bryant & Singerman (1985) extends the treatment to surfaces with a boundary.
https://en.wikipedia.org/wiki/Belyi%27s_theorem Belyi's theorem Contents 1 Quotients of the upper half-plane 2 Belyi functions 3 Applications Applications Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.
https://en.wikipedia.org/wiki/Dessin_d%27enfant Dessin d'enfant Contents 1 History 1.1 19th century 1.2 20th century 2 Riemann surfaces and Belyi pairs 3 Maps and hypermaps 4 Regular maps and triangle groups 5 Trees and Shabat polynomials 6 The absolute Galois group and its invariants
Contents 1 Examples 2 Problems 3 Some general results
Problems No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the dessins d'enfants of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields.
In mathematics, a pro-p group (for some prime number p) is a profinite group {\displaystyle G}G such that for any open normal subgroup {\displaystyle N\triangleleft G}N\triangleleft G the quotient group {\displaystyle G/N}G/N is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite.
Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of discrete finite p-groups.
The best-understood (and historically most important) class of pro-p groups is the p-adic analytic groups: groups with the structure of an analytic manifold over {\displaystyle \mathbb {Q} _{p}}\mathbb {Q} _{p} such that group multiplication and inversion are both analytic functions. The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i.e. there exists a positive integer {\displaystyle r}r such that any closed subgroup has a topological generating set with no more than {\displaystyle r}r elements. More generally it was shown that a finitely generated profinite group is a compact p-adic Lie group if and only if it has an open subgroup that is a uniformly powerful pro-p-group.
The Coclass Theorems have been proved in 1994 by A. Shalev and independently by C. R. Leedham-Green. Theorem D is one of these theorems and asserts that, for any prime number p and any positive integer r, there exist only finitely many pro-p groups of coclass r. This finiteness result is fundamental for the classification of finite p-groups by means of directed coclass graphs.
局所は、局所化：環に乗法逆元を機械的に添加する 局所環：In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The English term local ring is due to Zariski.[2]
https://ja.wikipedia.org/wiki/%E7%92%B0%E3%81%AE%E5%B1%80%E6%89%80%E5%8C%96 環の局所化（きょくしょか、英: localization）あるいは分数環 (ring of fraction)、商環 (ring of quotient)[注 1] は、環に乗法逆元を機械的に添加する方法である。すなわち、環 R とその部分集合 S が与えられたとき、環 R' と R から R' への環準同型を構成して、S の準同型像が R' における単元（可逆元）のみからなるようにする。さらに、R' が「可能な限りで最良な」あるいは「最も一般な」ものとなるようにするということを考える（こういった状況はふつうは普遍性によって表されるべきものである）。環 R の部分集合 S による局所化は S−1R で表され、あるいは S が素イデアル {p} の補集合であるときには R_ {p}} で表される。S−1R のことを RS と表すこともあるが、通常混乱の恐れはない。
局所化は完備化と重要な関係があり、環を局所化すると完備になるということがよくある。
用語について 「局所化」の名の起源は代数幾何学にある。R はある幾何学的対象（代数多様体）の上で定義された函数環とする。この多様体を点 p の近傍で「局所的に」調べようとするならば、p の近傍で 0 でないような函数全体の成す集合 S を考えることになる。その意味で、R を S に関して局所化して得られる環 S−1R は p の近傍における V の挙動についての情報のみをふくんでいる（局所環も参照）。
例 整数環を Z, 有理数体を Q と表す。
R = Z のとき、積閉集合 S = Z − {0} による局所化は S−1R = Q である。
https://en.wikipedia.org/wiki/Local_ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe.[1] The English term local ring is due to Zariski.[2]
Examples All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings. A nonzero ring in which every element is either a unit or nilpotent is a local ring.
>>34 メモ https://en.wikipedia.org/wiki/Klein_quartic Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to PSL(2, 7), the second-smallest non-abelian simple group. The quartic was first described in (Klein 1878b).
Closed and open forms It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The open and closed quartics have different metrics, though they are both hyperbolic and complete[1] – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.
Affine quartic The above is a tiling of the projective quartic (a closed manifold); the affine quartic has 24 cusps (topologically, punctures), which correspond to the 24 vertices of the regular triangular tiling, or equivalently the centers of the 24 heptagons in the heptagonal tiling, and can be realized as follows.
Considering the action of SL(2, R) on the upper half-plane model H2 of the hyperbolic plane by Möbius transformations, the affine Klein quartic can be realized as the quotient Γ(7)\H2. (Here Γ(7) is the congruence subgroup of SL(2, Z) consisting of matrices that are congruent to the identity matrix when all entries are taken modulo 7.)
Fundamental domain and pants decomposition
3-dimensional models An animation by Greg Egan showing an embedding of Klein’s Quartic Curve in three dimensions, starting in a form that has the symmetries of a tetrahedron, and turning inside out to demonstrate a further symmetry.
Dessin d'enfants The dessin d'enfant on the Klein quartic associated with the quotient map by its automorphism group (with quotient the Riemann sphere) is precisely the 1-skeleton of the order-3 heptagonal tiling.[10] That is, the quotient map is ramified over the points 0, 1728, and ∞; dividing by 1728 yields a Belyi function (ramified at 0, 1, and ∞), where the 56 vertices (black points in dessin) lie over 0, the midpoints of the 84 edges (white points in dessin) lie over 1, and the centers of the 24 heptagons lie over infinity. The resulting dessin is a "platonic" dessin, meaning edge-transitive and "clean" (each white point has valence 2).
https://en.wikipedia.org/wiki/Klein_quadric Klein quadric In mathematics, the lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a quadric, Q known as the Klein quadric.
If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ2V of V. The line coordinates obtained this way are known as Plücker coordinates. 以上
Punctured spheres These statements are clarified by considering the type of a Riemann sphere {\displaystyle {\widehat {\mathbf {C} }}}\widehat{\mathbf{C}} with a number of punctures. With no punctures, it is the Riemann sphere, which is elliptic. With one puncture, which can be placed at infinity, it is the complex plane, which is parabolic. With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. With three or more punctures, it is hyperbolic – compare pair of pants. One can map from one puncture to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity), but all maps from zero punctures to one or more, or one or two punctures to three or more are constant.
Isometries of Riemann surfaces The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry: ・the isometry group of the plane is the subgroup fixing infinity, and of the punctured plane is the subgroup leaving invariant the set containing only infinity and zero: either fixing them both, or interchanging them (1/z).
>>24 ありがとう （追加） https://en.wikipedia.org/wiki/Dessin_d%27enfant Dessin d'enfant Contents 1 History 1.1 19th century 1.2 20th century 2 Riemann surfaces and Belyi pairs 3 Maps and hypermaps 4 Regular maps and triangle groups 5 Trees and Shabat polynomials 6 The absolute Galois group and its invariants
Riemann surfaces and Belyi pairs Each triangle in the triangulation has three vertices labeled 0 (for the black points), 1 (for the white points), or ∞. For each triangle, substitute a half-plane, either the upper half-plane for a triangle that has 0, 1, and ∞ in counterclockwise order or the lower half-plane for a triangle that has them in clockwise order, and for every adjacent pair of triangles glue the corresponding half-planes together along the portion of their boundaries indicated by the vertex labels. The resulting Riemann surface can be mapped to the Riemann sphere by using the identity map within each half-plane. Thus, the dessin d'enfant formed from f is sufficient to describe f itself up to biholomorphism. However, this construction identifies the Riemann surface only as a manifold with complex structure; it does not construct an embedding of this manifold as an algebraic curve in the complex projective plane, although such an embedding always exists.
The same construction applies more generally when X is any Riemann surface and f is a Belyi function; that is, a holomorphic function f from X to the Riemann sphere having only 0, 1, and ∞ as critical values. A pair (X, f) of this type is known as a Belyi pair. From any Belyi pair (X, f) one can form a dessin d'enfant, drawn on the surface X, that has its black points at the preimages f-1(0) of 0, its white points at the preimages f-1(1) of 1, and its edges placed along the preimages f-1([0, 1]) of the line segment [0, 1]. Conversely, any dessin d'enfant on any surface X can be used to define gluing instructions for a collection of halfspaces that together form a Riemann surface homeomorphic to X; mapping each halfspace by the identity to the Riemann sphere produces a Belyi function f on X, and therefore leads to a Belyi pair (X, f). Any two Belyi pairs (X, f) that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface X defined over the algebraic numbers, there are a Belyi function f and a dessin d'enfant that provides a combinatorial description of both X and f.
Maps and hypermaps A vertex in a dessin has a graph-theoretic degree, the number of incident edges, that equals its degree as a critical point of the Belyi function.
Thus, any embedding of a graph in a surface in which each face is a disk (that is, a topological map) gives rise to a dessin by treating the graph vertices as black points of a dessin, and placing white points at the midpoint of each embedded graph edge. If a map corresponds to a Belyi function f, its dual map (the dessin formed from the preimages of the line segment [1, ∞]) corresponds to the multiplicative inverse 1/f.[5]
A dessin that is not clean can be transformed into a clean dessin in the same surface, by recoloring all of its points as black and adding new white points on each of its edges. The corresponding transformation of Belyi pairs is to replace a Belyi function β by the pure Belyi function γ = 4β(1 - β).
The absolute Galois group and its invariants The two choices of a lead to two Belyi functions f1 and f2. These functions, though closely related to each other, are not equivalent, as they are described by the two nonisomorphic trees shown in the figure.
However, as these polynomials are defined over the algebraic number field Q(√21), they may be transformed by the action of the absolute Galois group Γ of the rational numbers. An element of Γ that transforms √21 to -√21 will transform f1 into f2 and vice versa, and thus can also be said to transform each of the two trees shown in the figure into the other tree.
More generally, due to the fact that the critical values of any Belyi function are the pure rationals 0, 1, and ∞, these critical values are unchanged by the Galois action, so this action takes Belyi pairs to other Belyi pairs. One may define an action of Γ on any dessin d'enfant by the corresponding action on Belyi pairs; this action, for instance, permutes the two trees shown in the figure.
Due to Belyi's theorem, the action of Γ on dessins is faithful (that is, every two elements of Γ define different permutations on the set of dessins),[10] so the study of dessins d'enfants can tell us much about Γ itself.
The two Belyi functions f1 and f2 of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.[11] (引用終り) 以上
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves {\displaystyle {\mathcal {M}}_{g,n}}{\mathcal {M}}_{{g,n}} and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck[1] to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin.[2]
メモ https://en.wikipedia.org/wiki/Perverse_sheaf Perverse sheaf The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space X, which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces (intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations (microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahiro Kawai). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory. The properties characterizing perverse sheaves already appeared in the 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules.
Contents 1 Preliminary remarks 2 Definition and examples 3 Properties 4 Applications 5 String Theory
String Theory Massless fields in superstring compactifications have been identified with cohomology classes on the target space (i.e. four-dimensional Minkowski space with a six-dimensional Calabi-Yau (CY) manifold). The determination of the matter and interaction content requires a detailed analysis of the (co)homology of these spaces: nearly all massless fields in the effective physics model are represented by certain (co)homology elements. However, a troubling consequence occurs when the target space is singular. A singular target space means that only the CY manifold is singular as Minkowski space is smooth. Such a singular CY manifold is called a conifold as it is a CY manifold that admits conical singularities. Andrew Strominger observed (A. Strominger, 1995) that conifolds correspond to massless blackholes.
These singular target spaces, i.e. conifolds, correspond to certain mild degenerations of algebraic varieties which appear in a large class of supersymmetric theories, including superstring theory (E. Witten, 1982).
In the winter of 2002, T. Hubsch and A. Rahman met with R.M. Goresky to discuss this obstruction and in discussions between R.M. Goresky and R. MacPherson, R. MacPherson made the observation that there was such a perverse sheaf that could have the cohomology that satisfied Hubsch's conjecture and resolved the obstruction. R.M. Goresky and T. Hubsch advised A. Rahman's Ph.D. dissertation on the construction of a self-dual perverse sheaf (A. Rahman, 2009) using the zig-zag construction of MacPherson-Vilonen (R. MacPherson & K. Vilonen, 1986). This perverse sheaf proved the Hübsch conjecture for isolated conic singularities, satisfied Poincarè duality, and aligned with some of the properties of the Kähler package.
Satisfaction of all of the Kähler package by this Perverse sheaf for higher codimension strata is still an open problem.
See also Mixed Hodge module Mixed perverse sheaf (引用終り)
メモ https://ncatlab.org/nlab/show/Grothendieck+fibration Grothendieck fibration Last revised on January 13, 2021 Contents 1. Idea 2. Definition 3. Fibrations versus pseudofunctors 4. Fibrations versus presheaves of categories
https://arxiv.org/pdf/1806.06129.pdf CATEGORICAL NOTIONS OF FIBRATION FOSCO LOREGIAN AND EMILY RIEHL Date: Original version December 20, 2010; revised version February 19, 2019.
https://en.wikipedia.org/wiki/Fibred_category Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories (over a site) with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories. Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971).
