IUTðÇÞœßÌpê ..
106:PRQlÚÌf³ñ
21/04/17 18:23:51.69 cr30r3uy.net
âêâê
C³ŸÁÄæ
URLØÝž(www.kurims.kyoto-u.ac.jp)
] ÅVîñ
2021N0415ú
@Ei_¶jC³ÅðXV URLØÝž(www.kurims.kyoto-u.ac.jp)
@iC³ÓÌXgjF URLØÝž(www.kurims.kyoto-u.ac.jp)
EAdded an Introduction
EIn \S 1.3, added "(UndIg)", as well as a reference to "(Undig)" in \S 2.1
ERewrote various portions of \S 1.5
ERewrote Example 2.4.4
EModified the title of Example 2.4.5
EAdded Example 2.4.6
ESlightly modified the paragraph at the beginning of \S 3
ESlightly modified the final portion of \S 3.1 concerning (FxRng), (FxEuc), (FxFld)
EAdded Example 3.9.1 and made slight modifications to the surrounding text
EIn \S 3.10, rewrote the discussion preceding (Stp1)
EIn \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 "È"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0306ú
@Ei_¶jFÛ^Cq~
[[_ÉÖ·é_¶4ÑÌoÅðLOµÄA
@@V_¶ðfÚF
@@On the Essential Logical Structure of Inter-universal Teichmuller
@@@Theory in Terms of Logical AND "È"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
107:PRQlÚÌf³ñ
21/04/17 20:09:05.72 cr30r3uy.net
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URLØÝž(www.kurims.kyoto-u.ac.jp)
]ÅVîñ
2021N0415ú
@Ei_¶jC³ÅðXViC³ÓÌXgjF
@@On the Essential Logical Structure of Inter-universal Teichmuller
@@@Theory in Terms of Logical AND "È"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0115ú
@Ei_¶jC³ÅðXViC³ÓÌXgjF
2021N0306ú
@Ei_¶jFÛ^Cq~
[[_ÉÖ·é_¶4ÑÌoÅðLOµÄA
@@V_¶ðfÚF
@@On the Essential Logical Structure of Inter-universal Teichmuller
@@@Theory in Terms of Logical AND "È"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0115ú
@Ei_¶jC³ÅðXViC³ÓÌXgjF
@@Combinatorial Construction of the Absolute Galois Group of the Field of
@@@@Rational Numbers.
108:PRQlÚÌf³ñ
21/04/17 20:12:36.12 cr30r3uy.net
>>105
>O^F_à»ÌÞ¢Å
>ÌÌW_ÌhUhiPÈéSÌWjÆÍAÓ¡ªá€ÌÅ·
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109:PRQlÚÌf³ñ
21/04/25 18:03:40.36 x2gQxWeE.net
URLØÝž(www.youtube.com)
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.
110:PRQlÚÌf³ñ
21/05/01 08:46:56.11 4gUFX+vb.net
Inter-universal geometry Æ ABC\z (X) 54
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URLØÝž(www.nikkei.com)
wÌïâABC\z@uØŸvÉàwEÍâââ©
2021N430ú 11:00 [L¿ïõÀè] úo iÒWÏõ@ÂØTêj
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q[Oi[_
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OXEUMGÌè (Gross & Zagier 1986) ÍA_ s = 1 Éš¯éÈ~ÈüÌLÖÌ÷ªÌ±ÆÎÅq[Oi[_̳ðLq·éBÆÉÈ~ÈüÌiðÍIjKª 1 Å êÎq[Oi[_ͳÀÊiµœªÁÄ[fEFCQipêÅjÌKÍ1ÈãjÌÈüãÌL_ð\¬·éÌÉg€±ÆªÅ«éBæèêÊÉAGross, Kohnen & Zagier (1987) ÍAq[Oi[_Íe³® n ÉεÈüãÌL_ð\¬·éÌÉg€±ÆªÅ«±êçÌ_̳ÍEFCg 3/2 ÌW
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111:PRQlÚÌf³ñ
21/05/01 08:47:39.05 4gUFX+vb.net
>>110
Âë
R@MipêÅjÍãÉIC[nipêÅjð\¬·éœßÉq[Oi[_ðp¢A»êÉæÁÄK 1 ÌÈ~ÈüÉηéo[`EXEBi[g_C[\zÌœðØŸµœB?õipêÅjÍOXEULGÌèðÈ~Èü©çW
[A[xœlÌÌêÖÆêÊ»µœBuEͳWÌåæÌãÌK 1 ÌÈ~ÈüÌœÉεÄo[`EXEBi[g_C[\zðØŸµœ (Brown 1994)B
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URLØÝž(sub-asate.ssl-lolipop.jp)
miniwiki
Þâè
iñÌÌjKEXÌÞâè(Gauss class number problem)ÍAÊíÉð³êÄ¢éæ€ÉA eXÌ n ? 1 Éεު n Å éñÌÌ®SÈXgðàœçµœB±ÌâèÌœŒÍÌåÈwÒJ[Et[hqEKEX(Carl Friedrich Gauss)É¿ÈñÅ¢éB±ÌâèÍAÜœAãÌ̻ʮÌÅLq·é±ÆàÅ«éBÀñÌÉàÖAµœâèª èA»ÌUé¢Í
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Contents
1 ³XÌKEXÌ\z
2 {âèÌóµ
3 Þ 1 ̻ʮÌXgAbv
4 »ãÌW
5 ÀñÌ
ÂÃ
112:PRQlÚÌf³ñ
21/05/01 08:48:29.31 4gUFX+vb.net
>>111
Âë
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Þ 1 ÌñÌÌ®SXgÍAQ(k--ã) Å±Ì k ÍÌÌêÂÅ éB
-1,-2,-3,-7,-11,-19,-43,-67,-163.
