IUTđÇŢ˝ßĚpę ..
104:PRQlÚĚfłń
21/04/17 15:05:33.76 8MN6ablF.net
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IUTÍwƢ¤ŠO^F_ÉČÁÄéČ
105:PRQlÚĚfłń
21/04/17 17:29:17.92 cr30r3uy.net
>>104
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106:PRQlÚĚfłń
21/04/17 18:23:51.69 cr30r3uy.net
âęâę
CłžÁÄć
URLŘݸ(www.kurims.kyoto-u.ac.jp)
] ĹVîń
2021N0415ú
@Ei_śjCłĹđXV URLŘݸ(www.kurims.kyoto-u.ac.jp)
@iCłÓĚXgjF URLŘݸ(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 "Č"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0306ú
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[[_ÉÖˇé_ś4ŃĚoĹđLOľÄA
@@V_śđfÚF
@@On the Essential Logical Structure of Inter-universal Teichmuller
@@@Theory in Terms of Logical AND "Č"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
107:PRQlÚĚfłń
21/04/17 20:09:05.72 cr30r3uy.net
>>106 ÇÁ
d Ě÷šŞ
şLĚ
h2021N0115ú
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u2021N0415úvĚCłĹđĆŤĚ~XRs[iłçÉşĚh2021N0115úhĆSŻśŕej
i˝Ş{ÍsvČŞđAví¸mçˇRs[ľÄľÜÁ˝Ý˝˘j
˘ÂCt˘ÄCłˇéĚŠČHiOOG
URLŘݸ(www.kurims.kyoto-u.ac.jp)
]ĹVîń
2021N0415ú
@Ei_śjCłĹđXViCłÓĚXgjF
@@On the Essential Logical Structure of Inter-universal Teichmuller
@@@Theory in Terms of Logical AND "Č"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0115ú
@Ei_śjCłĹđXViCłÓĚXgjF
2021N0306ú
@Ei_śjFŰ^Cq~
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@@V_śđfÚF
@@On the Essential Logical Structure of Inter-universal Teichmuller
@@@Theory in Terms of Logical AND "Č"/Logical OR "É" Relations:
@@@Report on the Occasion of the Publication of the Four Main Papers
@@@on Inter-universal Teichmuller Theory.
2021N0115ú
@Ei_śjCłĹđXViCłÓĚXgjF
@@Combinatorial Construction of the Absolute Galois Group of the Field of
@@@@Rational Numbers.
108:PRQlÚĚfłń
21/04/17 20:12:36.12 cr30r3uy.net
>>105
>O^F_ŕťĚޢĹ
>ĚĚW_ĚhUhiPČéSĚWjĆÍAÓĄŞá¤Ěš
>ťąçŞA]vÉŹđľ˘Ä˘éć¤Év˘Üˇ
(âŤ)
EO^F_đA˘ÂŕěéH
EťĚĄĚO^F_ĚÔđsÁ˝č˝čH
EťąÜĹĺUžČbĹŕČłť¤ÉŠŚéŻÇiOO
109:PRQlÚĚfłń
21/04/25 18:03:40.36 x2gQxWeE.net
URLŘݸ(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:PRQlÚĚfłń