>>49 >Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory.
追加 https://en.wikipedia.org/wiki/Descent_(mathematics) Descent (mathematics) In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.
Contents 1 Descent of vector bundles 2 History 3 Fully faithful descent
Descent of vector bundles
Therefore, by going to a more abstract level one can eliminate the combinatorial side (that is, leave out the indices) and get something that makes sense for p not of the special form of covering with which we began. This then allows a category theory approach: what remains to do is to re-express the gluing conditions.
History The ideas were developed in the period 1955–1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem.
The difficulties of algebraic geometry with passage to the quotient are acute. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.
米田埋め込みとは，任意の局所小圏 C を C 上の前層 presheaf（Cop から Set への関手圏）に埋め込む関 手である．本稿では関手のいくつかの性質の定義を導入し，米田埋め込みを定義する．そしてそれが実際に埋 め込みになっていることを確認する．米田埋め込みとは直接関係はないが，第 1 節では圏同値の二つの定義を 紹介し，それらの定義が等しいことを確認する．
Antitone Galois connections Galois theory The motivating example comes from Galois theory: suppose L/K is a field extension. Let A be the set of all subfields of L that contain K, ordered by inclusion ⊆. If E is such a subfield, write Gal(L/E) for the group of field automorphisms of L that hold E fixed. Let B be the set of subgroups of Gal(L/K), ordered by inclusion ⊆. For such a subgroup G, define Fix(G) to be the field consisting of all elements of L that are held fixed by all elements of G. Then the maps E → Gal(L/E) and G → Fix(G) form an antitone Galois connection.
7 Connection to category theory 8 Applications in the theory of programming
Contents 1 In the school of Grothendieck 2 From pure category theory to categorical logic 3 Position of topos theory 4 Summary
In the school of Grothendieck During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of etale cohomology.
Summary The topos concept arose in algebraic geometry, as a consequence of combining the concept of sheaf and closure under categorical operations. It plays a certain definite role in cohomology theories. A 'killer application' is etale cohomology.
The subsequent developments associated with logic are more interdisciplinary. They include examples drawing on homotopy theory (classifying toposes). They involve links between category theory and mathematical logic, and also (as a high-level, organisational discussion) between category theory and theoretical computer science based on type theory. Granted the general view of Saunders Mac Lane about ubiquity of concepts, this gives them a definite status. The use of toposes as unifying bridges in mathematics has been pioneered by Olivia Caramello in her 2017 book.[1]
References Caramello, Olivia (2017). Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic `bridges. Oxford University Press. doi:10.1093/oso/9780198758914.001.0001. ISBN 9780198758914.
1. General Definitions, Examples and Applications 1.1 Definitions 1.2 Examples 1.3 Fundamental Concepts of the Theory 2. Brief Historical Sketch 3. Philosophical Significance Bibliography Academic Tools Other Internet Resources Related Entries
2. Brief Historical Sketch It is difficult to do justice to the short but intricate history of the field. In particular it is not possible to mention all those who have contributed to its rapid development. With this word of caution out of the way, we will look at some of the main historical threads.
Categories, functors, natural transformations, limits and colimits appeared almost out of nowhere in a paper by Eilenberg & Mac?Lane (1945) entitled “General Theory of Natural Equivalences.” We say “almost,” because their earlier paper (1942) contains specific functors and natural transformations at work, limited to groups. A desire to clarify and abstract their 1942 results led Eilenberg & Mac?Lane to devise category theory. The central notion at the time, as their title indicates, was that of natural transformation. In order to give a general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define functor, they borrowed the word ‘category’ from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it mathematically.
過去の論文のレベルでいうと、絶対遠アーベル幾何やエタール・テータ関数の様々な剛性性質に関する ・ Semi-graphs of Anabelioids ・ The Etale Theta Function ... ・ The Geometry of Frobenioids I, II ・ Topics in Absolute Anab. Geo. III の結果や理論を適用することによって主定理を帰結する: 主定理: θ-link の 左辺 に対して、軽微な不定性を除いて、右辺 の「異質」な環構造 しか用いない言葉により、明示的なアルゴリズム による記述を与えることができる。
In combinatorial mathematics, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle.[2][3] The cycle itself includes two out of the three adjacencies for each vertex, and the LCF notation specifies how far along the cycle each vertex's third neighbor is. A single graph may have multiple different representations in LCF notation.
LCF notation is a concise and convenient notation devised by Joshua Lederberg (winner of the 1958 Nobel Prize in Physiology and Medicine) for the representation of cubic Hamiltonian graphs (Lederberg 1965). The notation was subsequently modified by Frucht (1976) and Coxeter et al. (1981), and hence was dubbed "LCF notation" by Frucht (1976). Pegg (2003) used the notation to describe many of the cubic symmetric graphs. The notation only applies to Hamiltonian graphs, since it achieves its symmetry and conciseness by placing a Hamiltonian cycle in a circular embedding and then connecting specified pairs of nodes with edges.
Internal languages This can be seen as a formalization and generalization of proof by diagram chasing. One defines a suitable internal language naming relevant constituents of a category, and then applies categorical semantics to turn assertions in a logic over the internal language into corresponding categorical statements. This has been most successful in the theory of toposes, where the internal language of a topos together with the semantics of intuitionistic higher-order logic in a topos enables one to reason about the objects and morphisms of a topos "as if they were sets and functions".
Further reading Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic. Fairly accessible introduction, but somewhat dated. The categorical approach to higher-order logics over polymorphic and dependent types was developed largely after this book was published. Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, Elsevier. ISBN 0-444-50170-3. A comprehensive monograph written by a computer scientist; it covers both first-order and higher-order logics, and also polymorphic and dependent types. The focus is on fibred category as universal tool in categorical logic, which is necessary in dealing with polymorphic and dependent types.
P10 Bill Lawvere saw category theory as a new foundation for all mathematical thought. Mathematicians had been searching for foundations in the 19th century and were reasonably satisfied with set theory as the foundation. But Lawvere showed that the category of sets is simply a category with certain nice properties, not necessarily the center of the mathematical universe. He explained how whole algebraic theories can be viewed as examples of a single system. He and others went on to show that higher order logic was beautifully captured in the setting of category theory (more specifically toposes). It is here also that Grothendieck and his school worked out major results in algebraic geometry.
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.
The reverse of categorification is the process of decategorification. Decategorification is a systematic process by which isomorphic objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the representation theory of Lie algebras, modules over specific algebras are the principle objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications.[1]
Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'.[2]
Contents 1 Examples of categorification 2 Abelian categorifications 3 See also
Examples of categorification One form of categorification takes a structure described in terms of sets, and interprets the sets as isomorphism classes of objects in a category. For example, the set of natural numbers can be seen as the set of cardinalities of finite sets (and any two sets with the same cardinality are isomorphic). In this case, operations on the set of natural numbers, such as addition and multiplication, can be seen as carrying information about products and coproducts of the category of finite sets. Less abstractly, the idea here is that manipulating sets of actual objects, and taking coproducts (combining two sets in a union) or products (building arrays of things to keep track of large numbers of them) came first. Later, the concrete structure of sets was abstracted away - taken "only up to isomorphism", to produce the abstract theory of arithmetic. This is a "decategorification" - categorification reverses this step.
Other examples include homology theories in topology. Emmy Noether gave the modern formulation of homology as the rank of certain free abelian groups by categorifying the notion of a Betti number.[3] See also Khovanov homology as a knot invariant in knot theory.
An example in finite group theory is that the ring of symmetric functions is categorified by the category of representations of the symmetric group. The decategorification map sends the Specht module indexed by partition {\displaystyle \lambda }\lambda to the Schur function indexed by the same partition, (引用終り) 以上
In mathematics, the gluing axiom is introduced to define what a sheaf {F} on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor {F}: {O}(X)→ C to a category C}C which initially one takes to be the category of sets. Here {O}(X) is the partial order of open sets of X ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism U→ V if U is a subset of V}V, and none otherwise.
As phrased in the sheaf article, there is a certain axiom that F must satisfy, for any open cover of an open set of X. For example, given open sets U and V with union X and intersection W, the required condition is that {F}(X) is the subset of {F}(U) ｘ {F}(V) With equal image in {F}(W) In less formal language, a section s}s of F}F over X}X is equally well given by a pair of sections :(s',s'') on U and V respectively, which 'agree' in the sense that s' and s''have a common image in {F}(W) under the respective restriction maps {F}(U)→ {F}(W) and {F}(V)→ {F}. The first major hurdle in sheaf theory is to see that this gluing or patching axiom is a correct abstraction from the usual idea in geometric situations. For example, a vector field is a section of a tangent bundle on a smooth manifold; this says that a vector field on the union of two open sets is (no more and no less than) vector fields on the two sets that agree where they overlap.
Given this basic understanding, there are further issues in the theory, and some will be addressed here. A different direction is that of the Grothendieck topology, and yet another is the logical status of 'local existence' (see Kripke?Joyal semantics).
Sheafification To turn a given presheaf {P} into a sheaf {F}, there is a standard device called sheafification or sheaving. The rough intuition of what one should do, at least for a presheaf of sets, is to introduce an equivalence relation, which makes equivalent data given by different covers on the overlaps by refining the covers. One approach is therefore to go to the stalks and recover the sheaf space of the best possible sheaf {F} produced from {P}.
This use of language strongly suggests that we are dealing here with adjoint functors. Therefore, it makes sense to observe that the sheaves on X form a full subcategory of the presheaves on X. Implicit in that is the statement that a morphism of sheaves is nothing more than a natural transformation of the sheaves, considered as functors. Therefore, we get an abstract characterisation of sheafification as left adjoint to the inclusion. In some applications, naturally, one does need a description.
In more abstract language, the sheaves on X}X form a reflective subcategory of the presheaves (Mac Lane?Moerdijk Sheaves in Geometry and Logic p. 86). In topos theory, for a Lawvere?Tierney topology and its sheaves, there is an analogous result (ibid. p. 227). (引用終り) 以上
>>74 >Other examples include homology theories in topology. Emmy Noether gave the modern formulation of homology as the rank of certain free abelian groups by categorifying the notion of a Betti number.[3] See also Khovanov homology as a knot invariant in knot theory.
概要 結び目もしくは絡み目 L を表現する図形 D に、コバノフ括弧 [D]、これは次数付きベクトル空間の鎖複体、を割り当てる。すると、ジョーンズ多項式の構成の中でのカウフマン括弧の類似物となる。次に、[D] を（次数付きベクトル空間の中の）一連の次数シフトと（鎖複体の中の）高さシフトにより正規化して、新しい複体 C(D) を得る。この複体のホモロジーは L の不変量であることが分かり、その次数付きオイラー標数は L のジョーンズ多項式であることが分かる。
anser 45 One way to think of categorification is that it's a generalization of enumerative combinatorics. When a combinatorialist sees a complicated formula that turns out to be positive they think "aha! this must be counting the size of some set!" and when they see an equality of two different positive formulas they think "aha! there must be a bijection explaining this equality!" This is a special case of categorification, because when you decategorify a set you just get a number and when you decategorify a bijection you just get an equality. As a combinatorialist I'm sure you can come up with some examples that nicely illustrate how this sort of categorification is not totally well-defined. ("What exactly do Catalan numbers count?" has many answers rather than a single right answer.)
A more sophisticated kind of categorification in combinatorics is "Combinatorial Species" which categorify power series with positive coefficients.
Contents 1. Idea 2. Variants As a section of decategorification Examples As internalization in nCat Examples As homotopy coherent resolution Examples 3. Contrast to horizontal categorification 4. Homotopification versus laxification 5. Related entries 6. References
1. Idea
Roughly speaking, vertical categorification is a procedure in which structures are generalized from the context of set theory to category theory or from category theory to higher category theory.
What precisely that means may depend on circumstances and authors, to some extent. The following lists some common procedures that are known as categorification. They are in general different but may in cases lead to the same categorified notions, as discussed in the examples.
See also categorification in representation theory.
>>78 ＞One way to think of categorification is that it's a generalization of enumerative combinatorics. When a combinatorialist sees a complicated formula that turns out to be positive they think "aha! this must be counting the size of some set!" and when they see an equality of two different positive formulas they think "aha! there must be a bijection explaining this equality!" ＞A more sophisticated kind of categorification in combinatorics is "Combinatorial Species" which categorify power series with positive coefficients.
John Carlos Baez / Azimuthは、ちょっと大物かも David Roberts は、三流だと思うが
https://johncarlosbaez.wordpress.com/about/ About Hello! This is the official blog of the Azimuth Project. You can read about many things here: from math to physics to earth science and biology, computer science and the technologies of today and tomorrow—but in general, centered around the theme of what scientists, engineers and programmers can do to help save a planet in crisis.
https://johncarlosbaez.wordpress.com/2018/10/13/category-theory-course/ Azimuth Category Theory Course I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced.
