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100:PRQlÚÌf³ñ
21/04/17 11:59:07.65 cr30r3uy.net
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101:PRQlÚÌf³ñ
21/04/17 11:59:55.62 cr30r3uy.net
>>100
Âë
Ql¶£
], Vê (2008), gThe geometry of Frobenioids. I. The general theoryh, Kyushu Journal of Mathematics 62 (2): 293?400, doi:10.2206/kyushujm.62.293, ISSN 1340-6116, MR2464528
], Vê (2008), gThe geometry of Frobenioids. II. Poly-Frobenioidsh, Kyushu Journal of Mathematics 62 (2): 401?460, doi:10.2206/kyushujm.62.401, ISSN 1340-6116, MR2464529
], Vê (2009), gThe etale theta function and its Frobenioid-theoretic manifestationsh, 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
ON
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URLØÝž(mathoverflow.net)
What is an etale theta function?
asked Feb 6 '15 at 14:06
Minhyong Kim
(øpIè)
Èã
102:PRQlÚÌf³ñ
21/04/17 12:52:51.59 cr30r3uy.net
URLØÝž(www.kurims.kyoto-u.ac.jp)
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 gparallelograms, rectangles, and squaresh, 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 gfar from being an isomorphismh 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 gmassh represented by its domain as a gsomewhat smaller collection of massh
than the gmassh 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 gisomomorphism up to an appropriate renormalization operationh, in a fashion that is reminiscent of the isomorphism induced
on differentials by a Frobenius lifting, once one divides by p.
ÂÃ
103:PRQlÚÌf³ñ
21/04/17 12:53:20.91 cr30r3uy.net
>>102
Âë
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 TeichmNuller theory.
Contents:
Introduction
0. Notations and Conventions
1. Galois-theoretic Reconstruction Algorithms
2. Archimedean Reconstruction Algorithms
3. Nonarchimedean Log-Frobenius Compatibility
4. Archimedean Log-Frobenius Compatibility
5. Global Log-Frobenius Compatibility
Appendix: Complements on Complex Multiplication
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 TeichmNuller Theory
Acknowledgements
(øpIè)
Èã
104:PRQlÚÌf³ñ
21/04/17 15:05:33.76 8MN6ablF.net
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105:PRQlÚÌf³ñ
21/04/17 17:29:17.92 cr30r3uy.net
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106:PRQlÚÌf³ñ
21/04/17 18:23:51.69 cr30r3uy.net
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@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.
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@@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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@@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.
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@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0115ú
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@@@@Rational Numbers.
108:PRQlÚÌf³ñ
21/04/17 20:12:36.12 cr30r3uy.net
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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)
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2021N430ú 11:00 [L¿ïõÀè] úo iÒWÏõ@ÂØTêj
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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
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R@MipêÅjÍãÉIC[nipêÅjð\¬·éœßÉq[Oi[_ðp¢A»êÉæÁÄK 1 ÌÈ~ÈüÉηéo[`EXEBi[g_C[\zÌœðØŸµœB?õipêÅjÍOXEULGÌèðÈ~Èü©çW
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miniwiki
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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
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4 »ãÌW
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112:PRQlÚÌf³ñ
21/05/01 08:48:29.31 4gUFX+vb.net
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-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
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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
ºLªQlÉÈ軀
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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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_ÁÊu` II ] Vêisåwj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
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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).
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21/07/18 09:36:20.15 ycKpVVK0.net
prime-strip
ύtIASY
URLØÝž(nagasm.org)
FÛ TeichmNuller _üå
¯ TêY (såw ðÍ€)
2015 N 11
P19
6 ÅÍ v ž V(F) ðLÀf_Æ¢€±ÆɵĢܵœª, ±ÌÎÛ D?
v
(ÜœÍ F
?~
v
; F
?~Ê
v
; Dv;
Fv) ÉÍ g³Àf_Åh à è, »êçðWßé±ÆÅŸçêéÎÛ {D?