21/05/01 08:46:56.11 4gUFX+vb.net
Inter-universal geometry Ć ABC\z (X) 54
˝ÚŘݸ(mathÂ:253Ô)
URLŘݸ(www.nikkei.com)
wĚďâABC\z@uŘžvÉŕwEÍâââŠ
2021N430ú 11:00 [LżďőŔč] úo iŇWĎő@ÂŘTęj
w̢EĹÍAÔŞ˝ÁÄŠçŘžŞłľŠÁ˝ĆíŠéąĆŞ éBáŚÎAhCcĚq[Oi[Í1952NAjăĹĚwŇƢíęéKEXŞ\zľ˝uŢâčvÉÖˇéŘžđ\ľ˝Bˇ˘Ôłłę˝ŞA60NăăźÉĄĚwŇŞťęźę˘ľAęÉâčŞ éŕĚĚ{żIÉłľŠÁ˝ĆŘžłę˝BĄÍčƾğđcˇB
(řpIč)
iQlj
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q[Oi[_
q[Oi[_(w[Oi[_)ip: Heegner pointjĆÍAW
[ČüăĚ_Ĺ ÁÄAăź˝ĘĚ quadratic imaginary point ĚĆČÁĢéć¤ČŕĚĹ éBuCAEo[` (Bryan Birch) Éćčč`łęANgEw[Oi[ipęĹj (Kurt Heegner) Éöńşïçę˝Bq[Oi[ÍŢ 1 ĚńĚăĚKEXĚ\zđŘžˇé˝ßÉŢĚACfAđp˘˝B
OXEUMGĚč (Gross & Zagier 1986) ÍA_ s = 1 ɨŻéČ~ČüĚLÖĚ÷ŞĚąĆÎĹq[Oi[_ĚłđLqˇéBĆÉČ~ČüĚiđÍIjKŞ 1 Ĺ ęÎq[Oi[_ÍłŔĘiľ˝ŞÁÄ[fEFCQipęĹjĚKÍ1ČăjĚČüăĚL_đ\ŹˇéĚÉg¤ąĆŞĹŤéBćčęĘÉAGross, Kohnen & Zagier (1987) ÍAq[Oi[_ÍełŽ n ÉÎľČüăĚL_đ\ŹˇéĚÉg¤ąĆŞĹŤąęçĚ_ĚłÍEFCg 3/2 ĚW
[`ŽĚWĹ éąĆ𦵽B
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111:PRQlÚĚfłń
21/05/01 08:47:39.05 4gUFX+vb.net
>>110
ÂĂŤ
R@MipęĹjÍăÉIC[nipęĹjđ\Źˇé˝ßÉq[Oi[_đp˘AťęÉćÁÄK 1 ĚČ~ČüÉΡéo[`EXEBi[g_C[\zĚ˝đŘžľ˝B?őipęĹjÍOXEULGĚčđČ~ČüŠçW
[A[x˝lĚĚęÖĆęĘťľ˝BuEÍłWĚĺćĚăĚK 1 ĚČ~ČüĚ˝ÉÎľÄo[`EXEBi[g_C[\zđŘžľ˝ (Brown 1994)B
q[Oi[_ÍK 1 ĚČ~ČüăĚAPČű@ĹÍŠÂŻéąĆĚĹŤČŠÁ˝AńíÉ卢L_đvZˇéĚÉg¤ąĆŞĹŤéiT[xCÍ (Watkins 2006) đQĆjBASYĚŔÍAMagmaâPARI/GPĹÂ\Ĺ éB
URLŘݸ(sub-asate.ssl-lolipop.jp)
miniwiki
Ţâč
ińĚĚjKEXĚŢâč(Gauss class number problem)ÍAĘíÉđłęĢéć¤ÉA eXĚ n ? 1 Éξު n Ĺ éńĚĚŽSČXgđŕ˝çľ˝BąĚâčĚ˝źÍĚĺČwŇJ[Et[hqEKEX(Carl Friedrich Gauss)ÉżČńŢéBąĚâčÍAÜ˝AăĚ̝ʎĚĹLqˇéąĆŕĹŤéBŔńĚÉŕÖAľ˝âčŞ čAťĚUé˘Í
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ąĚâč̢ďČ_ÍAŔEĚLř(effective)ČvZĹ éB^Śç꽝ʎÉÎľAŢđvZˇéąĆÍŐľAŢĚńLř(ineffective)ČşEđßéű@͢Š éŞińLřĆÍAvZÍōȢŞAčĹ éƢ¤ąĆĚÝíŠéąĆđÓĄˇéjAľŠľLřČŔEđßiXgĚŽSČŘžjÍᄁB
Contents
1 łXĚKEXĚ\z
2 {âčĚóľ
3 Ţ 1 ̝ʎĚXgAbv
4 ťăĚW
5 ŔńĚ
ÂĂ
112:PRQlÚĚfłń
21/05/01 08:48:29.31 4gUFX+vb.net
>>111
ÂĂŤ
ťăĚW
ćčßNĚWÍAn = 1 ĚęŞNgEq[Oi[iEnglishĹj(Kurt Heegner)Éćčc_łęAW
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-1,-2,-3,-7,-11,-19,-43,-67,-163.