URLØÝž(en.wikipedia.org)
Class number problem
Contents
1 Gauss's original conjectures
2 Status
3 Lists of discriminants of class number 1
4 Modern developments
5 Real quadratic fields
(øpIè)
Èã
113:PRQlÚÌf³ñ
21/05/09 16:44:06.23 6xnjRD2S.net
URLØÝž(www.uvm.edu)
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 jmorrow4692@gmail.com.
The following topics were not covered during the workshop:
E mono-theta environments
E conjugacy synchronization
E log-shells (4 flavors)
E combinatorial versions of the Grothendieck conjecture
E Hodge theaters
E kappa-coric functions (the number field analog of etale theta) L
E log links
E theta links
E indeterminacies involved in [Moc15a, Corollary 3.12]
E elliptic curves in general position
E explicit log volume computations
CONTENTS
1. On Mochizukifs 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
114:PRQlÚÌf³ñ
21/07/05 06:06:22.96 tA3B4T+I.net
URLØÝž(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B76 (2019), 79?183
FÛ TeichmNuller _üå
(Introduction to Inter-universal TeichmNuller Theory)
By ¯ TêY (Yuichiro Hoshi)
P5
1. ~ªš
w ~ªšÆ͜ŵ倩. »êÍ Tate Pè gZb(1)ẖÆÅ·.
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Cyclotomic field
115:PRQlÚÌf³ñ
21/07/05 06:28:26.45 tA3B4T+I.net
>>114
>Tate Pè
ºLTate twist Ýœ¢ŸË
AµAºLÍhan operation on Galois moduleshÆ éÌÅ
¯æ¶ÌLqÆÍ¿åÁÆá€æ€È
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URLØÝž(en.wikipedia.org)
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 Ï : GK š 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
116:PRQlÚÌf³ñ
21/07/05 06:48:13.60 tA3B4T+I.net
>>115
>Tate twist
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»êÆhWhat is the intuition behind the concept of Tate twists?hÆ¿â·épšÍ©K€×«Åµå€Ë
URLØÝž(math.stackexchange.com)
About the definition of l-adic Tate-twist asked Sep 20 '18 at 6:30 Elvis Torres Perez
(²)
Zl(0)=Zl , Zl(1)=lim©?(Êli), Zl(n+1)=Zl(n)?ZlZl(1) for n0
URLØÝž(math.stackexchange.com)
What is the intuition behind the concept of Tate twists? asked Aug 16 '11 at 4:06 Nicole
117:PRQlÚÌf³ñ
21/07/05 20:32:45.22 tA3B4T+I.net
>>114Âë
URLØÝž(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B76 (2019), 79?183
FÛ TeichmNuller _üå
(Introduction to Inter-universal TeichmNuller Theory)
By ¯ TêY (Yuichiro Hoshi)
P9
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118:PRQlÚÌf³ñ
21/07/05 20:33:07.84 tA3B4T+I.net
>>117
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êû, M Í O?k̯^šÅ·©ç, n {ÊÌjM[n]def = Ker(n: M š M) Í Ên(k) ̯^šÆÈè, »Ì n ÉÖ·étÉÀðæé±Æ
Å, ©(M)def = lim©?nM[n] Æ¢€ ©(k) ̯^š, ÂÜè, ~ªšªŸçêÜ·. G ? ©(G)
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±Ì ({ÍÜÁœ³ÖWÈ) 2 ÂÌ~ªšÉÖµÄ, ȺÌÀªmçêÄ¢Ü
·. ([10], Remark 3.2.1, ðQÆŸ³¢.)
G ? M Æ¢€f[^©ç, ÖèIÉ, G ¯Ïȯ^ ©(M)?š ©(G) ?@ÂÜ
è, Frobenius I~ªšÆG^[I~ªšÆÌÔÌ~ª«¯^@?@ð\¬
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gÃTIÈ~ª«¯^h ÈÇÆÄÎêĢܷ.