David Roberts says: 14 October, 2018 at 10:09 pm Amusingly, that example on the first page on lecture one about fd vector spaces having skeleton the standard R^ns is one that Mochizuki (and Go Yamashita, acting as a proxy) claim shouldn’t do! See eg the bottom of page 2 in this FAQ by Yamashita http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20IUTeich.pdf namely the dialogue in A4. Odd…
Reply Todd Trimble says: 15 October, 2018 at 12:00 am I’m somewhat sympathetic to the sentiment that working with a skeleton can be occasionally confusing. Mainly because it can cause one to “see” things which are not actually there! One of my favorite examples is the conceptual distinction between linear orderings of the set \{1, 2, \ldots, n\} and permutations thereon. Because it’s hard not to notice the usual ordering there, it’s very tempting to conflate the two — an urge which goes away when one works not with this skeleton of finite sets, but finite sets more generally, where the distinction becomes totally clear. I gather that Mochizuki (or Yamashita) is driving at something similar.
Reply David Roberts says: 15 October, 2018 at 11:04 am I agree that blind reduction to the skeleton is not the way to do things, but I have taught first-year linear algebra a number of times, and our course uses exclusively the skeleton :-). Not to mention in physics, where everything is R^3 or R^4, and one just makes sure the not-standard basis is explicit.
https://en.wikipedia.org/wiki/John_C._Baez John Carlos Baez (/ˈbaɪɛz/; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR)[2] in Riverside, California. He has worked on spin foams in loop quantum gravity, applications of higher categories to physics, and applied category theory.
Blogs Baez runs the blog "Azimuth", where he writes about a variety of topics ranging from This Week's Finds in Mathematical Physics to the current focus, combating climate change and various other environmental issues.[11] (引用終り) 以上
Q8. Can you give examples of further research or results that arose from inter-universal Teichmüller theory? A8. I myself am interested in pursuing the possibility of applying various ideas that appear in inter-universal Teichmüller theory to the study of the Riemann zeta function. At the present
time, I have obtained some interesting observations, but no substantive results. Hoshi is studying an application of inter-universal Teichmüller theory to the birational section conjecture in birational anabelian geometry, while Porowski and Minamide are studying numerical improvements of certain height inequalities in inter-universal Teichmüller theory. I also hear that Dimitrov is studying the possibility of applying inter-universal Teichmüller theory to the study of Siegel-zeroes. References [pGC] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves. Invent. Math. 138 (1999), p.319423. [EtTh] S. Mochizuki, The Étale Theta Function and its Frobenioid-theoretic Manifestations. Publ. Res. Inst. Math. Sci. 45 (2009), p.227349. [AbsTopII] S. Mochizuki, Topics in Absolute Anabelian Geometry II: Decomposition Groups. J. Math. Sci. Univ. Tokyo 20 (2013), p.171269. [AbsTopIII] S. Mochizuki, Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms. J. Math. Sci. Univ. Tokyo 22 (2015), p.9391156. [FAQ] G. Yamashita, FAQ on Inter-Universality, an informal note available at http://www.kurims.kyoto-u.ac.jp/~motizuki/research-english.html [Y] G. Yamashita, A proof of the abc conjecture after Mochizuki, preprint available at http://www.kurims.kyoto-u.ac.jp/~gokun/myworks.html
1995 John Baez-James Dolan Opetopic sets (opetopes) based on operads. Weak n-categories are n-opetopic sets. 1995 John Baez-James Dolan Introduced the periodic table of mathematics which identifies k-tuply monoidal n-categories. It mirrors the table of homotopy groups of the spheres. 1995 John Baez?James Dolan Outlined a program in which n-dimensional TQFTs are described as n-category representations. 1995 John Baez?James Dolan Proposed n-dimensional deformation quantization. 1995 John Baez?James Dolan Tangle hypothesis: The n-category of framed n-tangles in n + k dimensions is (n + k)-equivalent to the free weak k-tuply monoidal n-category with duals on one object.
1995 John Baez-James Dolan Cobordism hypothesis (Extended TQFT hypothesis I): The n-category of which n-dimensional extended TQFTs are representations, nCob, is the free stable weak n-category with duals on one object. 1995 John Baez-James Dolan Stabilization hypothesis: After suspending a weak n-category n + 2 times, further suspensions have no essential effect. The suspension functor S: nCatk→nCatk+1 is an equivalence of categories for k = n + 2. 1995 John Baez-James Dolan Extended TQFT hypothesis II: An n-dimensional unitary extended TQFT is a weak n-functor, preserving all levels of duality, from the free stable weak n-category with duals on one object to nHilb.
https://en.wikipedia.org/wiki/John_C._Baez John Carlos Baez (/?ba??z/; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR)[2] in Riverside, California. He has worked on spin foams in loop quantum gravity, applications of higher categories to physics, and applied category theory.
Baez is also the author of This Week's Finds in Mathematical Physics,[3] an irregular column on the internet featuring mathematical exposition and criticism. (引用終り) 以上
自明に自明 一つ目は、通常の数学について、 どこが特殊性を使っているところで、 どこが一般論から従うところかを切り分ける手段を提供するところでしょう。 スローガンで言えば、Jon Peter Mayによる
Perhaps the purpose of categorical algebra is to show that which is formal is formally formal. や、その元となったPeter John Freydによる
Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial. になるかと思います（これらの出典や意味については、Mathematics Stack Exchangeの「“The purpose of being categorical is to make that which is formal, formally formal” what does it mean?」が参考になります）。
Pavel Samuilovich Urysohn (February 3, 1898 ? August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are fundamental results in topology. His name is also commemorated in the terms Urysohn universal space, Frechet?Urysohn space, Menger?Urysohn dimension and Urysohn integral equation. He and Pavel Alexandrov formulated the modern definition of compactness in 1923.
参考文献 望月, 新一 (2008), “The geometry of Frobenioids. I. The general theory”, Kyushu Journal of Mathematics 62 (2): 293?400, doi:10.2206/kyushujm.62.293, ISSN 1340-6116, MR2464528 望月, 新一 (2008), “The geometry of Frobenioids. II. Poly-Frobenioids”, Kyushu Journal of Mathematics 62 (2): 401?460, doi:10.2206/kyushujm.62.401, ISSN 1340-6116, MR2464529 望月, 新一 (2009), “The etale theta function and its Frobenioid-theoretic manifestations”, Kyoto University. Research Institute for Mathematical Sciences. Publications 45 (1): 227?349, doi:10.2977/prims/1234361159, ISSN 0034-5318, MR2512782 Mochizuki, Shinichi (2011), Comments
メモ https://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20III.pdf TOPICS IN ABSOLUTE ANABELIAN GEOMETRY III: GLOBAL RECONSTRUCTION ALGORITHMS Shinichi Mochizuki November 2015
Abstract. In the present paper, which forms the third part of a three-part series on an algorithmic approach to absolute anabelian geometry, we apply the absolute anabelian technique of Belyi cuspidalization developed in the second part, together with certain ideas contained in an earlier paper of the author concerning the category-theoretic representation of holomorphic structures via either the topological group SL2(R) or the use of “parallelograms, rectangles, and squares”, to develop a certain global formalism for certain hyperbolic orbicurves related to a oncepunctured elliptic curve over a number field. This formalism allows one to construct certain canonical rigid integral structures, which we refer to as log-shells, that are obtained by applying the logarithm at various primes of a number field. Moreover, although each of these local logarithms is “far from being an isomorphism” both in the sense that it fails to respect the ring structures involved and in the sense [cf. Frobenius morphisms in positive characteristic!] that it has the effect of exhibiting the “mass” represented by its domain as a “somewhat smaller collection of mass” than the “mass” represented by its codomain, this global formalism allows one to treat the logarithm operation as a global operation on a number field which satisfies the property of being an “isomomorphism up to an appropriate renormalization operation”, in a fashion that is reminiscent of the isomorphism induced on differentials by a Frobenius lifting, once one divides by p.
More generally, if one thinks of number fields as corresponding to positive characteristic hyperbolic curves and of once-punctured elliptic curves on a number field as corresponding to nilpotent ordinary indigenous bundles on a positive characteristic hyperbolic curve, then many aspects of the theory developed in the present paper are reminiscent of [the positive characteristic portion of] p-adic Teichm¨uller theory.
Introduction §I1. Summary of Main Results §I2. Fundamental Naive Questions Concerning Anabelian Geometry §I3. Dismantling the Two Combinatorial Dimensions of a Ring §I4. Mono-anabelian Log-Frobenius Compatibility §I5. Analogy with p-adic Teichm¨uller Theory Acknowledgements (引用終り) 以上
2021年04月15日 ・（論文）修正版を更新 https://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf （修正箇所のリスト）： https://www.kurims.kyoto-u.ac.jp/~motizuki/2021-04-15-ess-lgc-iut.txt ・Added an Introduction ・In \S 1.3, added "(UndIg)", as well as a reference to "(Undig)" in \S 2.1 ・Rewrote various portions of \S 1.5 ・Rewrote Example 2.4.4 ・Modified the title of Example 2.4.5 ・Added Example 2.4.6 ・Slightly modified the paragraph at the beginning of \S 3 ・Slightly modified the final portion of \S 3.1 concerning (FxRng), (FxEuc), (FxFld) ・Added Example 3.9.1 and made slight modifications to the surrounding text ・In \S 3.10, rewrote the discussion preceding (Stp1) ・In \S 3.11, slightly modified the discussion following ({\Theta}ORInd)
On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND "∧"/Logical OR "∨" Relations: Report on the Occasion of the Publication of the Four Main Papers on Inter-universal Teichmuller Theory.
2021年03月06日 ・（論文）宇宙際タイヒミューラー理論に関する論文4篇の出版を記念して、 新論文を掲載： On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND "∧"/Logical OR "∨" Relations: Report on the Occasion of the Publication of the Four Main Papers on Inter-universal Teichmuller Theory.
2021年04月15日 ・（論文）修正版を更新（修正箇所のリスト）： On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND "∧"/Logical OR "∨" Relations: Report on the Occasion of the Publication of the Four Main Papers on Inter-universal Teichmuller Theory. 2021年01月15日 ・（論文）修正版を更新（修正箇所のリスト）： 2021年03月06日 ・（論文）宇宙際タイヒミューラー理論に関する論文4篇の出版を記念して、 新論文を掲載： On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND "∧"/Logical OR "∨" Relations: Report on the Occasion of the Publication of the Four Main Papers on Inter-universal Teichmuller Theory. 2021年01月15日 ・（論文）修正版を更新（修正箇所のリスト）： Combinatorial Construction of the Absolute Galois Group of the Field of Rational Numbers.
IUT overview: What papers are involved? Where does it start? Taylor Dupuy 20151217 In this video I give an overview of what papers are involved in Mochizuki's work on ABC. Hopefully this is useful to get a scope of things.
http://www.uvm.edu/~tdupuy/anabelian/VermontNotes_20.pdf KUMMER CLASSES AND ANABELIAN GEOMETRY Date: April 29, 2017. JACKSON S. MORROW
ABSTRACT. These notes comes from the Super QVNTS: Kummer Classes and Anabelian geometry. Any virtues in the notes are to be credited to the lecturers and not the scribe; however, all errors and inaccuracies should be attributed to the scribe. That being said, I apologize in advance for any errors (typo-graphical or mathematical) that I have introduced. Many thanks to Taylor Dupuy, Artur Jackson, and Jeffrey Lagarias for their wonderful insights and remarks during the talks, Christopher Rasmussen, David Zureick-Brown, and a special thanks to Taylor Dupuy for his immense help with editing these notes. Please direct any comments to [email protected]. The following topics were not covered during the workshop: ・ mono-theta environments ・ conjugacy synchronization ・ log-shells (4 flavors) ・ combinatorial versions of the Grothendieck conjecture ・ Hodge theaters ・ kappa-coric functions (the number field analog of etale theta) ´ ・ log links ・ theta links ・ indeterminacies involved in [Moc15a, Corollary 3.12] ・ elliptic curves in general position ・ explicit log volume computations CONTENTS 1. On Mochizuki’s approach to Diophantine inequalities Lecturer: Kiran Kedlaya . . 2 2. Why the ABC Conjecture? Lecturer: Carl Pomerance . 3 3. Kummer classes, cyclotomes, and reconstructions (I/II) Lecturer: Kirsten Wickelgren . 3 4. Kummer classes, cyclotomes, and reconstructions (II/II) Lecturer: David Zureick-Brown . 6 5. Overflow session: Kummer classes Lecturer: Taylor Dupuy . 8 6. Introduction to model Frobenioids Lecturer: Andrew Obus . 11 7. Theta functions and evaluations Lecturer: Emmanuel Lepage . . 13 8. Roadmap of proof Notes from an email from Taylor Dupuy . . 17
（参考：文字化けは面倒なので修正しませんので、原文ご参照） https://en.wikipedia.org/wiki/Tate_twist Tate twist In number theory and algebraic geometry, the Tate twist,[1] named after John Tate, is an operation on Galois modules.