URLŘݸ(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
(řpIč)
Čă
113:PRQlÚĚfłń
21/05/09 16:44:06.23 6xnjRD2S.net
URLŘݸ(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:PRQlÚĚfłń
21/07/05 06:06:22.96 tA3B4T+I.net
URLŘݸ(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B76 (2019), 79?183
FŰ TeichmNuller _üĺ
(Introduction to Inter-universal TeichmNuller Theory)
By Ż TęY (Yuichiro Hoshi)
P5
1. ~ި
w ~ިĆͽž夊. ťęÍ Tate Pč gZb(1)h̹ƚ.
(řpIč)
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˝žAuĚvĹÍȢŠŕmęȢ
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URLŘݸ(ja.wikipedia.org)
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URLŘݸ(en.wikipedia.org)
Cyclotomic field
115:PRQlÚĚfłń
21/07/05 06:28:26.45 tA3B4T+I.net
>>114
>Tate Pč
şLTate twist Ý˝˘žË
AľAşLÍhan operation on Galois moduleshĆ éĚĹ
ŻćśĚLqĆÍżĺÁĆá¤ć¤Č
ÂÜčAŻćśĚLqÍAhan operation hĹÍČAťęŞWÜÁ˝AáŚÎQĚć¤ČWđÓĄľÄ˘éCŞˇé
iQlFśťŻÍĘ|ČĚĹCłľÜšńĚĹA´ś˛QĆj
URLŘݸ(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 Ď : 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
116:PRQlÚĚfłń
21/07/05 06:48:13.60 tA3B4T+I.net
>>115
>Tate twist
şLŞQlÉČ蝤
ú{ęĹÍAł|IÉîńʪȢ
ťęĆhWhat is the intuition behind the concept of Tate twists?hĆżâˇép¨ÍŠK¤×ŤĹľĺ¤Ë
URLŘݸ(math.stackexchange.com)
About the definition of l-adic Tate-twist asked Sep 20 '18 at 6:30 Elvis Torres Perez
(˛)
Zl(0)=Zl , Zl(1)=limŠ?(Ęli), Zl(n+1)=Zl(n)?ZlZl(1) for n0
URLŘݸ(math.stackexchange.com)
What is the intuition behind the concept of Tate twists? asked Aug 16 '11 at 4:06 Nicole
117:PRQlÚĚfłń
21/07/05 20:32:45.22 tA3B4T+I.net
>>114ÂĂŤ
URLŘݸ(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B76 (2019), 79?183
FŰ TeichmNuller _üĺ
(Introduction to Inter-universal TeichmNuller Theory)
By Ż TęY (Yuichiro Hoshi)
P9
2. txjIChĚ~ŞŤŻ^
É, ĘQěptŤmCh Gk ? O?k
ĚŻ^¨ G ? M đl@ľÜľĺ¤. ąĚ
f[^ G ? M Í, txjICh (Frobenioid ? cf. [6], Definition 1.3) ĆÄÎę
éwIÎŰĚ éęáĆżČf[^ĆČÁĢܡ. ą¤˘Á˝txjICh (Ě
éęáĆżČf[^@?@ČPĚ˝ß, Čş, हęđtxjIChĆž˘ŘÁ
ľܢܡŞ) Ş^Śçę˝ĆŤ, ťĚ gGh ĚŞđ G^[I (Letale-like ? cf.,
e.g., [6], Introduction, I4) ŞĆÄŃ, ťľÄ, ťĚă, gMh ĚŞđ Frobenius I
(Frobenius-like ? cf., e.g., [6], Introduction, I4) ŞĆÄŃܡ. (ąĚęĚ) G
^[IŞÍ, ĘQĹ, oŠÍ Galois QšŠç, ÂÜč, gÎĚŤh Ĺ č, ´oĆ
ľÄÍ gżĘĚȢh, gŔĚĚȢh (ˇČíż, g˛Ěć¤Čh, gźzIČh) ÎŰš. ę
ű, (ąĚęĚ) Frobenius IŞÍ, ĘmChĹ, oŠÍKČĚWÜ蚊ç,
´oĆľÄÍ gżĘĚ éh, gŔĚđÂh (ˇČíż, gťŔɜݡéh, gŔݡéh) Î
Űš.