(øpIè)
119:PRQlÚÌf³ñ
21/07/05 20:47:40.66 tA3B4T+I.net
URLØÝž(www.math.nagoya-u.ac.jp)
QOOPNxu`àevñ
wwÈ
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åw@
_ÁÊu` II ] Vêisåwj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
i11 19 ú`23 új uÈ~ÈüÌ Hodge-Arakelov _Éš¯éA[xôœv
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[Hh] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag (1977).
[Mn] J. S. Milne, Etale Cohomology L , Princeton Mathematical Series 33, Princeton University Press (1980).
ÂÃ
120:PRQlÚÌf³ñ
21/07/05 20:47:58.33 tA3B4T+I.net
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121:PRQlÚÌf³ñ
21/07/05 23:15:45.59 tA3B4T+I.net
FAinter-universal
URLØÝž(www.kurims.kyoto-u.ac.jp)(Muroran%202002-08).pdf
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ÍdvÅ·. (ÚµÍ [8], Remark 4.7.3, (iii), ðQÆŸ³¢.)
134:PRQlÚÌf³ñ
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Corollary 3.12, ÌØŸÖA
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URLØÝž(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B72 (2018), 209?307
±EFÛ TeichmNuller _üå
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25. Š
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21/07/18 23:36:38.51 ycKpVVK0.net
Legendre form
È~Èü gy^2 = x(x - 1)(x - É)h
URLØÝž(en.wikipedia.org)
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-É)
URLØÝž(www.kurims.kyoto-u.ac.jp)
INTER-UNIVERSAL TEICHMULLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS
Shinichi Mochizuki April 2020
P41
Corollary 2.2. (Construction of Suitable Initial Š-Data) Suppose that
X = P1Q is the projective line over Q, and that D º X is the divisor consisting of
the three points g0h, g1h, and gh. We shall regard X as the gÉ-lineh - i.e.,
we shall regard the standard coordinate on X = P1
Q as the gÉh in the Legendre
form gy2 = x(x-1)(x-É)h of the Weierstrass equation defining an elliptic curve -
and hence as being equipped with a natural classifying morphism UX š (Mell)Q
[cf. the discussion preceding Proposition 1.8]. Let
ÂÃ
137:PRQlÚÌf³ñ
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>>136
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P94
Q ÍLÌ Q ÌãÂï@-@ÆÌÔÉ, ©RÈSPË
ª¶ÝµÜ·. e³ É ž Q \ {0, 1} ÉεÄ, ûö® gy^2 = x(x - 1)(x - É)h ðlŠé±
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ð¿Ü·. ÁÉ, e³ É ž Q \ {0, 1} Éš¢Ä,
E È~Èü EÉ Ì q p[^ªèßé FÉ ãÌ_Iöq qÉ Ì deg(qÉ),
E _Iöq qÉ ªèßé FÉ ãÌ gíñh È_Iöq fÉ Ì deg(fÉ),
E Ì FÉ Ìâ΀ð·Ïªèßé FÉ ãÌ_Iöq dÉ Ì deg(dÉ),
E è]Ì Q(É) ÌLÌãÌgå dÉ def = [Q(É) : Q]
Æ¢€ 4 ÂÌlðlŠé±ÆªÅ«Ü·. ±êç 4 ÂÌlÍ, É ž Q\ {0, 1} ð»Ì GQ €
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Öh ÆlŠé±ÆªÅ«Ü·. ±ÌÝèÌàÆ, Belyi Êðp¢œc_ðKp·é±Æ
ÉæÁÄ, ±Ì 26 Ì`ªÅqל gDiophantus ôœwIs®h ðØŸ·éœßÉÍ,
ȺÌå£ðØŸ·êÎ[ªÅ é±Æªí©èÜ· ([5], Theorem 2.1; [10], Corollary
2.2, (i); [10], Corollary 2.3, ÌØŸðQÆ):
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Èã
138:PRQlÚÌf³ñ
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URLØÝž(blog.livedoor.jp)
ywzABC\zj
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EF. Tan and K. ChenÉæé[NVbv¿(2015.7ÉkÅJóêœuWorkshop on Inter-Universal Teichmuller Theoryvæè) (pê)
URLØÝž(wiutt.csp.escience.cn)
Note on the theory of Absolute Anabelian Geometry of Mochizuki URLØÝž(wiutt.csp.escience.cn)
EMinhyong KimÉæéðày[p[(pê)
URLØÝž(people.maths.ox.ac.uk)
E¯TêYÉæéT[xC(2015.12JÃÌ€WïàuFÛ Teichmuller _üåvÅÌu`¿)(ú{ê)
URLØÝž(www.kurims.kyoto-u.ac.jp)
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21/08/17 16:51:56.02 nT2E/2XT.net
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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 egeneralised geometric objectf 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 galgebro-geometric anabelioidsh (i.e., anabelioids arising from (anabelian) varieties) and gabstract anabelioidsh (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 ggeometric objectsh 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.
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