For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist of V, denoted V(1), is the representation on the tensor product V?Qp(1), where Qp(1) is the p-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure Ks of K). More generally, if m is a positive integer, the mth Tate twist of V, denoted V(m), is the tensor product of V with the m-fold tensor product of Qp(1). Denoting by Qp(?1) the dual representation of Qp(1), the -mth Tate twist of V can be defined as {\displaystyle V\otimes \mathbf {Q} _{p}(-1)^{\otimes m}.}{\displaystyle V\otimes \mathbf {Q} _{p}(-1)^{\otimes m}.} References [1] 'The Tate Twist', in Lecture Notes in Mathematics', Vol 1604, 1995, Springer, Berlin p.98-102
1．Robertとか、woitとか、間違った人のサイトを見ても、間違った情報しかないと思うよ 2．それよか、IUTを読むための用語集資料スレ2 http://2chb.net/r/math/1606813903/ に情報を集めているので、そこらも見てちょうだい 3．あと、下記を見る方が良いと思うよ 望月サイトのhttp://www.kurims.kyoto-u.ac.jp/~motizuki/ https://www.kurims.kyoto-u.ac.jp/~motizuki/papers-japanese.html 望月論文 講演のアブストラクト・レクチャーノート [1] 実複素多様体のセクション予想と測地線の幾何. PDF [2] p進Teichmuller理論. PDF [3] Anabelioidの幾何学. PDF [4] Anabelioidの幾何学とTeichmuller理論. PDF [5] 離散付値環のalmost etale extensions（学生用のノート）. PDF [6] 数体と位相曲面に共通する「二次元の群論的幾何」（2012年8月の公開講座）. PDF
Animation 2 - https://www.kurims.kyoto-u.ac.jp/~motizuki/2020-01%20Computation%20of%20q-pilot%20(animation).mp4 第二の、IUTeichに関するアニメーション（＝[IUTchIII], Theorem Bの内容に対応） "Computation of the log-volume of the q-pilot via the multiradial representation" を公開。
https://en.wikipedia.org/wiki/Legendre_form Legendre form In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because[1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity {\displaystyle \scriptstyle {k}}\scriptstyle {k} (the ellipse being defined parametrically by {\displaystyle \scriptstyle {x={\sqrt {1-k^{2}}}\cos(t)}}\scriptstyle{x = \sqrt{1 - k^{2}} \cos(t)}, {\displaystyle \scriptstyle {y=\sin(t)}}\scriptstyle{y = \sin(t)}). In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals. The Legendre form of an elliptic curve is given by y^{2}=x(x-1)(x-λ)
https://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 April 2020 P41 Corollary 2.2. (Construction of Suitable Initial Θ-Data) Suppose that X = P1Q is the projective line over Q, and that D ⊆ X is the divisor consisting of the three points “0”, “1”, and “∞”. We shall regard X as the “λ-line” - i.e., we shall regard the standard coordinate on X = P1 Q as the “λ” in the Legendre form “y2 = x(x-1)(x-λ)” of the Weierstrass equation defining an elliptic curve - and hence as being equipped with a natural classifying morphism UX → (Mell)Q [cf. the discussion preceding Proposition 1.8]. Let
本体リンク切れで、キャッシュ貼る https://webcache.googleusercontent.com/search?q=cache:k2PgzayvOKEJ:https://ncatlab.org/nlab/show/anabelioid+&cd=3&hl=ja&ct=clnk&gl=jp nLab anabelioid Contents 1. Introduction 2. Details 3. Associated notions 4. References Introduction 0.1 An anabelioid is a category intended to play the role of a ‘generalised geometric object’ in algebraic/arithmetic geometry. Its definition is simple: a finite product of Galois categories, or in other words of classifying topoi of profinite groups. The significance comes from the fact that in anabelian geometry, an algebraic variety is essentially determined by its algebraic fundamental group, which arises from a Galois category associated to the algebraic variety. The idea, due to Shinichi Mochizuki, is that one can develop the geometry of these Galois categories themselves, and products of Galois categories in general; thus, develop a form of categorical algebraic geometry.
To quote from Remark 1.1.4.1 of Mochizuki2004:
The introduction of anabelioids allows us to work with both “algebro-geometric anabelioids” (i.e., anabelioids arising from (anabelian) varieties) and “abstract anabelioids” (i.e., those which do not necessarily arise from an (anabelian) variety) as geometric objects on an equal footing.
The reason that it is important to deal with “geometric objects” as opposed to groups, is that:
We wish to study what happens as one varies the basepoint of one of these geometric objects.
Details 0.2 The following definitions follow Mochizuki2004.
Definition 0.3. A connected anabelioid is exactly a Galois category.
Definition 0.4. An anabelioid is a category equivalent to a finite product of connected anabelioids, that is, to a finite product of Galois categories.
Remark 0.5. An anabelioid is also known as a multi-Galois category.
Associated notions 0.6 finite etale morphism of anabelioids References 0.7 The geometry of anabelioids, Shinichi Mochizuki, 2004, Publ. Res. Inst. Math. Sci., 40, No. 3, 819-881. paper Zentralblatt review Created on April 17, 2020 at 18:29:54. See the history of this page for a list of all contributions to it. (引用終り) 以上
1 はじめに 集合論とはなにか? 自然数の全体 N を調べる理論を自然数論というのと同じよう に，集合論とはすべての集合のなす宇宙 V の構造を調べる理論である．この宇宙 V は代数や微積分などあらゆる数学の展開に十分なほど広大であることが知られてい る．本ノートは現代数学の標準言語でもある公理的集合論ZFC を紹介する．
高校数学でもおなじみの関係・関数の概念は，数学全般においても基本的かつ必要 不可欠である．数学だけではない．たとえば，数理論理学のモデル論は，述語記号は 関係を表し，関数記号は関数を表すとして構成されるので，関係・関数の概念は必要 不可欠である．本ノートの目標は V の構造の基本を述べることであるが，関係・関 数概念をきちんと定義するために必要な範囲の構造に限定される．したがって V 自 身の構造の深い性質についてはふれない．
http://www.uvm.edu/~tdupuy/anabelian/VermontNotes_20.pdf KUMMER CLASSES AND ANABELIAN GEOMETRY Date: April 29, 2017 JACKSON S. MORROW ABSTRACT. These notes comes from the Super QVNTS: Kummer Classes and Anabelian geometry. Any virtues in the notes are to be credited to the lecturers and not the scribe; however, all errors and inaccuracies should be attributed to the scribe. That being said, I apologize in advance for any errors (typo-graphical or mathematical) that I have introduced. Many thanks to Taylor Dupuy, Artur Jackson, and Jeffrey Lagarias for their wonderful insights and remarks during the talks, Christopher Rasmussen, David Zureick-Brown, and a special thanks to Taylor Dupuy for his immense help with editing these notes.
The following topics were not covered during the workshop: ・ mono-theta environments ・ conjugacy synchronization ・ log-shells (4 flavors) ・ combinatorial versions of the Grothendieck conjecture ・ Hodge theaters ・ kappa-coric functions (the number field analog of etale theta) ´ ・ log links ・ theta links ・ indeterminacies involved in [Moc15a, Corollary 3.12] ・ elliptic curves in general position ・ explicit log volume computations
p進Teichmuller理論 ”An Introduction to p-adic Teichmuller Theory”は、目を通しておくのが良い
https://www.kurims.kyoto-u.ac.jp/~motizuki/papers-japanese.html 望月 論文 p進Teichmuller理論 [3] An Introduction to p-adic Teichmuller Theory. PDF （これは、次のAsterisque, tome 278 (2002)と同じですね） https://www.kurims.kyoto-u.ac.jp/~motizuki/An%20Introduction%20to%20p-adic%20Teichmuller%20Theory.pdf
講演のアブストラクト・レクチャーノート [2] p進Teichmuller理論. PDF （Hokudai 2001-01 か） https://www.kurims.kyoto-u.ac.jp/~motizuki/p-shin%20Teichmuller%20riron%20no%20kaisetsu%20(Hokudai%202001-01).pdf An Introduction to p-adic Teichm¨uller Theory 望月 新一 TX 近藤智
P16 Here are some relations between the three generalisations of CFT and their further developments:
2dLC?−− 2dAAG−−− IUT l ／ ｜ ｜ l ／ ｜ ｜ l／ ｜ ｜ LC 2dCFT anabelian geometry ＼ ｜ ／ ＼ ｜ ／ ＼ ｜ ／ CFT 注）記号： Class Field Theory (CFT), Langlands correspondences (LC), 2dAAG = 2d adelic analysis and geometry, two-dimensional (2d) (P8 "These generalisations use fundamental groups: the etale fundamental group in anabelian geometry, representations of the etale fundamental group (thus, forgetting something very essential about the full fundamental group) in Langlands correspondences and the (abelian) motivic A1 fundamental group (i.e. Milnor K2) in two-dimensional (2d) higher class field theory.")
Problem 7. Find more direct relations between the generalisations of CFT. Use them to produce a single unified generalisation of CFT.23
現代的な定式化 現代的な言葉で言えば、基礎体 K の最大アーベル拡大 A は存在して、その拡大次数は K 上無限大となり得るから、その時 A に対応するガロワ群 G は副有限群となり、従ってコンパクト位相群かつまたアーベル群になる。類体論の中心定な目的は、この群 G を基礎体 K の言葉で記述することである。特に、K の有限次アーベル拡大と K に対する適当な（有限な剰余体を持つ局所体の場合の乗法群や大域体の場合のイデール類群のような）対象におけるノルム群との間の一対一対応を確立し、それらのノルム群を（例えば、指数有限な開部分群といったように）直截的に記述することである。そのような部分群に対応する有限次アーベル拡大を類体と呼び、これが理論の名称の由来となっている。
類体論の基本的な結果は「最大アーベル拡大のガロワ群 G は、基礎体 K のイデール類群 CK の（基礎体 K の特定の構造に関係して CK に入る自然な位相に関する）副有限完備化に自然同型である」ことを主張する。同じことだが、K の任意の有限次ガロワ拡大 L に対し、この拡大のガロワ群の最大アーベル商（アーベル化）と、K のイデール類群を L のイデール類群のノルム写像による像で割ったものとの間に、同型
素イデアル G の抽象的な記述だけではなくて、そのアーベル拡大においてどのように素イデアルが分解するかを理解することが数論の目的にとってより本質的である。この記述はフロベニウス元を用いて、二次体における素数の因数分解の様子を完全に与える二次の相互律を非常に広範に一般化するものである。つまり、類体論の内容には、（三次の相互律といったような）より高次の「冪剰余の相互律」についての理論が含まれるのである。
https://en.wikipedia.org/wiki/Complex_multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers;[1] and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
体 K 上定義されたアーベル多様体 A がCM-タイプ(CM-type)であるとは、自己準同型環 End(A) の中で十分に大きな部分可換環を持つことをいう。この用語は虚数乗法 (complex multiplication) 論から来ていて、虚数乗法論は19世紀に楕円曲線の研究のため開発された。20世紀の代数的整数論と代数幾何学の主要な成果のひとつに、アーベル多様体の次元 d > 1 の理論の正しい定式化が発見されたことがある。この問題は、多変数複素函数論を使うことが非常に困難であるため、非常に抽象的である。
K が複素数体であれば、任意のCM-タイプの A は、実は、数体である定義体（英語版）(field of definition)を持っている。自己準同型環の可能なタイプは、対合（ロサチの対合（英語版）(Rosati involution)）をもつ環として既に分類されていて、CM-タイプのアーベル多様体の分類を導き出す。楕円曲線と同じような方法でCM-タイプの多様体を構成するには、Cd の中の格子 Λ から始め、アーベル多様体のリーマンの関係式を考えに入れる必要がある。
CM-タイプ(CM-type)は、単位元における A の正則接空間上の、EndQ(A) の（極大）可換部分環 L の作用を記述したものである。単純な種類のスペクトル理論が適応され、L が固有ベクトルの基底を通して作用することを示すことができる。言い換えると、L は A の正則ベクトル場の上の対角行列を通した作用を持っている。L 自体が複数の体の積ではなく数体であるという単純な場合には、CM-タイプは L の複素埋め込み(complex embedding)のリストである。複素共役をペアとして、2d 個の複素埋め込みがあり、CM-タイプは各々のペアのから一つを選択する。そのようなCM-タイプの全てが実現されることが知られている。
1 CFT and its generalisations 2 Back to the root: CFT 3 Back to the root: CFT 4 CFT mechanism 5 CFT mechanism 6 Anabelian geometry 7 ‘Pre-Takagi’ LC 8 2D objects of HAT 9 HCFT 10 Zeta functions 11 Classical 1D theory of Iwasawa and Tate 12 HAT and elliptic curves 13 Measure and integration on 2D local fields 14 Two adelic structures in dimension 2 15 The triangle diagrammes 16 Higher zeta integral 17 HAT and meromorphic continuation and FE of the zeta function 18 HAT and GRH 19 HAT and the Tate?BSD conjecture
P29 Anabelian geometry and IUT
P33 Powerful restoration results in absolute mono-anabelian geometry were established by Mochizuki and applied in the IUT theory.