łÄ, ăĚć¤ČtxjICh G ? M Ş^ŚçęܡĆ, łŤŮÇq׽ƨ
č, (G Í Gk ĚŻ^¨ĹˇĚĹ) PA[xô˝wIÉ G Šç G ? Š(G) Ƣ¤~
ިđł/\ŹˇéąĆŞĹŤÜˇ.
ÂĂ
118:PRQlÚĚfłń
21/07/05 20:33:07.84 tA3B4T+I.net
>>117
ÂĂŤ
ęű, M Í O?kĚŻ^¨ĹˇŠç, n {ĘĚjM[n]def = Ker(n: M ¨ M) Í Ęn(k) ĚŻ^¨ĆČč, ťĚ n ÉÖˇétÉŔđćéąĆ
Ĺ, Š(M)def = limŠ?nM[n] Ƣ¤ Š(k) ĚŻ^¨, ÂÜč, ~Ş¨Şžçęܡ. G ? Š(G)
ĚűÍG^[IŞŠç\Źľ˝ĚĹ gG^[I~ިh ĆÄŃ, G ? Š(M) Ěű
Í Frobenius IŞŠç\Źľ˝ĚĹ gFrobenius I~ިh ĆÄÔąĆɾܾĺ¤.
ąĚl@Éćč, 1 ÂĚtxjICh G ? M Šç, G^[I~ި G ? Š(G) Ć
Frobenius I~ި G ? Š(M) Ƣ¤ 2 ÂĚ~Ş¨ŞžçęÜľ˝.
ąĚ ({ÍÜÁ˝łÖWČ) 2 ÂĚ~ިÉÖľÄ, ČşĚŔŞmçęĢÜ
ˇ. ([10], Remark 3.2.1, đQĆžł˘.)
G ? M Ƣ¤f[^Šç, ÖčIÉ, G ŻĎČŻ^ Š(M)?¨ Š(G) ?@ÂÜ
č, Frobenius I~ިĆG^[I~ިĆĚÔĚ~ŞŤŻ^@?@đ\Ź
ˇéąĆŞĹŤé. Ü˝, ąĚ~ŞŤŻ^Í, G ? M Ş gÂ_IČÝčh Šç
śśÄ˘éęÉÍ, ]Ě~ިĚÔĚŻęĆęvˇé.
ąąÉoęˇé~ŞŤŻ^Í, ľÎľÎ gÇŢĚ_đp˘˝~ŞŤŻ^h, é˘Í,
gĂTIČ~ŞŤŻ^h ČÇĆÄÎęĢܡ.
(řpIč)
119:PRQlÚĚfłń
21/07/05 20:47:40.66 tA3B4T+I.net
URLŘݸ(www.math.nagoya-u.ac.jp)
QOOPNxu`ŕevń
wwČ
˝łČw¤Č
ĺw@
_ÁĘu` II ] Vęisĺwj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
i11 19 ú`23 új uČ~ČüĚ Hodge-Arakelov _ɨŻéA[xô˝v
P278
ČÚź _ÁĘu` II SłŻ@] Vę
Tu^Cg@ Č~ČüĚ Hodge-Arakelov _ɨŻéA[xô˝
ÎŰwN ĺw@ QPĘ Iđ
łČ Čľ
Ql ăqĚuQlśŁvQĆ
\őmŻ
[Hh] öxĚXL[_ĆC[Mn] ÉđŕľÄ éG^[ETCgâăIî{QĚîbD
[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).
ÂĂ
120:PRQlÚĚfłń
21/07/05 20:47:58.33 tA3B4T+I.net
ÂĂŤ
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Grothendieck ĚuA[xNwvĆÍCĚĚć¤Č_IČĚĚăĹč`łęCŠÂ éô˝IČ
đđ˝ˇă˝lĚĚô˝ÍCťĚu_Iî{QvÉŔÉ˝fłęéĹ ë¤Ć˘¤lŚűđo
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21/07/06 07:31:37.30 TlVKjijJ.net
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[4] AnabelioidĚô˝wĆTeichmuller_. PDF
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đĘ˝ľÜˇ. 8 Ĺs¤FŰ TeichmNuller _ĚĺčĚ g~j`
AĹh ĚŕžĚ˝ßÉ, ąĚ 7
ĹÍ, ťĚ g˝çtIASYh Ƣ¤TOÉ¢ÄĚČPČŕžđs˘Üˇ. (ÚľÍ, áŚÎ, [11] Ě
Example 1.7 Šç Remark 1.9.2 ÜĹĚŞđQĆžł˘.)