v }vžV(F )
, (ÜœÍ {F?~
v }vžV(F )
;
{F?~Ê
v }vžV(F )
; {Dv}vžV(F )
; {Fv}vžV(F )) ̯^šÍ, D? (ÜœÍ F?~; F?~Ê; D; F) f_È (D?-
(respectively, F
?~-; F
?~Ê-; D-; F-) prime-strip ? cf. [10], Definition 4.1, (iii) (respectively, [11],
Definition 4.9, (vii); [11], Definition 4.9, (vii); [10], Definition 4.1, (i); [10], Definition 5.2, (i)) ÆÄÎê
Ü·. (³mÉÍ, F ð»ÌKÈgåÌÉæèÖŠœè, Üœ, æèdvȱÆƵÄ, YÌ gvh ÌÍÍð,
»ÌgåÌÌ·×ÄÌf_Æ·éÌÅÍÈ, »ÌKȪWɧÀ·é, Æ¢ÁœC³ðs€Kvª
éÌÅ·ª@?@±êÉ¢ÄÍ 17 ÅüßÄàŸµÜ·.) ÈÆàLÀf_ÅÍ, gF nh ÌÎÛÍ (t
Á\¢t«) txjIChÅ è, gD nh ÌÎÛÍÊQ (Æ¿Èf[^) Å·. Üœ, g?h Æ¢€L
Í, FÛ TeichmNuller _ÅÍ, gPðIh ð\·LÆÈÁĢܷ4
ÂÃ
132:PRQlÚÌf³ñ
21/07/18 09:36:41.02 ycKpVVK0.net
>>131
Âë
7 ύtIASY
FÛ TeichmNuller _ÅÍ, gœçtIASYh Æ¢€ÁÊÈ«¿ðœ·ASYªdvÈð
ðÊœµÜ·. 8 Ås€FÛ TeichmNuller _ÌåèÌ g~j`
AÅh ÌàŸÌœßÉ, ±Ì 7
ÅÍ, »Ì gœçtIASYh Æ¢€TOÉ¢ÄÌÈPÈàŸðs¢Ü·. (ÚµÍ, áŠÎ, [11] Ì
Example 1.7 ©ç Remark 1.9.2 ÜÅ̪ðQÆŸ³¢.)
ÜžÅÉ, Ìæ€ÈÝèðl@µÜµå€. çtIf[^ (radial data ? cf. [11], Example 1.7, (i))
ÆÄÎêé éwIÎÛª^ŠçêÄ¢éƵܷ. É, »ÌçtIf[^©çASYIÉ\¬Å«
é (ºI) ÎÛÅ é RAIf[^ (coric data ? cf. [11], Example 1.7, (i)) ª^ŠçêÄ¢éƵ
Ü·. ±Ìæ€ÈÝèð çtI« (radial environment ? cf. [11], Example 1.7, (ii)) ÆÄÑÜ·. ïÌ
IÉÍ, áŠÎ, ȺÌæ€ÈçtI«ÌáðlŠé±ÆªÅ«Ü·:
(a) gçtIf[^h ƵÄ, 1 ³¡fü^óÔ C (̯^š) ð, gRAIªh ƵÄ, çtIf[^Å
é C (̯^š) ©ç g»Ì³¥\¢ðYêéh Æ¢€ASYÉæÁÄŸçêéº 2 ³Àü^óÔ
R
?2
(̯^š) ðÌp·é.
(øpIè)
Èã
133:PRQlÚÌf³ñ
21/07/18 11:26:10.63 ycKpVVK0.net
URLØÝž(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B72 (2018), 209?307
±EFÛ TeichmNuller _üå
¯ TêY
P227
6. si
µ©µÈªç, ȺÌRÉæÁÄ, äXÍ, ±Ì gàÁÆàÀŒÈAv[`h ð
Ìp·é±ÆªÅ«Ü¹ñ. ±ÌAv[`ðÌp·éÆ, ŒOÌ}ªŠ·æ€É, F
?
l =
{|1|, . . . , |l
?|} Ìe³ÉεÄ, ηé J ̳ƵÄ, òJ = l
? ÊèÌÂ\«ðl¶µÈ
¯êÎÈçÈÈèÜ·. »ÌÊ, SÌƵÄ, J Æ F
?
l ÆÌÖAƵÄ, òJòJ = (l
?)
l
?
ÊèÌÂ\«ðl¶µÈ¯êÎÈèܹñ. êû, ±ÌÂ\«ÌÂ@?@ÂÜè, sè
«@?@Í, äXÌÚWÌÏ_©çܷ߬͜. ÁÉ, È~Èü̳Ì]¿ÌÏ_©
çlŠÜ·Æ, ±ÌßåÈsè«ðeµÄµÜ€Æ, ]Ìs®æèà gã¢s®h
µ©Ÿé±ÆªÅ«ÈÈÁĵ܀ÌÅ·.
ãqÌâèðð·éœßÉ, si (procession ? cf. [7], Definition 4.10) Æ¢
€TOð±üµÜµå€.
siðlŠœêÌûª, œŸÌÛIÈWÆ©ôµœêæèà, xÌ
WÉÖ·és諪¬³Èé
Æ¢€dvÈÀðÏ@µÜµœ. siÆ¢€TOðp¢é±ÆÌÊÌ_ƵÄ,
ëxÌu£
Æ¢€_à°çêÜ·. |T| ðœŸÌWÆ©ô·, ÂÜè, |T| ð, |T| Ì©ÈSPËS
ÌÌÈ·QÌìpÆ¢€sè«ÌàÆŵ€ê, ëx 0 ž |T| ƻ̳̌ ž T
?
