Introduction to Inter-universal Teichmuller Theory IIyuichiro/talk20151202.pdf · 2016-02-23 · Theory II | An Etale Aspect of the Theory of Etale Theta Functions | Yuichiro Hoshi

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Introduction to Inter-universal Teichmuller

Theory II

— An Etale Aspect of the Theory of Etale Theta Functions —

Yuichiro Hoshi

RIMS, Kyoto University

December 2, 2015

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, 2015 1 / 26

Notation and Terminology

For an odd prime number l,

F⋊±l

def= (Fl)+ ⋊ {±1}, F⋇

l

def= F×

l /{±1}, l⋇def= ♯F⋇

l = l−12

an F±l -group

def⇔ a set S equipped with a {±1}-orbit of S ∼→ Fl

For a topological group G,

∞H i(G,A)def= lim−→H⊆G: open subgps of finite index

H i(H,A)

For a p-adic local field k, the ×µ-Kummer structure of Gk ↷ O×µ

kdef⇔ {Im

((O×

k)H = O×

kH ↪→ O×

k↠ O×µ

k

)}H⊆Gk: open subgps

a poly-(iso)morphism A→ Bdef⇔ a set consisting of (iso)morphisms A→ B

the full poly-isomorphism A∼→ B

def⇔ the poly-isom. Isom(A,B)


Fundamental Strategy (cf. p.23 of I)

□ is, for instance, a log-shell, a theta function, or a κ-coric function.

Start with a usual/existing □ (i.e., a Frobenius-like □).

Construct links by means of such Frobenius-like objects.

Take an etale-like object closely related to □(e.g., “πtemp

1 (Xv)” for a theta function — cf. II and III).

Give a multiradial mono-anabelian algorithm of reconstructing □from the etale-like object, i.e., construct a suitable etale-like □.

Establish “multiradial Kummer-detachment” of □, i.e.,

a suitable Kummer isomorphism “Frob.-like □ ∼→ etale-like □”.


Fundamental Strategy (cf. p.23 of I)

□ is, for instance, a log-shell, a theta function, or a κ-coric function.

Start with a usual/existing □ (i.e., a Frobenius-like □).

Construct links by means of such Frobenius-like objects.

Take an etale-like object closely related to □(e.g., “πtemp

1 (Xv)” for a theta function — cf. II and III).

Give a multiradial mono-anabelian algorithm of reconstructing □from the etale-like object, i.e., construct a suitable etale-like □.

Establish “multiradial Kummer-detachment” of □, i.e.,

a suitable Kummer isomorphism “Frob.-like □ ∼→ etale-like □”.


p, l: prime numbers

k: a p-adic local field, i.e., [k : Qp] <∞, s.t.√−1 ∈ k

E: an elliptic curve/k which has split multiplicative reduction/Ok

q ∈ O▷k : the q-parameter of E

X log def= (E, {o} ⊆ E): the smooth log curve/k determined by E

{±1}↷ E, hence also ↷ X log ⇒ X log → C log def= [X log/{±1}]

.Assumptions..

......

2, p, and l are distinct prime numbers

E[2l](k) = E[2l](k) (⇔ µl(k) ⊆ k and qdef= q1/2l ∈ k)

C log is a k-core (⇒ one may apply elliptic cuspidalization)


One obtains a comm. diagram of conn’d log etale tempered coverings

Ylog (1)−−−→

µl

Y log

(2)

yµ2 (3)

yµ2

Y log (4)−−−→µl

Y log

(5)

yl·Z (6)

yl·Z

X log (7)−−−→µl

X log (8)−−−→Fl

X log

(9)

y{±1} (10)

y{±1}

C log (11), deg=l−−−−−−→ C log

Csp(−) def= the set of cusps of “(−)”

Irr(−) def= the set of irreducible comp. of the special fiber of “(−)”


The three squares are cartesian.

The composite Y log (6)→ X log (8)→ X log: the covering determined

by the dual semi-graph of the special fiber of X log.