ܸĹÉ, Ěć¤ČÝčđl@ľÜľĺ¤. çtIf[^ (radial data ? cf. [11], Example 1.7, (i))
ĆÄÎęé éwIÎŰŞ^ŚçęĢéƾܡ. É, ťĚçtIf[^ŠçASYIÉ\ŹĹŤ
é (şI) ÎŰĹ é RAIf[^ (coric data ? cf. [11], Example 1.7, (i)) Ş^ŚçęĢéĆľ
ܡ. ąĚć¤ČÝčđ çtIÂŤ (radial environment ? cf. [11], Example 1.7, (ii)) ĆÄŃܡ. ďĚ
IÉÍ, áŚÎ, ČşĚć¤ČçtIÂŤĚáđlŚéąĆŞĹŤÜˇ:
(a) gçtIf[^h ĆľÄ, 1 łĄfü^óÔ C (ĚŻ^¨) đ, gRAIŞh ĆľÄ, çtIf[^Ĺ
é C (ĚŻ^¨) Šç gťĚłĽ\˘đYęéh Ƣ¤ASYÉćÁÄžçęéş 2 łŔü^óÔ
R
?2
(ĚŻ^¨) đĚpˇé.
(řpIč)
Čă
133:PRQlÚĚfłń
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URLŘݸ(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B72 (2018), 209?307
ąEFŰ TeichmNuller _üĺ
Ż TęY
P227
6. si
ľŠľČŞç, ČşĚRÉćÁÄ, äXÍ, ąĚ gŕÁĆŕŔźČAv[`h đ
ĚpˇéąĆŞĹŤÜšń. ąĚAv[`đĚpˇéĆ, źOĚ}ŞŚˇć¤É, F
?
l =
{|1|, . . . , |l
?|} ĚełÉÎľÄ, Ρé J ̳ƾÄ, ňJ = l
? ĘčĚÂ\ŤđlśľČ
ŻęÎČçČČčܡ. ťĚĘ, SĚĆľÄ, J Ć F
?
l ĆĚÖAĆľÄ, ňJňJ = (l
?)
l
?
ĘčĚÂ\ŤđlśľČŻęÎČčÜšń. ęű, ąĚÂ\ŤĚÂ@?@ÂÜč, sč
Ť@?@Í, äXĚÚWĚĎ_Šçͽߏܡ. ÁÉ, Č~ČüĚłĚ]żĚĎ_Š
çlŚÜˇĆ, ąĚßĺČsčŤđeľÄľÜ¤Ć, ]ĚsŽćčŕ gă˘sŽh
ľŠžéąĆŞĹŤČČÁľܤĚš.
ăqĚâčđđˇé˝ßÉ, si (procession ? cf. [7], Definition 4.10) Ƣ
¤TOđąüľÜľĺ¤.
siđlŚ˝ęĚűŞ, ˝žĚŰIČWĆŠôľ˝ęćčŕ, xĚ
WÉÖˇés荪ŹłČé
Ƣ¤dvČŔđĎ@ľÜľ˝. siƢ¤TOđp˘éąĆĚĘĚ_ĆľÄ,
ëxĚuŁ
Ƣ¤_ŕ°çęܡ. |T| đ˝žĚWĆŠôˇ, ÂÜč, |T| đ, |T| ĚŠČSPËS
ĚĚȡQĚěpƢ¤sčŤĚŕĆĹľ¤ę, ëx 0 ¸ |T| Ɲ̟̳ ¸ T
?
đćʡéąĆÍsÂ\š. ęű, siđlŚ˝ę, (gS
}
1
h Ƣ¤f[^ÉćÁÄ)
0 ¸ |T| Í gÁĘČłh Ƣ¤ąĆÉČč, ťĚźĚł ¸ T
? ĆĚćĘŞÂ\ĆČčܡ.