・J.-P. Serre, Local fields, Translated from the French by Marvin Jay Greenberg. Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979. ・J.-P. Serre, Local class field theory, 1967 Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 128-161 Thompson, Washington, D.C. ・J.-P. Serre, Abelian l-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute W. A. Benjamin, Inc., New York-Amsterdam 1968.
をそれぞれ挙げます．また，この講義でその説明を目標としている定理は，
・望月新一, A version of the Grothendieck conjecture for p-adic local fields, Internat. J. Math. 8 (1997), no. 4, 499-506. ・望月新一, Topics in absolute anabelian geometry I: generalities, J. Math. Sci. Univ. Tokyo 19 (2012), no. 2, 139-242. ・星裕一郎, A note on the geometricity of open homomorphisms between the absolute Galois groups of p-adic local fields, to appear in Kodai Math. J.
下記”Introducing anabelian geometry, a general talk” IVAN FESENKO これ、結構いいね
https://ivanfesenko.org/?page_id=126 IVAN FESENKO Research ? Ivan Fesenko L Anabelian geometry and IUT theory of Shinichi Mochizuki, and applications Introducing anabelian geometry, a general talk
https://en.wikipedia.org/wiki/Jean-Louis_Verdier Jean-Louis Verdier (French: [v??dje]; 2 February 1935 ? 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothendieck, notably contributing to SGA 4 his theory of hypercovers and anticipating the later development of etale homotopy by Michael Artin and Barry Mazur, following a suggestion he attributed to Pierre Cartier. Saul Lubkin's related theory of rigid hypercovers was later taken up by Eric Friedlander in his definition of the etale topological type.
https://www.kurims.kyoto-u.ac.jp/~motizuki/research-japanese.html 望月 過去と現在の https://www.kurims.kyoto-u.ac.jp/~motizuki/Kako%20to%20genzai%20no%20kenkyu.pdf ・過去と現在の研究の報告 (2008-03-25 現在） 初期の歩み 学位を取得した 1992 年夏から 2000 年夏までの私の研究の主なテーマは次の三つ に分類することができます： (a) p 進 Teichm¨uller 理論：(1993 年〜1996 年) この理論は、複素数体上の双曲的リーマン面に対する Koebe の上半平面に よる一意化や、そのモジュライに対する Bers の一意化の p 進的な類似と見る こともでき、また Serre-Tate の通常アーベル多様体に対する標準座標の理論の 双曲曲線版と見ることもできる。詳しくは、 A Theory of Ordinary p-adic Curves や An Introduction to p-adic Teichm¨uller Theory をご参照下さい。 (b) p 進遠アーベル幾何：(1995 年〜1996 年) この理論の代表的な定理は、「劣 p 進体」（＝ p 進局所体上有限生成な体の部 分体）上の相対的な設定において、双曲的曲線への任意の多様体からの非定数 的な射と、それぞれの数論的基本群の間の開外準同型の間に自然な全単射が存 在するというものである。詳しくは、 The Local Pro-p Anabelian Geometry of Curves をご参照下さい。 (c) 楕円曲線の Hodge-Arakelov 理論：(1998 年〜2000 年) この理論の目標は、複素数体や p 進体上で知られている Hodge 理論の類似 を、数体上の楕円曲線に対して Arakelov 理論的な設定で実現することにある。 代表的な定理は、数体上の楕円曲線の普遍拡大上のある種の関数空間と、楕円 曲線の等分点上の関数からなる空間の間の、数体のすべての素点において計量 と（ある誤差を除いて）両立的な全単射を主張するものである。この理論は、 古典的なガウス積分 ∫ ∞ ?∞ e?x2dx = √π の「離散的スキーム論版」と見ることもできる。詳しくは、 A Survey of the Hodge-Arakelov Theory of Elliptic Curves I, II をご参照下さい。
https://www.kurims.kyoto-u.ac.jp/~yuichiro/talk20140311_report.pdf 絶対 Galois 群による数体の復元 星 裕一郎 (京都大学 数理解析研究所) 2014 年 5 月 本稿は, 早稲田大学で開催された “第 18 回早稲田整数論研究集会” において 2014 年 3 月 11 日 に星が行った講演 “Reconstruction of a Number Field from the Absolute Galois Group” の報告原稿である. P1 ・ K を体とする. K が Q のある有限次拡大と同型であるとき, K は NF (= Number Field) であると言うことにする. ある素数 p が存在して K が Qp のある有限次拡大と同型であるとき, K は MLF (= Mixed-characteristic Local Field) であると言うことにする.
Peter Scholze君のIUTに対する批判（下記） ”the reader will not find any proof that is longer than a few lines ・・ which is in line with the amount of mathematical conten ” https://zbmath.org/pdf/07317908.pdf Mochizuki, Shinichi Inter-universal Teichmuller theory. I: Construction of Hodge theaters. (English) Publ. Res. Inst. Math. Sci. 57, No. 1-2, 3-207 (2021). Reviewer: Peter Scholze (Bonn) In parts II and III, with the exception of the critical Corollary 3.12, the reader will not find any proof that is longer than a few lines; the typical proof reads “The various assertions of Corollary 2.3 follow immediately from the definitions and the references quoted in the statements of these assertions.”, which is in line with the amount of mathematical content. (引用終り)
つまり ”the reader will not find any proof that is longer than a few lines”、”which is in line with the amount of mathematical content”
原文：Esaki's “five don’ts” rules 1.Don’t allow yourself to be trapped by your past experiences. 2.Don’t allow yourself to become overly attached to any one authority in your field ? the great professor, perhaps. 3.Don’t hold on to what you don’t need. 4.Don’t avoid confrontation. 5.Don’t forget your spirit of childhood curiosity.
https://en.wikipedia.org/wiki/Tate_module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G.
Contents 1 Definition 2 Examples 2.1 The Tate module 2.2 The Tate module of an abelian variety 3 Tate module of a number field
Examples The Tate module When the abelian group A is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module of A is sometimes referred to as the Tate module (where the choice of p and K are tacitly understood). It is a free rank one module over Zp with a linear action of the absolute Galois group GK of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character of K. It can also be considered as the Tate module of the multiplicative group scheme Gm,K over K.
Proposition 2.3 The construction A → A＾ satisfies the following properties. 1. The＾-construction defines an equivalence of categories from the category of compact topological abelian groups to the opposite of the category of discrete abelian groups. The＾-construction is its own inverse. 2. For a profinite group G, G＾ is isomorphic to Homc(G, μ∞), where μ∞ ⊆ S1 is the group of all roots of unity, isomorphic to Q/Z. If G is a p-profinite group, then μ∞ can be replaced by μp∞, the group of all p-power roots of unity, isomorphic to Z[1/p]/Z. 3. The functor A → A＾ is exact. 4. For G a profinite abelian group, G is torsion free if and only if G＾ is divisible. Similarly for “p-torsion free” and “p-divisible”.
Proof: Statement (1) is one version of the statement of the Pontrjagin duality theorem, (2) is an immediate consequence, and (3) follows immediately from (1). It remains to prove (4). To prove (4), we note that G is torsion free if and only if the sequence 0 → G ー(×n) -→ G is exact. The exactness proves that this occurs if and only if G＾ G＾ ×n ー(×n) -→G＾-→ 0 is exact, so ×n is surjective. This is the result.
We now have the main result of this section. Theorem 2.1 Let F be any field containing all roots of unity. Then the absolute Galois group GF of F is totally torsion free. Remark 2.3 Class field theory shows, for example, that one cannot expect this result to hold for absolute Galois groups of number fields, so that some condition on the field is necessary.
そこで、当時数人が集まってやっていた圏論勉強会に参加して圏論の勉強を始めました。当時読んでいた書籍は Conceptual Mathematics: A First Introduction to Categories でした。この本は圏論の初学者向けに書かれた本で、数学的な知識をほとんど仮定せずに理解できるように書かれている非常によい本です。一方で全く数学の素養がない状態で読むと、証明もちゃんと追えているのかあやふやでなんとなく分かった気にさせられる本でもあります。私がまさにそのような状態でした。
Abstract. We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to conclude that the “ABC Conjecture”, or, equivalently, the so-called “Effective Mordell Conjecture”, holds for arbitrary rational points of the projective line minus three points if and only if it holds for rational points which are in “sufficiently general position” in the sense that the following properties are satisfied: (a) the rational point under consideration is bounded away from the three points at infinity at a given finite set of primes; (b) the Galois action on the l-power torsion points of the corresponding elliptic curve determines a surjection onto GL2(Zl), for some prime number l which is roughly of the order of the sum of the height of the elliptic curve and the logarithm of the discriminant of the minimal field of definition of the elliptic curve, but does not divide the conductor of the elliptic curve, the rational primes that are absolutely ramified in the minimal field of definition of the elliptic curve, or the local heights [i.e., the orders of the q-parameter at primes of [bad] multiplicative reduction] of the elliptic curve.
Introduction In the classical intersection theory of subvarieties, or cycles, on algebraic varieties, various versions of the “moving lemma” allow one to replace a given cycle by another cycle which is equivalent, from the point of view of intersection theory, to the given cycle, but is supported on subvarieties which are in a “more convenient” position ? i.e., typically, a “more general” position, which is free of inessential, exceptional pathologies ? within the ambient variety.
＜q-parameter についてメモ＞ https://ivanfesenko.org/wp-content/uploads/2021/10/notesoniut.pdf ARITHMETIC DEFORMATION THEORY VIA ARITHMETIC FUNDAMENTAL GROUPS AND NONARCHIMEDEAN THETA-FUNCTIONS, NOTES ON THE WORK OF SHINICHI MOCHIZUKI IVAN FESENKO This text was published in Europ. J. Math. (2015) 1:405?440. P9 If v is a bad reduction valuation and Fv is the completion of F with respect to v, then the Tate curve F× v /hqvi, where qv is the q-parameter of EF at v and hqvi is the cyclic group generated by qv, is isomorphic to EF(Fv), hqvi → the origin of EF, see Ch.V of [44] and §5 Ch.II of [43]. P10 Define an idele qEF ∈ lim -→ A×k: its components at archimedean and good reduction valuations are taken to be 1. Its components at places where EF has split multiplicative reduction are taken to be qv, where qv is the q-parameter of the Tate elliptic curve EF(Fv) = F×v /hqvi. The ultimate goal of the theory is to give a suitable bound from above on deg(qEF). Fix a prime integer l > 3 which is relatively prime to the bad reduction valuations of EF, as well as to the value nv of the local surjective discrete valuation of the q-parameter qv for each bad reduction valuation v. P13 Let q ∈ L be a non-zero element of the maximal ideal of the ring of integers of L (this q will eventually be taken to be the q-parameter qv of the Tate curve EF(Fv) ' F×v /hqvi, where L = Fv, for bad reduction primes v of E, see Ch.5 of [44]).