⇒ Z def= Gal(Y log/X log) ∼= Z; Csp(Y log), Irr(Y log): Z-torsors

X log (8)→ X log: the intermediate covering corresp’g to l · Z ⊆ Z

⇒ Fldef= Gal(X log/X log) ∼= Fl; Csp(X

log), Irr(X log): Fl-torsors

Fix an ∈ Csp(X log), i.e., the zero cusp of X log.

⇒ a structure of elliptic curve on the underlying scheme of X log

X log (9)→ C log def= [X log/{±1}]

zero cusp of X log ⇒ a cusp of C log, i.e., the zero cusp of C log


X log (7)→ X log totally ramifies at ∀ ∈ Csp(X log), Gal ∼= µl

⇒ Csp(X log)∼→ Csp(X log), Irr(X log)

∼→ Irr(X log)

zero cusp of X log ⇒ a cusp of X log, i.e., the zero cusp of X log

Y log (3)→ Y log: the double covering determined by “u = u1/2”

⇒ Csp(Y log)2:1↠ Csp(Y log), Irr(Y log)

∼→ Irr(Y log)

Autk(Clog) = Autk(C

log) = {id}

Autk(Xlog) = Gal(X log/C log) = {±1}

Autk(Xlog) = Gal(X log/X log)⋊Gal(X log/C log) = Fl ⋊ {±1}

Autk(Xlog) = Gal(X log/X log)×Gal(X log/C log) = µl × {±1}

(⇒ Csp(X log)Autk(Xlog) = {the zero cusp})


Labels of Cusps and Components

⇒ geometry of X log/k det. a natural str. of F±l -gp on Csp(X log),

hence also on Irr(X log), Csp(X log), Irr(X log),

i.e., each element of these sets is labeled by an ∈ Fl up to {±1}..

...... LabCusp± def= Csp(X log)

{±1}↷∼→ Fl (↶ (F⋊±

l∼=) Autk(X

log))

Fix a lifting ∈ Irr(Y log) of 0 ∈ Irr(X log).

(Such liftings form an (l · Z)-torsor.)

⇒ Such a lifting det. Z ∼→ Irr(Y log)∼← Irr of Y

log, Y log, Y log,

i.e., each ∈ Irr of Y log, Ylog, Y log, Y log is labeled by an ∈ Z.


Evaluation Points

⇒∃!µ− ∈ X(k): a 2-torsion whose closure intersects 0 ∈ Irr(X log)

µY− ∈ Y (k): a unique lifting of µ− whose closure inter. 0 ∈ Irr(Y log)

ξYa ∈ Y (k): the image of µY− by a ∈ Z = Gal(Y log/X log)

.Definition..

......

an evaluation point of Ylog

(resp. Y log) labeled by a ∈ Zdef⇔ a (necessarily k-rat’l) lifting ∈ Y (resp. Y ) of ξYa ∈ Y (k)

an evaluation point of X log labeled by a ∈ LabCusp±

def⇔ the image ∈ X of an evaluation point of ∈ Ylog

labeled by

a lifting ∈ Z of a ∈ LabCusp±


Theta Functions

The function

Θ(u) = q−18 ·

∑n∈Z

(−1)n · q12(n+ 1

2)2 · u2n+1

on 0 ∈ Irr(Y log) uniquely extends to a meromorphic function Θ on

the stable model of Y .

the zero divisor of Θ =∑

c∈Csp(Y log) [c]

the pole divisor of Θ =∑

a∈Z∼=Irr(Y log)a2·ordk(q)

2· [a]

⇒ ∃An “l-th root” Θ on Y of Θ(an evaluation pt labeled by 0) · Θ−1

(Note: Θ(an ev. pt labeled by 0) = −Θ(another ev. pt labeled by 0))


Special Values

Θ on Y : Θ(an evaluation point labeled by j) ∈ µ2l · qj2

−2 −1 0 1 2 j

· · · − • − • − • − • − • − · · · • · · ·

ev. ⇓ pt

q4 q 1 q q4 qj2

mod µ2l

−l⋇ − l⋇ + 1 · · · − 1 0 1 · · · l⋇ − 1 l⋇︸︷︷︸LabCusp±

{±1}↷∼= Fl


Π(−): the log etale π1 of “(−)log”