ťľÄ, ŔŰ, FŰ TeichmNuller _ɨ˘Ä,
ëxÍPI/RAIČx, ńëxÍlQI/çtIČx
Ƣ¤Ď@Ěƨč, ëxĆńëxÍ, ÜÁ˝ŮČéđđĘ˝ľÜˇ. (4,
(d), â [2], 21, ĚOźĚc_đQĆžł˘.) ąĚĎ_Šç, gëxĚuŁÂ\Ťh
Ídvš. (ÚľÍ [8], Remark 4.7.3, (iii), đQĆžł˘.)
134:PRQlÚĚfłń
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Corollary 3.12, ĚŘžÖA
sŽĚąo
URLŘݸ(repository.kulib.kyoto-u.ac.jp)
RIMS K?oky?uroku Bessatsu
B72 (2018), 209?307
ąEFŰ TeichmNuller _üĺ
Ż TęY
P297
25. Ś
~Ę
LGP NĆź§IČ˝çtI\ŚĆťĚA
P301
ąĚ 25 ĚĹăÉ, ăqĚ˝çtI Kummer ŁEđp˘˝ q WÎŰĚĚvZÉ
¢Ä, ČPÉྵܵĺ¤. (ÚľÍ, [9], Corollary 3.12, ĚŘžđQĆžł˘.)
ąĚ 25 Ě`ŞĚ Ś
~Ę
LGP NŞčßéŻ^ ő 0
C
?
LGP
?¨ ö 0
C
?
˘ Í,
ő 0Ś WÎŰđ ö 0
q W
ÎŰÉھܡ. (24, (a), đQĆžł˘.) ľ˝ŞÁÄ, 14, (e), (i), Šç, ]Ě
deg(ö 0
q WÎŰ) đ,
ő 0Ś WÎŰĚ@? gő ̤h ĚłĽ\˘ĚĎ_ŠçĹÍČ@?
gö ̤h ĚłĽ\˘ĚĎ_ŠçĚÎĚĎđp˘ÄvZˇéąĆŞÂ\š. ęű, ˝çt
I Kummer ŁEÉćÁÄ, sčŤ (Ind1), (Ind2), (Ind3) đFßęÎ, Ś~Ę
LGP NŞU
ąˇéŻ^ ő 0F
?~Ę
˘
?¨ ö 0F
?~Ę
˘ (24, (b), đQĆ) Ćź§ˇéŻ^ ő 0RFrob
?¨ ö 0RFrob
Şžçęܡ.
vol(ö 0Ś) ¸ R ž {}
đ, sčŤ (Ind1), (Ind2), (Ind3) ĚěpÉćé ö 0Ś WÎŰĚOšĚaWĚ (gö ̤h
ĚłĽ\˘Éćé) łĽď (holomorphic hull ? cf. [9], Remark 3.9.5) ([2], 12, Ě
ăźĚc_đQĆ) ĚsiłKťÎĚĎĆľÄč`ľÜľĺ¤. ˇéĆ, ź§IŻ^
ő 0RFrob
?¨ ö 0RFrob ̜݊ç,
ő 0Ś WÎŰĚÎĚĎÍ, vol(ö 0Ś) ČşĆČç´éđž
Üšń. ľ˝ŞÁÄ, _ĆľÄ, sŽ
vol(ö 0Ś) deg(ö 0q WÎŰ)
Şžçęܡ.