Just as in the classical complex theory, elliptic functions on L with period q can be expressed in terms of θ, a property which highlights the central role of nonarchimedean theta-functions in the theory of functions on the Tate curve. For more information see §2 Ch.I and §5 Ch.II of [43] and p. 306-307 of [38]. ・・ via the change of variables q = exp(2πiτ),u = exp(2πiz)
P24 54 In IUT, the two combinatorial dimensions of a ring, which are often related to two ring-theoretic dimensions (one of which is geometric, the other arithmetic), play a central role. These two dimensions are reminiscent of the two parameters (one of which is related to electricity, the other to magnetism) which are employed in a subtle fashion in the study of graphene to establish a certain important synchronisation for hexagonal lattices. (引用終り) 以上
メモ （最新版） https://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf ON THE ESSENTIAL LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMULLER THEORY IN TERMS ¨ OF LOGICAL AND “∧”/LOGICAL OR “∨” RELATIONS: REPORT ON THE OCCASION OF THE PUBLICATION OF THE FOUR MAIN PAPERS ON INTER-UNIVERSAL TEICHMULLER THEORY ¨ Shinichi Mochizuki April 2022 P140版
（元） https://www.kurims.kyoto-u.ac.jp/~motizuki/On%20the%20Essential%20Logical%20Structure%20of%20IUT%20IV,%20V%20(marked%20up%20version).pdf ON THE ESSENTIAL LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMULLER THEORY I, II, III, IV, V ¨ Shinichi Mochizuki (RIMS, Kyoto University) September 2021 P42版
定義（等角同型）． ふたつのリーマン面 S と R が等角同型 (conformally isomorphic) または単に 同型 (isomorphic) であるとは，ある正則（等角）な同相写像 h : S → R が存在するときをいう． 定理 7.1 (一意化定理) 任意のリーマン面は，次のような形のリーマン面 R と等角同 型である： R = X/Γ ただし X = C?, C, もしくは D であり，Γ は P SL(2, C) のある離散部分群． まだ P SL(2, C) が X がどのように作用するのかが説明されていないので，現時点ではかなりあいま い主張であるが，この X/Γ がモデルに相当するリーマン面である．とりあえず，「任意のリーマン面 は，ごくごく簡単なリーマン面を，P SL(2, C) という比較的素性のよくわかっている群の部分群で 割ったものと同等だ」という部分に意味がある．1 以下ではその構成方法を概観するが，その手順は はあたかも，地球から地球儀を構成するかのようである．地表をくまなく歩いて地図帳を作り，それ を使い慣れた材質に写し取りながら模型を構成していく． まずは準備段階として，定理の証明に必要な「基本群と被覆空間」の用語を復習しつつ，リーマン 面の普遍被覆空間を構成する．2
8 リーマン面の一意化定理 一意化定理の証明を終わらせよう．手順としては，
8.2 商リーマン面の構成
8.3 リーマン面の一意化
単連結リーマン面の一意化定理. まず次の定理は証明無しで用いよう： 定理 8.5 (ケーベ，ポアンカレ) 任意の単連結リーマン面 X は，C?, C，もしくは D と 等角同型である． 証明は簡単ではない．まずコンパクトな場合（C? ）とそうでないでない場合に分け，さらにグリーン 関数が構成できる（D）かできない（C）かで区別される．
9 タイヒミュラー空間の定義 今回の目標はとにかく，タ空間を定義することにある．最初に前回の補足として例外型・双曲型 リーマン面について解説したあと，言葉の準備（写像の持ち上げ，リーマン面上の擬等角写像）をし て，定義に取り掛かる．定義の意味については，次回に． 以下，S, R をリーマン面とする．
9.2 写像の持ち上げ
9.3 リーマン面間の擬等角写像の定義
9.5 タイヒミュラー空間の定義 いよいよ，「リーマン面 S のタイヒミュラー空間」を定義する．とりあえず，形式的に定義を済ま せてしまおう． S とそのアトラス A を固定する．つぎに，別のリーマン面 R で，S からの向きを保つ擬等角写像 f : S → R が存在するようなもの全体を考える．もう少し形式的に，そのような f と R のペアとし て (R, f) の形のもの全体を考えるのである．この写像 f をマーキング (marking) と呼び，(R, f) を マークされたリーマン面 (marked Riemann surface) と呼ぶ． その全体の集合に，次の同値関係を考えよう：
It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6g-6 for a surface of genus g >= 2. In this way Teichmuller space can be viewed as the universal covering orbifold of the Riemann moduli space.
Contents 1 History 2 Definitions 2.1 Teichmuller space from complex structures 2.2 The Teichmuller space of the torus and flat metrics 2.3 Finite type surfaces 2.4 Teichmuller spaces and hyperbolic metrics 2.5 The topology on Teichmuller space 2.6 More examples of small Teichmuller spaces 2.7 Teichmuller space and conformal structures 2.8 Teichmuller spaces as representation spaces 2.9 A remark on categories 2.10 Infinite-dimensional Teichmuller spaces 3 Action of the mapping class group and relation to moduli space 3.1 The map to moduli space 3.2 Action of the mapping class group 3.3 Fixed points 4 Coordinates 4.1 Fenchel?Nielsen coordinates 4.2 Shear coordinates 4.3 Earthquakes 5 Analytic theory 5.1 Quasiconformal mappings 5.2 Quadratic differentials and the Bers embedding 5.3 Teichmuller mappings 6 Metrics 6.1 The Teichmuller metric 6.2 The Weil?Petersson metric 7 Compactifications 7.1 Thurston compactification 7.2 Bers compactification 7.3 Teichmuller compactification 7.4 Gardiner?Masur compactification 8 Large-scale geometry 9 Complex geometry 9.1 Metrics coming from the complex structure 9.2 Kahler metrics on Teichmuller space 9.3 Equivalence of metrics 10 See also 11 References 12 Sources 13 Further reading
History Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann (1826-1866), who knew that 6g-6 parameters were needed to describe the variations of complex structures on a surface of genus g >= 2. The early study of Teichmuller space, in the late nineteenth?early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincare, Paul Koebe, Jakob Nielsen, Robert Fricke and Werner Fenchel.
The main contribution of Teichmuller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmuller space (introduced by Bers).
The geometric vein in the study of Teichmuller space was revived following the work of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmuller space, and this is a very active subject of research in geometric group theory. (引用終り) 以上
似ているが、ちょっと違う Quasiregular map：between Euclidean spaces Rn of the same dimension or, more generally,・・ https://en.wikipedia.org/wiki/Quasiregular_map Quasiregular map In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces Rn of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable.
Contents 1 Motivation 2 Definition 3 Properties 4 Rickman's theorem 5 Connection with potential theory
http://www.misojiro.t.u-tokyo.ac.jp/~hirai/ Hiroshi Hirai Associate Professor Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan.
https://ncatlab.org/nlab/show/Teichm%C3%BCller+theory Teichmuller theory nLab Contents 1. Idea 2. Properties Complex structure on Teichmuller space Relation to moduli stack of complex curves / Riemann surfaces 3. Related concepts 4. References
3. Related concepts Kodaira-Spencer theory moduli space of curves Grothendieck-Teichmuller group quantum Teichmuller theory p-adic Teichmuller theory inter-universal Teichmuller theory Outer space for version in supergeometry see at super Riemann surface
Quantization of Teichmuller spaces and the quantum dilogarithm RM Kashaev Letters in Mathematical Physics 43 (2), 105-115 引用246 1998年
http://sciencewise.info/resource/Teichm_ller_modular_group/Teichm%C3%BCller_modular_group_by_Wikipedia ScienceWISE Mapping class group of a surface From Wikipedia, the free encyclopedia In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmuller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.
The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups.
Contents 1 History 2 Definition and examples 2.1 Mapping class group of orientable surfaces 2.2 The mapping class groups of the sphere and the torus 2.3 Mapping class group of surfaces with boundary and punctures 2.4 Mapping class group of an annulus 2.5 Braid groups and mapping class groups 2.6 The Dehn?Nielsen?Baer theorem 2.7 The Birman exact sequence 3 Elements of the mapping class group 3.1 Dehn twists 3.2 The Nielsen?Thurston classification 3.3 Pseudo-Anosov diffeomorphisms 4 Actions of the mapping class group 4.1 Action on Teichmuller space 4.2 Action on the curve complex 4.3 Other complexes with a mapping class group action 4.3.1 Pants complex 4.3.2 Markings complex 5 Generators and relations for mapping class groups 5.1 The Dehn?Lickorish theorem 5.2 Finite presentability 5.3 Other systems of generators 5.4 Cohomology of the mapping class group 6 Subgroups of the mapping class groups 6.1 The Torelli subgroup 6.2 Residual finiteness and finite-index subgroups 6.3 Finite subgroups 6.4 General facts on subgroups 7 Linear representations (引用終り) 以上
またRiemann面のmoduli spaceはmapping class groupの分類空間にかなり近いものであることも分か る。実際, Harerは[Har86]で, 「割る前」のTeichmuller空間を mapping class groupの作用を込めて考え, mapping class groupのvirtual cohomological dimensionの評価を得ている。
座標変換はまず φ?1 で M に戻してから ψ によって座標のある集合 V ' に写す写像である。間に座標が決められていない空間 M を挟む形になっているものの、座標変換全体はユークリッド空間の部分集合 U ' からユークリッド空間の部分集合 V ' への写像になっている。すなわち M を経由しているという事実を無視し、座標変換を合成写像としてではなく全体で 1 つの写像として捉えると、それは普通のユークリッド空間からユークリッド空間への写像である。
m 次元座標近傍の族 S = {(Uλ, φλ) | λ ∈ Λ} が M 全体を覆っているとする:
極大座標近傍系 m 次元位相多様体 M に対し Cn 級座標近傍系として S と T の 2つを取るとする。和集合 S ∪ T が再び M のCn 級座標近傍系になるとき、 S と T は同値であるという。これは同値関係を定める。これは S に属する座標近傍と T に属する座標近傍の間にも座標変換が存在し S での計算と T での計算に違いが無いという性質を保証するための同値関係である。
こうして座標近傍系の取り方に依存しない Cn 級多様体が定義される。m 次元位相多様体 M 上に互いに微分同相でない複数の微分構造が存在することもある。
多様体上の関数 m 次元 Cn 級多様体 M 上で定義された実数値関数 f を考える。
f: M → R これは、多様体上の点 p ∈ M に対して実数値 f(p) を対応させる関数である。特定の局所座標を考えているわけではないので、この関数の変数は (x1, x2, ..., xm) のように数を並べた座標ではなく単に点を表している。
{ φ(t) ∈ M | t ∈ I} という点の集合を曲線というのではなく、写像 φ を曲線というのである。なお、φ の変数 t を媒介変数という。
a ? c < d ? b とする。φ が 開区間 I = (a,b) で定義された Cr 級曲線であるとき、 I に含まれる閉区間 [c,d] や 半開区間 [c,d), (c,d] に φ の定義域を制限して得られる写像も Cr 級曲線という。
歴史 多様体の歴史はゲッティンゲンで行われたリーマンの講演に始まる。
多様体論は、ロバチェフスキーの双曲幾何学によって始まった非ユークリッド幾何学やガウスの曲面論を背景として様々な幾何学を統一し、 n 次元の幾何学へと飛躍させた。発見当初はカント哲学に打撃を与えた非ユークリッド幾何学も多様体論の一例でしかなくなってしまった。
リーマンがゲッティンゲン大学の私講師に就任するために行った講演『幾何学の基礎に関する仮説について』の中で「何重にも拡がったもの」と表現した概念が n 次元多様体のもとになり n 次元の幾何学に関する研究が始まった。この講演を聴いていたガウスがその着想に夢中になり、（ガウスは普段はあまり表立って他人を褒めることはなかったが、）リーマンの着想がいかに素晴らしいかを同僚に語り続けたり、帰り道にうわの空で道端の溝に落ちたりしたと言われている。
原文 Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911?1912, opening the road to the general concept of a topological space that followed shortly. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Notably, the Whitney embedding theorem[6] showed that the intrinsic definition in terms of charts was equivalent to Poincare's definition in terms of subsets of Euclidean space. (引用終り) 以上
ヘンゼルの補題（ヘンゼルのほだい、英: Hensel's lemma）とは、1変数多項式が素数 p を法として単根（英語版）を持つならば、その根は p の任意の冪乗を法とする根に一意的に持ち上げられるという、合同算術における補題である。この補題は、多項式が法 p で2つの互いに素な多項式（英語版）に因数分解できるならば、その因数分解は p の任意の冪乗を法とする因数分解に持ち上げることができるという補題に一般化できる。因数分解に現れる多項式の次数が1の場合が根の場合に相当する。ヘンゼルの持ち上げ補題（英: Hensel's lifting lemma）とも呼ばれる。名称はクルト・ヘンゼルに因む。
p の冪指数を無限に大きくしていったときの（射影極限の意味での）極限を取ることにより、法 p での根（または因数分解）を p 進整数上での根（または因数分解）に持ち上げることができる。
還元と持ち上げ R を可換環、I を R のイデアルとする。R の元を標準写像 R\→ R/I による像で置き換えることを、I を法とする還元、または法 I での還元と呼ぶ。 持ち上げとは還元の逆の操作である。つまり、R/I の元を使って表されている対象があったとき、持ち上げとは対象の性質を保ったまま還元するとこの対象に等しくなるように R（もしくはある k > 1 に対する R/I^{k}の元に置き換えることをいう。
Power series Main article: Formal power series Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the ring completion of the polynomial ring with respect to the ideal generated by x.
https://en.wikipedia.org/wiki/Formal_power_series Formal power series Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to p-adic integers, which can be defined as formal series of the powers of p.
https://arxiv.org/abs/math-ph/0504035 Mathematical Physics [Submitted on 10 Apr 2005] Riemann Hypothesis and Short Distance Fermionic Green's Functions Michael McGuigan
www.kurims.kyoto-u.ac.jp IUT I: CONSTRUCTION OF HODGE THEATERS Shinichi Mochizuki May 2020 Abstract. This data determines various hyperbolic orbicurves that are related via finite ´etale coverings to the once-punctured elliptic curve XF determined by EF.
https://researchmap.jp/Hiroaki_NAKAMURA/ 中村 博昭 On Arithmetic Monodromy Representations of Eisenstein Type in Fundamental Groups of Once Punctured Elliptic Curves Hiroaki Nakamura PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 49(3) 413-496 2013年9月 査読有り
このページは 1999 年8月〜12月にカリフォルニア大学・バークレーの 数理科学研究所 (MSRI) で行われた Program on Galois Groups and Fundamental Groups Organizers: Eva Bayer, Michael Fried, David Harbater, Yasutaka Ihara, B. Heinrich Matzat, Michel Raynaud, John Thompson の紹介ページ http://msri.org/activities/programs/9900/galois/ の日本語訳をもとに 中村が加工を施して作成したものです。(2000/10/1) (引用終り)
Articles on Anabelian Geometry H.Nakamura, A.Tamagawa, S.Mochizuki: ``The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves'' Copyright 1999 American Mathematical Society ``Sugaku Expositions'' (AMS), Volume 14 (2001), 31--53 English translation (by S.Mochizuki) from ``Sugaku'' 50(2), 1998, pp. 113-129 (Japanese). pdf http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf
H.Nakamura: "On Galois rigidity of fundamental groups of algebraic curves" in "Nonabelian Fundamental Groups and Iwasawa Theory" (J.Coates, M.Kim, F.Pop, M.Saidi, P.Schneider eds.) London Math. Soc. Lecture Note Series, 393 (2012), 56--71 (Cambridge UP). pdf http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/monkey/02nakamura.pdf This is a translation into English of an old Japanese article published in "Report Collection of the 35th Algebra Symposium held at Hokkaido University in 1989" + 8 complementary notes newly added in English.