∆(−)def= Ker(Π(−) ↠ Gk), i.e., the geom. log etale π1 of “(−)log”

Πtp(−): the log tempered π1 of “(−)log”

∆tp(−)

def= Ker(Πtp

(−) ↠ Gk), i.e., the geom. log temp’d π1 of “(−)log”

For N ≥ 1, if “J∃↠ Gk”, then J [µN ]

def= µN(k)⋊ J .

a tautological splitting sJ : J → J [µN ] of J [µN ] ↠ J

a natural homomorphism H1(J,µN(k))→ Out(J [µN ])

Thus, by the Kummer theory, we have:

k× ↠ k×/(k×)N ↪→ H1(ΠtpY ,µN(k)) → Out(Πtp

Y [µN ])

.

......DY

def= ⟨Im(k×), Gal(Y log/X log) ∼= l · Z⟩ ⊆ Out(Πtp

Y [µN ])


∆Θdef= [∆X ,∆X ]/[∆X , [∆X ,∆X ]]

⇒ ∆Θ∼= ∆ab

X ∧∆abX , i.e., “∼= Z(1) (def= lim←−n

µn(k))”

⇒ l ·∆Θ∼= Z(1)

ηΘ ∈ H1(Πtp

Y,∆Θ): the Kummer class of a suitable ∈ O×

k · Θ

⇒ ∃ηΘ ∈ H1(Πtp

Y, l ·∆Θ) s.t. η

Θ|Y = Im(ηΘ) in H1(Πtp

Y, ∆Θ),

i.e., the Kummer class of an ∈ O×k ·Θ

−1

ηΘ,l·Z×µ2 ⊆ H1(Πtp

Y, l ·∆Θ): the orbit of ηΘ by

Gal(Ylog/X log) = Πtp

X/Πtp

Y= l · Z× µ2

(⇒ indep. of the choice of a lifting ∈ Irr(Y log) of 0 ∈ Irr(X log))


Thus, relative to (l ·∆Θ)⊗Z Z/NZby scheme

∼→theory

µN(k),

each ∈ ηΘ,l·Z×µ2 mod N ⊆ H1(Πtp

Y, (l ·∆Θ)⊗Z Z/NZ) can be

obtained as “sΘY− sΠtp

Y

” for some sΘY:

1 −→ µN(k) −→ Πtp

Y[µN ]

sΘY

, sΠtp

Y↶−→ Πtp

Y−→ 1

.

......sΘY: Πtp

Y

sΘY

↪→ Πtp

Y[µN ] ↪→ Πtp

Y [µN ] : a (mod N) theta section


Mono-theta Environments and Associated Cyclotomes.Definition..

......

A (mod N) model mono-theta environmentdef⇔ a triple

(ΠtpY [µN ], DY ⊆ Out(Πtp

Y [µN ]), {γ · Im(sΘY) · γ−1}γ∈µN (k))

A (mod N) mono-theta environmentdef⇔ an isomorph MΘ

N =

(Π, DΠ ⊆ Out(Π), sΘΠ) of a mod N model mono-theta env.

The subgroup of Π (of MΘN) corresp’g to “µN(k) ⊆ Πtp

Y [µN ]”

is group-theoretic. ⇒ Πµ(MΘN): the exterior cyclotome

The subquotient of Π (of MΘN) corresponding to “l ·∆Θ” is

group-theoretic. ⇒ (l ·∆Θ)(MΘN): the interior cyclotome


Algorithmic Reconstruction

Π•: an isomorph of ΠtpX

MΘN : a mod N mono-theta environment

Π•∃func’l⇒

algorithmtopological gp corresponding to the topological gp

Πtp

Y, Πtp

Y, Πtp

Y , ΠtpY , Πtp

X , ΠtpX , Πtp

C , ΠtpC , Gk, l ·∆Θ

Π•∃func’l⇒

algorithma subset corresponding to the subset

(l · Z× µ2)-orbit of O×k ·Θ ⊆ H1(Πtp

Y, l ·∆Θ)

Π•∃func’l⇒

algorithma mod N mono-theta environment

MΘN

∃func’l⇒algorithm

a topological group corresponding to ΠtpX


.Rigidity Properties of Mono-theta Environments..