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URLŘݸ(www.youtube.com)
FŰ^Cq~
[_(IUT_)ÉÖˇé2ÂĚAj[V
1,213 ńŽ2020/04/11
îęóÔĚZVEłń
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URLŘݸ(www.kurims.kyoto-u.ac.jp)
EŽćłURL
Animation 1 - URLŘݸ(www.kurims.kyoto-u.ac.jp)
IUTeichÉÖˇéAj[Vi[IUTchIII], Theorem AĚŕeÉÎj
@"The Multiradial Representation of Inter-universal Teichmuller Theory"đöJB
ÎčĹF@ułv@tF[hAEgĹ@iavi wmvj@
Animation 2 - URLŘݸ(www.kurims.kyoto-u.ac.jp)(animation).mp4
ćńĚAIUTeichÉÖˇéAj[Vi[IUTchIII], Theorem BĚŕeÉÎj
@"Computation of the log-volume of the q-pilot via the multiradial representation"
@đöJB
136:PRQlÚĚfłń
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Legendre form
Č~Čü gy^2 = x(x - 1)(x - É)h
URLŘݸ(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)
URLŘݸ(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 Ś-Data) Suppose that
X = P1Q is the projective line over Q, and that D ş 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 ¨ (Mell)Q
[cf. the discussion preceding Proposition 1.8]. Let
ÂĂ
137:PRQlÚĚfłń
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>>136
ÂĂŤ
ąEFŰ Teichmuller _üĺ PDF (2018) iIndex čj URLŘݸ(repository.kulib.kyoto-u.ac.jp)
P94
Q ÍLĚ Q ĚăÂď@-@ĆĚÔÉ, ŠRČSPË
ŞśÝľÜˇ. eł É ¸ Q \ {0, 1} ÉÎľÄ, űöŽ gy^2 = x(x - 1)(x - É)h đlŚéą
ĆÉćÁÄ, Q(É) ăĚČ~Čü (EÉ)Q(É) Şžçęܡ. Ü˝, č]Ě Q(É) ĚgĺĚ FÉ
đ FÉdef= Q(É, ă-1,(EÉ)Q(É)[3 E 5](Q)) Ćč`ˇéĆ, ÇmçęĢéƨč, FÉ ăĚ
Č~Čü EÉ def = (EÉ)Q(É) ~Q(É) FÉ Í, FÉ Ěˇ×ÄĚf_ɨ˘ÄXŞôć@IŇł
đżÜˇ. ÁÉ, eł É ¸ Q \ {0, 1} ɨ˘Ä,
E Č~Čü EÉ Ě q p[^Şčßé FÉ ăĚ_Iöq qÉ Ě deg(qÉ),
E _Iöq qÉ Şčßé FÉ ăĚ gíńh Č_Iöq fÉ Ě deg(fÉ),
E Ě FÉ ĚâΤđˇĎŞčßé FÉ ăĚ_Iöq dÉ Ě deg(dÉ),
E č]Ě Q(É) ĚLĚăĚgĺ dÉ def = [Q(É) : Q]
Ƣ¤ 4 ÂĚlđlŚéąĆŞĹŤÜˇ. ąęç 4 ÂĚlÍ, É ¸ Q\ {0, 1} đťĚ GQ ¤
đÉćčÖŚÄŕĎíçȢ˝ß, ÁÉ, ąęç 4 ÂĚlđ gUP ĚÂ_ĚȡWĚăĚ
Öh ĆlŚéąĆŞĹŤÜˇ. ąĚÝčĚŕĆ, Belyi Ęđp˘˝c_đKpˇéąĆ
ÉćÁÄ, ąĚ 26 Ě`ŞĹq×˝ gDiophantus ô˝wIsŽh đŘžˇé˝ßÉÍ,
ČşĚĺŁđŘžˇęÎ[ŞĹ éąĆŞíŠčܡ ([5], Theorem 2.1; [10], Corollary
2.2, (i); [10], Corollary 2.3, ĚŘžđQĆ):
(řpIč)
Čă
138:PRQlÚĚfłń
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URLŘݸ(blog.livedoor.jp)
ywzABC\zj
[XyĹVîńz
2018N0124ú
FŰ^Cq~
[_ĚÜĆßWiki
(2018.1.24XV)
EF. Tan and K. ChenÉćé[NVbvż(2015.7ÉkĹJĂłę˝uWorkshop on Inter-Universal Teichmuller Theoryvćč) (pę)
URLŘݸ(wiutt.csp.escience.cn)
Note on the theory of Absolute Anabelian Geometry of Mochizuki URLŘݸ(wiutt.csp.escience.cn)
EMinhyong KimÉćéđŕy[p[(pę)
URLŘݸ(people.maths.ox.ac.uk)
EŻTęYÉćéT[xC(2015.12JĂ̤WďŕuFŰ Teichmuller _üĺvĹĚu`ż)(ú{ę)
URLŘݸ(www.kurims.kyoto-u.ac.jp)
139:PRQlÚĚfłń
21/08/17 16:51:56.02 nT2E/2XT.net
{ĚNŘęĹALbV
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URLŘݸ(webcache.googleusercontent.com)
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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