Galois-Teichmueller theory: H.Nakamura : ``Limits of Galois representations in fundamental groups along maximal degeneration of marked curves II'' Proc. Symp. Pure Math., 70 (2002), 43--78 ps / pdf http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/anteater/naka-lim.pdf
H.Nakamura, H.Tsunogai, S.Yasuda: "Harmonic and equianharmonic equations in the Grothendieck-Teichmueller group, III" Journal Inst. Math. Jussieu 9 (2010), 431-448. NTY2010jimj.pdf (Copyright: Cambridge University Press) http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/squirrel/NTY2010jimj.pdf available from Cambridge Journals Online
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.
History Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann (1826-1866), who knew that 6g-6 parameters were needed to describe the variations of complex structures on a surface of genus g ≥ 2. The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke and Werner Fenchel.
The main contribution of Teichmüller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers).
The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this is a very active subject of research in geometric group theory.
A History and Survey of the Subject by Pierre Lochak International Centre for Theoretical Sciences 2024/02/26 DISCUSSION MEETING : GROTHENDIECK TEICHMÜLLER THEORY
ORGANIZERS : Pierre Lochak (CNRS and IMJ-PRG, Paris, France) and Devendra Tiwari (Bhaskaracharya Pratishthana, Pune, India) DATE : 26 February 2024 to 01 March 2024 VENUE : Madhava Lecture Hall, ICTS Bengaluru and Online Beyond “dessins d’enfant”, the theory nowadays referred to as Grothendieck-Teichmüller theory (Galois-Teichmüller in Grothendieck’s manuscripts) may well represent the main new theme in the Esquisse d'un Programme, as confirmed in the Promenade à travers une œuvre (which is part of Récoltes et semailles). Simplifying a great deal one may say that Grothendieck’s main ideas were taken up especially by Y. Ihara, V. Drinfeld and P. Deligne in the mid and late eighties.They derive in large part from the elementary remark that the fundamental group remains the only invariant in classical algebraic topology which is not a priori abelian .Making this remark fruitful probably required the genius of Alexandre Grothendieck . The fact is that out of it Grothendieck-Teichmüller theory (on which we will concentrate) and Anabelian Geometry (including the so-called “section conjecture”) were born.
In Grothendieck’s Esquisse, he is dealing with the full étale fundamental group, which is profinite almost by definition, or say by a form of the GAGA principle. It leads to the original version of the Grothendieck-Teichmüller group which again by definition (or by functoriality) and using the famous Belyi theorem, contains the absolute Galois group Gal(Q) of the field Q (the prime field in charateristic zero, as Grothendieck likes to put it).
A significant bifurcation occurred in Deligne’s 1989 paper on Le groupe fondamental de la droite projective moins trois points,in which the author brings in the rich toolbox of rational homotopy theory and motives (at least what we nowadays call mixed Tate motives),at the expense of using the prounipotent (not profinite) fundamental group. The ensuing version of the Grothendieck-Teichmüller group of course does not contain the Galois group anymore but this linearized version of the theory lends itself more easily to computations (e.g. those involving Multiple Zeta Values) and has become largely prevalent (including lately in deformation theory).
In this week long meeting we will discuss both versions (which could also be termed “linear” and “nonlinear”), including in particular an introduction to the profinite (nonlinear) version of the theory, which seems much closer to what Grothendieck initially had in mind and has been hitherto much less publicized. There will be mini-courses by subject experts of introductory nature for younger researchers, who were not exposed to these topics before.There will also be a few research talks by active researchers to explain the current state of the art in the subject of the meeting.
Accommodation will be provided for outstation participants at our on campus guest house. ICTS is committed to building an environment that is inclusive, non discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups. Eligibility Criteria: Senior Ph.D. students, postdocs, and faculties working on topics related to the theme of the meeting. (引用終り)
This multi-volume set deals with Teichmüller theory in the broadest sense, namely, as the study of moduli space of geometric structures on surfaces, with methods inspired or adapted from those of classical Teichmüller theory. The aim is to give a complete panorama of this generalized Teichmüller theory and of its applications in various fields of mathematics.
The volumes consist of chapters, each of which is dedicated to a specific topic. The present volume has 19 chapters and is divided into four parts:
The metric and the analytic theory (uniformization, Weil–Petersson geometry, holomorphic families of Riemann surfaces, infinite-dimensional Teichmüller spaces, cohomology of moduli space, and the intersection theory of moduli space). The group theory (quasi-homomorphisms of mapping class groups, measurable rigidity of mapping class groups, applications to Lefschetz fibrations, affine groups of flat surfaces, braid groups, and Artin groups). Representation spaces and geometric structures (trace coordinates, invariant theory, complex projective structures, circle packings, and moduli spaces of Lorentz manifolds homeomorphic to the product of a surface with the real line). The Grothendieck–Teichmüller theory (dessins d'enfants, Grothendieck's reconstruction principle, and the Teichmüller theory of the soleniod). This handbook is an essential reference for graduate students and researchers interested in Teichmüller theory and its ramifications, in particular for mathematicians working in topology, geometry, algebraic geometry, dynamical systems and complex analysis.
Y.Ihara, H.Nakamura: ``Some illustrative examples for anabelian geometry in high dimensions'' in `Geometric Galois Actions I' (L.Schneps, P.Lochak eds.) London Math. Soc. Lect. Note Series 242 (1997), pp. 127--138. http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/lion/INanabel.pdf
H.Nakamura: ``Galois representations in the profinite Teichmueller modular groups'' in `Geometric Galois Actions I' (L.Schneps, P.Lochak eds.) London Math. Soc. Lect. Note Series 242 (1997), pp. 159--173. http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/lion/Gaction.pdf
Galois-Teichmueller theory: P.Lochak, H.Nakamura, L.Schneps: "Eigenloci of 5 point configurations on the Riemann sphere and the Grothendieck-Teichmueller group" Math. J. Okayama Univ. 46 (2004), 39--75. http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/deer/_09_Lochak-Nakamura-Schneps.pdf
The Teichmüller space of a surface was introduced by O. Teichmüller in the 1930s. It is a basic tool in the study of Riemann's moduli spaces and the mapping class groups. These objects are fundamental in several fields of mathematics, including algebraic geometry, number theory, topology, geometry, and dynamics.
The original setting of Teichmüller theory is complex analysis. The work of Thurston in the 1970s brought techniques of hyperbolic geometry to the study of Teichmüller space and its asymptotic geometry. Teichmüller spaces are also studied from the point of view of the representation theory of the fundamental group of the surface in a Lie group
＜IUT最新文書＞ https://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html 2024年03月24日 望月新一 ・（過去と現在の研究）2024年4月に開催予定のIUGCの研究集会での講演の スライドを公開。https://www.kurims.kyoto-u.ac.jp/~motizuki/IUT%20as%20an%20Anabelian%20Gateway%20(IUGC2024%20version).pdf P8 In this context, it is important to remember that, just like SGA, IUT is formulated entirely in the framework of “ZFCG” (i.e., ZFC + Grothendieck’s axiom on the existence of universes), especially when considering various set-theoretic/foundational subtleties (?) of “gluing” operations in IUT (cf. [EssLgc], §1.5,§3.8,§3.9, as well as [EssLgc],§3.10, especially the discussion of “log-shift adjustment” in (Stp 7)): (引用終り)
ある特定の文脈において おそらく最も単純なバージョンは、研究対象が特定の集合で閉じている限り、任意の集合が宇宙であるというものである。 もし研究対象が実数として形式化されていれば、実数の集合である実数直線 R は考察下において宇宙になりうる。 これは1870年代から1880年代にかけてゲオルク・カントールが実解析の応用として、初の現代的な集合論と濃度の開発に用いた宇宙である。 カントールが当時興味を持っていた集合は、R の部分集合だった。
この宇宙の概念はベン図の使用に反映されている。 ベン図において、作用は伝統的に宇宙 U を表す大きな四角形の内部に生じる。 一般的に集合が U の部分集合であれば、それは円によって表現される。集合 A の補集合は A の円の外側の四角形の部分によって与えられている。
通常の数学 与えられた X (カントールの場合には、 X = R) の部分集合を考えれば、宇宙は X の部分集合の集合の存在を要請する。 (例えば、X の位相は X の部分集合の集合である。) X の様々な部分集合の集合は、それ自体は X の部分集合にならないが、代わりに X の冪集合 PX の要素はX の部分集合になる。 これに続き、研究対象は宇宙が P(PX) になるような場合における X の部分集合の集合などを構成する。
集合論 SNは通常の数学の宇宙であるという主張に正確な意味を与えることは可能である。すなわち、それはツェルメロ集合論のモデルである。 Vi のすべての和集合は次のようにフォン・ノイマン宇宙 V となる これらの和集合 V は真の類である。 置換公理と同時期にZFにを加られた正則性公理は、すべての 集合が V に属することを主張している。
クルト・ゲーデルの構成可能集合 L と構成可能公理 到達不能基数は ZF のモデルと加法性公理を生じ、さらにグロタンディーク宇宙の集合の存在と等価である。
アレクサンドル・グロタンディーク(Alexander Grothendieck)のアプローチは、固定された射有限群 G に対して有限 G-集合の圏を特徴付ける圏論的性質に関係している。例えば、G として ˆZ と表記される群が考えられる。この群は巡回加法群 Z/nZ の逆極限である。あるいは同じことであるが、有限指数の部分群の位相に対する無限巡回群の完備化である。すると、有限 G-集合は G が商有限巡回群を通して作用している有限集合 X であり、X の置換を与えると特定することができる。
上の例では、古典的なガロア理論との関係は、 ˆZ を任意の有限体 F 上の代数的閉包 F の射有限ガロア群 Gal(F/F) と見なすことである。すなわち、F を固定する F の自己同型は、 F 上の大きな有限分解体をとるように、逆極限により記述される。幾何学との関係は、原点を取り除いた複素平面内の単位円板の被覆空間として見なすことができる。複素変数 z と考えると、円板の zn 写像により実現される有限被覆は、穴あき円板の基本群の部分群 n.Z に対応する。
SGA1[1]で出版されたグロタンディークの理論は、どのようにして G-集合の圏をファイバー函手(fibre functor) Φ から再構成するかが示されている。ファイバー函手は、幾何学的な設定では、（集合として）固定されたベースポイント上の被覆のファイバーを持つ。実際、タイプ G ≅ Aut(Φ) として証明された同型が存在する。右辺は、Φ の自己同型群（自己自然変換）である。集合の圏への函手をもつ圏の抽象的な分類は、射有限な G に対する G-集合の圏を認識することによって与えられる。
PART I: Introduction and motivation The term “anabelian” was invented by Grothendieck, and a possible translation of it might be “beyond Abelian”. The corresponding mathematical notion of “anabelian Geometry” is vague as well, and roughly means that under certain “anabelian hypotheses” one has: ∗ ∗ ∗Arithmetic and Geometry are encoded in Galois Theory ∗ ∗ ∗ It is our aim to try to explain the above assertion by presenting/explaining some results in this direction. For Grothendieck’s writings concerning this the reader should have a look at [G1], [G2].