......

Cyclotomic Rigidity

Discrete Rigidity

Constant Multiple Rigidity

Isomorphism Class Compatibility

Frobenioid Structure Compatibility

Cyclotomic Rigidity

MΘN


a canonical (l ·∆Θ)(MΘN)⊗Z Z/NZ ∼→ Πµ(MΘ

N),

i.e., “(l ·∆Θ)⊗Z Z/NZ ∼→ µN(k) by scheme theory”

(cf. “a suitable Kmm isom. Frob.-like □ ∼→ etale-like □” of p.3)


Discrete Rigidity

By means of the var. surj. µN(k) ↠ µM(k) (M |N), one may define

the notion of a “projective system of mono-theta env. {MΘN}N≥1”.

∀proj. system ∼= the natural proj. system of model mono-theta env.

Constant Multiple Rigidity

MΘN


the subset corresponding to the subset

θdef= (l · Z× µ2)-orbit of µl ·Θ ⊆ H1(Πtp

Y, l ·∆Θ)

of

(l · Z× µ2)-orbit of O×k ·Θ ⊆ H1(Πtp

Y, l ·∆Θ)


Pointed Inversion Automorphisms

Consider a pair (ιX , µX

− ) of

a unique ιX ∈ Autk(X) of order two, i.e., “−1 ↷ LabCusp±”

an evaluation point µX

− of X log labeled by 0 ∈ LabCusp±

.Definition..

......

A pointed inversion automorphism of Ylog

def⇔ a lifting (ιY , µY

−) on Ylog

of (ιX , µX

− ) s.t. ι2 = id, ι(µ) = µ

A group-theoretic pointed inversion automorphismdef⇔ a “group-theoretic pair (ι, D)” associated to a (ιY , µ

Y

−)


Thus, such a (ι, D) det. a “lifting ∈ Irr(Y log) of 0 ∈ Irr(X log)”.

In particular:

θ = (l · Z× µ2)-orbit of µl ·Θ ⊇ θι = µ2l-multiples of a “Θ”

(where (−)ι is the “set of ι-invariants”), which thus implies that

H1(Πtp

Y, l ·∆Θ)

restriction to D−→ H1(D, l ·∆Θ)

θι∼→ (Kummer class of) µ2l,

as well as, for a decomp. subgp Dj ⊆ Πtp

Ylabeled by j (for (ι,D)),

H1(Πtp

Y, l ·∆Θ)

restriction to Dj−→ H1(Dj, l ·∆Θ)

θι∼→ (Kummer class of) µ2l · qj

2.

(The operation of Galois evaluation w.r.t. D, as well as Dj)Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT II December 2, 2015 21 / 26

Reconstructions of Etale Theta Functions via Mono-theta Env.

MΘ∗ = {MΘ

N}N : a projective system of mono-theta environments

(l ·∆Θ)(MΘ∗ )

def= lim←−(· · ·

∼→ (l ·∆Θ)(MΘN)

∼→ (l ·∆Θ)(MΘM)

∼→ · · · )

Πµ(MΘ∗ )

def= lim←−N

Πµ(MΘN)

By the cycl. rig.: (l ·∆Θ)(MΘ∗ )

∼→ Πµ(MΘ∗ )

By the cons. mult. rig.: θ(MΘ∗ ) ⊆ H1(Πtp

Y(MΘ

∗ ), (l ·∆Θ)(MΘ∗ ))

(θ(MΘ∗ ) ⊆) ∞θ(MΘ

∗ ) ⊆ ∞H1(Πtp

Y(MΘ

∗ ), (l ·∆Θ)(MΘ∗ ))

defined by { η ∈ ∞H1 |n · η ∈ θ for some n ≥ 1 }

By (l ·∆Θ)(MΘ∗ )