ðæÊ·é±ÆÍsÂ\Å·. êû, siðlŠœê, (gS
}
1
h Æ¢€f[^ÉæÁÄ)
0 ž |T| Í gÁÊȳh Æ¢€±ÆÉÈè, »ÌŒÌ³ ž T
? ÆÌæʪÂ\ÆÈèÜ·.
»µÄ, ÀÛ, FÛ TeichmNuller _Éš¢Ä,
ëxÍPI/RAIÈx, ñëxÍlQI/çtIÈx
Æ¢€Ï@ÌÆšè, ëxÆñëxÍ, ÜÁœÙÈéððÊœµÜ·. (4,
(d), â [2], 21, ÌOŒÌc_ðQÆŸ³¢.) ±ÌÏ_©ç, gëxÌu£Â\«h
ÍdvÅ·. (ÚµÍ [8], Remark 4.7.3, (iii), ðQÆŸ³¢.)
134:PRQlÚÌf³ñ
21/07/18 12:35:54.18 ycKpVVK0.net
Corollary 3.12, ÌØŸÖA
s®Ì±o
URLØÝž(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B72 (2018), 209?307
±EFÛ TeichmNuller _üå
¯ TêY
P297
25. Š
~Ê
LGP NÆŒ§IÈœçtI\ŠÆ»ÌA
P301
±Ì 25 ÌÅãÉ, ãqÌœçtI Kummer £Eðp¢œ q WÎÛÌÌvZÉ
¢Ä, ÈPÉàŸµÜµå€. (ÚµÍ, [9], Corollary 3.12, ÌØŸðQÆŸ³¢.)
±Ì 25 Ì`ªÌ Š
~Ê
LGP Nªèßé¯^ õ 0
C
?
LGP
?š ö 0
C
?
¢ Í,
õ 0Š WÎÛð ö 0
q W
ÎÛÉڵܷ. (24, (a), ðQÆŸ³¢.) µœªÁÄ, 14, (e), (i), ©ç, ]Ì
deg(ö 0
q WÎÛ) ð,
õ 0Š WÎÛÌ@? gõ Ì€h ̳¥\¢ÌÏ_©çÅÍÈ@?
gö Ì€h ̳¥\¢ÌÏ_©çÌÎÌÏðp¢ÄvZ·é±ÆªÂ\Å·. êû, œçt
I Kummer £EÉæÁÄ, sè« (Ind1), (Ind2), (Ind3) ðFßêÎ, Š~Ê
LGP NªU
±·é¯^ õ 0F
?~Ê
¢
?š ö 0F
?~Ê
¢ (24, (b), ðQÆ) ÆŒ§·é¯^ õ 0RFrob
?š ö 0RFrob
ªŸçêÜ·.
vol(ö 0Š) ž R Ÿ {}
ð, sè« (Ind1), (Ind2), (Ind3) ÌìpÉæé ö 0Š WÎÛÌO¹ÌaWÌ (gö Ì€h
̳¥\¢Éæé) ³¥ï (holomorphic hull ? cf. [9], Remark 3.9.5) ([2], 12, Ì
ãŒÌc_ðQÆ) Ìsi³K»ÎÌÏƵÄè`µÜµå€. ·éÆ, Œ§I¯^
õ 0RFrob
?š ö 0RFrob ̶ݩç,
õ 0Š WÎÛÌÎÌÏÍ, vol(ö 0Š) ȺÆÈçŽéðŸ
ܹñ. µœªÁÄ, _ƵÄ, s®
vol(ö 0Š) deg(ö 0q WÎÛ)
ªŸçêÜ·.
135:PRQlÚÌf³ñ
21/07/18 15:26:20.19 ycKpVVK0.net
URLØÝž(www.youtube.com)
FÛ^Cq~
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1,213 ñ®2020/04/11
îêóÔÌZVE³ñ
J[(khara,inc.)§ìÌIUTeichÖWÌCG®æyµÝ
URLØÝž(www.kurims.kyoto-u.ac.jp)
E®æ³URL
Animation 1 - URLØÝž(www.kurims.kyoto-u.ac.jp)
IUTeichÉÖ·éAj[Vi[IUTchIII], Theorem AÌàeÉÎj
@"The Multiradial Representation of Inter-universal Teichmuller Theory"ðöJB
ÎèÅF@u³v@tF[hAEgÅ@iavi wmvj@
Animation 2 - URLØÝž(www.kurims.kyoto-u.ac.jp)(animation).mp4
æñÌAIUTeichÉÖ·éAj[Vi[IUTchIII], Theorem BÌàeÉÎj
@"Computation of the log-volume of the q-pilot via the multiradial representation"
@ðöJB
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