PART II: Grothendieck’s Anabelian Geometry The natural context in which the above result appears as a first prominent example is Grothendieck’s anabelian geometry, see [G1], [G2]. We will formulate Grothendieck’s anabelian conjectures in a more general context later, after having presented the basic facts about ´etale fundamental groups. But it is easy and appropriate to formulate here the so called birational anabelian Conjectures, which involve only the usual absolute Galois group.
P22 The result above by Mochizuki is the precursor of his much stronger result concerning hyperbolic curves over sub-p-adic fields as explained below.
PART III: Beyond Grothendieck’s anabelian Geometry
References Ihara, Y., On beta and gamma functions associated with the Grothendieck-Teichmller group II, J. reine angew. Math. 527 (2000), 1–11. Mochizuki, Sh., The profinite Grothendieck Conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Univ Tokyo 3 (1966), 571–627. Mochizuki, The absolute anabelian geometry of hyperbolic curves, Galois theory and modular forms, 77–122, Dev. Math., 11, Kluwer Acad. Publ., Boston, MA, 2004. Nagata, M., A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85–91. Nakamura, H., Galois rigidity of the ´ etale fundamental groups of punctured projective lines, J. reine angew. Math. 411 (1990) 205–216.
P10 The conclusion of this discussion is that with consistent identifications of copies of real numbers, one must in (1.5) omit the scalars j^2 that appear, which leads to an empty inequality. We voiced these concerns in this form at the end of the fourth day of discussions. On the fifth and final day,
Mochizuki tried to explain to us why this is not a problem after all. In particular, he claimed that up to the “blurring” given by certain indeterminacies the diagram does commute; it seems to us that this statement means that the blurring must be by a factor of at least O(l^2) rendering the inequality thus obtained useless. （google訳） 望月氏は、結局のところ、なぜこれが問題にならないのかを説明しようとしました。 特に、特定の不確定性によって与えられる「ぼやけ」までは、図は可換であると彼は主張した。 このステートメントは、ぼかしは少なくとも O(l^2) 倍でなければならず、こうして得られた不等式を役に立たなくすることを意味しているように私たちには思えます。
P9 2.2. Proof of [IUTT-3, Corollary 3.12]. As we indicated earlier, there is no clear distinction between abstract and concrete pilot objects in Mochizuki’s work, so it is argued in [IUTT-3, Corollary 3.12] that the multiradial algorithm [IUTT-3, Theorem 3.11]*12 implies that up to certain indeterminacies, e.g. (Ind 1,2,3) (without which the conclusion would be obviously false), this becomes an identification of concrete Θ-pilot objects and concrete q-pilot objects (encoded via their action on processions of tensor packets of log-shells), and then the inequality follows directly. 注） *12 We pause to observe that with the simplifications outlined above, such as identifying identical copies of objects along the identity, the critical [IUTT-3, Theorem 3.11] does not become false, but trivial.
https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf 望月新一 [3] Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice. PDF NEW !! (2020-05-18)
P154 for the collection of data (a), (b), (c) regarded up to indeterminacies of the following two types:
(Ind1) the indeterminacies induced by the automorphisms of the procession of D-prime-strips Prc(n,◦DT);
(Ind2) for each vQ ∈ Vnon Q (respectively, vQ ∈ Varc Q ), the indeterminacies induced by the action of independent copies of Ism [cf. Proposition 1.2, (vi)] (respectively, copies of each of the automorphisms of order 2 whose orbit constitutes the poly-automorphism discussed in Proposition 1.2, (vii)) on each of the direct summands of the j+1 factors appearing in the tensor product used to define IQ(S± j+1;n,◦DvQ ) [cf. (a) above; Proposition 3.2, (ii)] —where we recall that the cardinality of the collection of direct summands is equal to the cardinality of the set of v ∈ V that lie over vQ.
(Ind3) as one varies m ∈ Z, the isomorphisms of (a) are “upper semicompatible”, relative to the log-links of the n-th column of the LGPGaussian log-theta-lattice under consideration, in a sense that involves certain natural inclusions “⊆” at vQ ∈ Vnon Q and certain natural surjections “↠” at vQ ∈ Varc Q —cf. Proposition 3.5, (ii), (a), (b), for more details.
・プロ数学者が考えていることは、IUTを乗り越えていくこと ・"Arithmetic and Homotopic Galois Theory”は、IUTの復習セミナーにあらず ・みんな自分の次の論文を狙っています（下記は一例）
(参考) https://ahgt.math.cn...ry%20RIMS%202024.pdf A RIMS- Kyoto University & “Arithmetic and Homotopic Galois Theory” lecture BERKOVICH METHODS FOR ANABELIAN RECONSTRUCTIONS AND THE RESOLUTION OF NONSINGULARITIES E. LEPAGE- April. 08, 10, & 12, 2024
RESOLUTION OF NON-SINGULARITIES AND LOG-DIFFERENTIALS TALK 2 This talk will focus on Mochizuki and Tsujimura’s proof of the absolute anabelian conjecture: every isomorphism between the étale fundamental groups of hyperbolic curves over finite extensions of Qp is geometric. The new input of their work is the proof of resolution of non-singularities: given a hyperbolic curve X over a finite extensions of Qp is geometric, every divisorial valuations on K(X) comes from some irreducible component of the special fiber of the stable model after replacing X by some finite étale cover. If Mochizuki and Tsujimura’s proof is written in a purely scheme-theoretic framework, some of its intuition comes from previous work using analytic methods: resolution of non-singularities can be reduced to the study of the vanishing of differentials appearing in the image of the Hodge-Tate map H1(XCp ,Zp(1)) → H0(XCp ,Ω1). I will reformulate their proof using analytic geometry. ID:Agzcnutl(3/3)
なるほど ・Will Sawinのコメントは２カ所あり i)Cite Improve this answer Follow edited Dec 12, 2013 at 18:49 Will Sawin ii)2 That is quite a list of authors. – Will Sawin Oct 5, 2012 at 18:39 ですね。 ・補足すると、上記”ii)2”は、”answered Oct 5, 2012 at 7:45 Niels”へのコメントで ”i)Cite Improve ”は、Dec 12, 2013で １年後に思い出したようにFollowしている
追記 ・”1 users.ictp.it/~pub_off/lectures/lns001/Matsumoto/Matsumoto.pdf – Junyan Xu May 7, 2013 at 23:11 Add a comment” があるが、リンク切れ
(c)楕円曲線のHodge-Arakelov理論: (1998年〜2000年） この理論は、 古典的なガウス積分 ∫-∞〜∞ exp(-x^2)dx＝√π の「離散的スキーム論版」と見ることもできる。詳しくは、 A Survev of the Hodge-Arakelov TheolEV of ElliDtic Curves I.II をご参照下さい。
P5 因みに、2000年夏まで研究していたスキーム論的なHodge-Arakelov理論がガウス 積分 ∫-∞〜∞ exp(-x^2)dx＝√π の「離散的スキーム論版」だとすると、IUTbichは、 このガウス積分の「大域的ガロア理論版ないしはIU版」 と見ることができ、また古典的なガウス積分の計算に出てくる「直交座標」と「極座 標」の間の座標変換は、(IU版では）ちょうど「The geometry of Frobenioids l II」 で研究した「Frobenius系構造」と「etale系構造」の間の「比較理論」に対応して いると見ることができる。この「本体」の理論は、現在のところ二篇の論文に分けて 書く予定である。
(3) 圏の幾何：これについては、私の論文 ・Categorical representation of locally noetherian log schemes ・Categories of log schemes with archimedean structures ・Conformal and Quasiconformal Categorical Representation of Hyperbolic Riemann Surfaces
それから、講演のレクチャーノート ・「A Brief Survey of the Geometry of Categories (岡山大学 2005年5月)」 を参照して下さい。簡単にまとめると、スキーム（または、log schemeやarchimedeanな構造付きのlog scheme）や双曲的リーマン面の構造は、そのような対象たちが定義する圏（＝‘category'）の圏論的構造 だけで決まるという話です。
因みに、IUTeich関係の話では、p進Teichmuller理論に登場する「標準的なFrobenius持ち上げの微分を とる」という操作の「抽象的パターン的類似物」が主役です。p進Teichmuller理論の解説としては、 ・An Introduction to p-adic Teichmuller Theory ・「An Introduction to p-adic Teichmuller Theory」 (和文） が挙げられます。 (引用終り) 以上
https://ja.wikipedia.org/wiki/%E4%B8%80%E5%85%83%E4%BD%93 一元体（field with one element）あるいは標数1の体とは、「ただひとつの元からなる有限体」と呼んでもおかしくない程に有限体と類似の性質を持つ数学的対象を示唆する仮想的な呼称である しばしば、一元体を F1 あるいは Fun[note 1] で表す 通常の抽象代数学的な意味での「ただひとつの元からなる体」は存在せず、「一元体」の呼称や「F1」といった表示はあくまで示唆的なものでしかないということには留意すべきである
そういった新しい枠組みにおける理論で一元体を実現しているようなものは未だ存在していないが、標数 1 の体に類似した対象についてはいくつか知られており、それらの対象もやはり用語を流用して象徴的に一元体 F1 と呼ばれている なお、一元体上の数学は日本の黒川信重ら一部の数学者によって、絶対数学と呼ばれている
F1 が旧来の意味の体にならないことは、体が通常加法単位元 0 と乗法単位元 1 という二つの元を持つことから明らかである 制限を緩めて、ただひとつの元からなる環を考えても、それは 0 = 1 のみからなる零環 (trivial ring) であり、零環の振舞いと有限体の振る舞いは大きく違うものになってしまう 提案されている多くの F1 理論では抽象代数学をすっかり書き換えることが行われており、ベクトル空間や多項式環といった旧来の抽象代数学でしばしば扱われる数学的対象は、その抽象化された性質とよく似た性質を持つ新しい理論における対応物で置き換えられている このような理論によって新しい基礎付けのもと可換環論や代数幾何学の展開が可能となる こういった F1 についての理論の決定的な特徴のひとつは、新しい基礎付けのもとで古典的な抽象代数学で扱ったものよりも多くの数学的対象が扱えるようになり、そのなかに標数 1 の体であるかのように振舞う対象があるということである
Monoid schemes Deitmar's construction of monoid schemes[25] has been called "the very core of F1‑geometry",[16] as most other theories of F1‑geometry contain descriptions of monoid schemes. Morally, it mimicks the theory of schemes developed in the 1950s and 1960s by replacing commutative rings with monoids. The effect of this is to "forget" the additive structure of the ring, leaving only the multiplicative structure. For this reason, it is sometimes called "non-additive geometry". （google訳） モノイドスキーム Deitmar のモノイド スキームの構築[25] は、 F 1幾何学の他のほとんどの理論にモノイド スキームの記述が含まれているため、「 F 1幾何学のまさに核心」と呼ばれています[16]。道徳的には、可換環をモノイドに置き換えることによって 1950 年代と 1960 年代に開発されたスキーム理論を模倣しています。この効果は、環の加法構造を「忘れ」、乗法構造だけを残すことです。このため、「非加算ジオメトリ」と呼ばれることもあります
一般の幾何空間のアブストラクトナンセンスな定義 Cは圏で、その部分圏Lが与えられているとします。C=(可換環の圏)、L=（局所環の圏）が典型的な例です。圏Cの対象は、空間の上に棲んでいる関数達の集合を表現するモノです。部分圏Lの対象は特に、1点での関数芽の集合を表現するのに適したモノ、Lの射は1点の周辺の対応を記述するモノですね。茎や芽の概念を定義するために、圏Cでは、有向系（directed family of objects）の極限が取れる必要があります。
https://ja.wikipedia.org/wiki/%E4%B8%80%E5%85%83%E4%BD%93 一元体 しばしば、一元体を F1 あるいは Fun[note 1] で表す 一元体上の数学は日本の黒川信重ら一部の数学者によって、絶対数学と呼ばれている https://en.wikipedia.org/wiki/Field_with_one_element Field with one element Monoid schemes Deitmar's construction of monoid schemes[25] has been called "the very core of F1‑geometry",[16] as most other theories of F1‑geometry contain descriptions of monoid schemes. Morally, it mimicks the theory of schemes developed in the 1950s and 1960s by replacing commutative rings with monoids.
リンク切れ ”Characteristic 1, entropy and the absolute point” で下記のarxivヒットしたので貼る
https://arxiv.org/abs/0911.3537 [Submitted on 18 Nov 2009] Characteristic one, entropy and the absolute point Alain Connes, Caterina Consani
https://arxiv.org/pdf/0911.3537 CHARACTERISTIC 1, ENTROPY AND THE ABSOLUTE POINT arXiv:0911.3537v1 [math.AG] 18 Nov 2009 ALAIN CONNES AND CATERINA CONSANI