∼→ Πµ(MΘ∗ ):

θenv

(MΘ∗ ) ⊆ ∞θ

env(MΘ

∗ ) ⊆ ∞H1(Πtp

Y(MΘ

∗ ),Πµ(MΘ∗ ))


Reconstructions of Constant Portions via Mono-theta Environments

Π•: an isomorph of ΠtpX

func’l⇒algorithm

M def= MΘ

∗ (Π•): a proj. system

Since X log is of strictly Belyi type, by Belyi cuspidalization,

∃a functorial algorithm for reconstructing, from Π•, an isomorph

Π• ↠ G•def= Gk(Π•) ↷ k(Π•) ⊇ k(Π•)

× ↪→ ∞H1(G•, (l ·∆Θ)(Π•))

of the “ΠtpX ↠ Gk ↷ k ⊇ k

× Kummer↪→ ∞H1(Gk, l ·∆Θ)”

Oµ

k(Π•)⊆ O×

k(Π•)⊆ O▷

k(Π•)⊆ k(Π•)

× ⊆ ∞H1(G•, (l ·∆Θ)(Π•))

By (l ·∆Θ)(Π•)∼→ (l ·∆Θ)(M)

∼→ Πµ(M):

k(M)

∪

Oµ

k(M)⊆ O×

k(M)⊆ O▷

k(M)⊆ k(M)× ⊆ ∞H1(G•,Πµ(M))


Reconstructions of Splittings via Mono-theta Environments

(O× ·∞θenv

)(M)def= O×

k(M)+ ∞θ

env(M) ⊆ ∞H1(Πtp

Y(M),Πµ(M))

In particular, for a gp-th’c pt’d inv. aut. (ι,D) for Πtp

Y(M),

∞H1(Πtp

Y(M),Πµ(M))

Gal. ev. w.r.t. D−→ ∞H1(D,Πµ(M))

(O× · ∞θenv

)(M)ι ↠ O×k(M)

(i.e., Gal. ev. labeled by 0 ∈ LabCusp±) determines a splitting

(O× · ∞θenv

)(M)ι/Oµ

k(M)= O×µ

k(M)×

(∞θ

env(M)ι/Oµ

k(M)

).


Thus, in summary, we obtain:.A Local Multiradial Algorithm Related to Etale Theta Functions..

......

∃A multiradial algorithm as follows:

coric data: an isomorph (G ↷ O×µ, ×µ-Kmm) of Gk ↷ O×µ

k

radial data: (Π• ↷ Πµ(MΘ∗ (Π•)), a coric data, αµ,×µ)

for an isomorph Π• of ΠtpX ,

where αµ,×µ is the pair of the full poly-isomorphism G•∼→“G”

and Πµ(MΘ∗ (Π•))⊗Z (Q/Z) zero→“O×µ”

output: the radial data and:


.A Local Multiradial Algorithm Related to Etale Theta Functions..

......

The proj. system of mono-theta environments M def= MΘ

∗ (Π•)

The subsets

O×k(M)

∪ (O× · ∞θenv

)(M) ⊆ ∞H1(Πtp

Y(M),Πµ(M))

The set of group-th’c pointed inversion automorphisms {(ι,D)}

The splittings for the various “(ι,D)”

(O× · ∞θenv

)(M)ι/Oµ

k(M)= O×µ

k(M)×

(∞θ

env(M)ι/Oµ

k(M)

)via the operation of Galois evaluation w.r.t. “D”

The diagram

Πµ(M)⊗Z (Q/Z)nat’l∼→ Oµ

k(M)

nat’l∼→ Oµ

k(Π•)

zero→ O×µ

k(Π•)

full∼→

poly“O×µ”


Documents

Introduction to Inter-universal Teichmuller Theory IIyuichiro/talk20151202.pdf · 2016-02-23 · Theory II | An Etale Aspect of the Theory of Etale Theta Functions | Yuichiro Hoshi