Introduction to Inter-universal Teichm¨ uller Theory I

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.

...

Introduction to Inter-universal Teichm¨ uller Theory I

— An Approximate Statement of the Main Theorem —

Yuichiro Hoshi

RIMS, Kyoto University

December 1, 2015

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 1 / 23

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Notation and Terminology

O^µ ^def= (O^×)tor ⊆ O^× ^def= {|z|= 1}

⊆ O^▷ ^def= {0<|z| ≤1} ⊆ O ^def= {|z| ≤1} O^×^µ ^def= O^×/O^µ

an isomorphof A ^def⇔an object which is isomorphic to A R₊: the underlying additive module of a ring R

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F: a number field, i.e., [F :Q]<∞, s.t.√

−1∈F V(−): the set of primes of (−)

E: an elliptic curve over F which has

either good or split multiplicative reduction at ∀v ∈V(F) q_v ∈ O^▷_F_v: the q-parameter of E at v ∈V(F)

q_E ^def= (q_v)_v_∈V_(F₎∈∏

v∈V(F)O^▷Fv

⇒degq_E (= [F :Q]⁻¹log(∏

♯(OFv/q_vOFv))) (≈6·ht_E) .The Szpiro Conjecture for Elliptic Curves over Number Fields ..

...A certain upper bound of ht_E, i.e., degq_E

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Suppose that the following(∗) holds:

. ...

(∗): ∃N ≥ 2, ∃C ≥ 0 s.t. degq_E^N ≤ degq_E +C Then since degq_E^N =N ·degq_E, one may conclude that

degq_E ≤ C N −1. In order to establish(∗), let us

take two isomorphs^†S,^‡S of (a part of) scheme theory, consider a “link” between these two isomorphs

Θ_naive: ^†S ∋ ^†q_E^N 7→ ^‡q_E ∈ ^‡S, and

compare, via Θ_naive, the computation of deg of ^†q_E^N (in^†S) with the computation of deg of ^‡q_E (in ^‡S).

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Θ_naive: ^†S→^‡S: ^†q_E^N 7→^‡q_E

Very roughly speaking, the main theorem of IUT asserts that:

. ...

Relative to such a link, the computation of deg^†q^N_E is, up to mild indeterminacies,compatible with the computation of deg^‡q_E. (⇒ degq_E^N ^ind.= deg^↷ q_E ⇒ (∗) ⇒ the Szpiro Conjecture) Terminology

a(n) (arithmetic)holomorphic structure

def⇔a (structure which determines a) ring structure a mono-analytic structure

def⇔an “underlying” (“non-holomorphic”) structure of a hol. str.

(e.g.: Qp, π^´₁^et(P¹_Q_p\ {0,1,∞}): hol.; (Qp)₊,Q^×_p, G_Q_p: mono-an.)

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F_mod ⊆F: the field of moduli of E l: a prime number

K ^def= F(E[l](F))

V⊆V(K): the image of a splitting of V(K)↠Vmod

def= V(Fmod) Suppose thatF/F_mod and K/F_mod are Galois.

S^def= [

SpecOK/Gal(K/Fmod)]

(the stack-theoretic quotient)

⇒ The arith. div. onOF determined by q_E can be descended to an arith. div. onS, i.e., by considering the arith. div. on S det’d by

q ^def= (q_v ^def= q_v_|_F ∈ OF^▷_v|

F ⊆ OK^▷v)_v_∈V ∈ ∏

v∈V

OK^▷v.

Note that degq^def= [F_mod :Q]⁻¹log(∏

v∈V♯(OKv/q_vOKv)) = degq_E.

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Recall An arithmetic line bundle on OK

= a certain pair of a l.b. L on OK and a metric on L ×ZC 1→µ(K)→K^× ^ADiv→ ⊕

w∈V(K)(K_w^×/O^×_K_w)→APicOK →1 Categories of Arithmetic Line Bundles on S

F_mod^⊛ : the Frobenioid of arithmetic line bundles onS .Module-theoretic Description

..

...

Fmod^⊛ : the Frobenioid of collections {avOKv}v∈V s.t.

a_v ∈K_v^×, a_v ∈ O_K^×_v for almost v ∈V .Multiplicative Description

..

...

F_MOD^⊛ : the Frobenioid of pairs (T, {t_v}v∈V) s.t.

T: an F_mod^× -torsor, t_v ∈T ×^F^mod^× K_v^×/OK^×v

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⇒The holomorphic structure of F_mod determines Fmod^⊛

−→ F∼ mod^⊛

−→∼ FMOD^⊛

{avOKv} 7→ (F_mod^× , {ordv(av)}) Fmod^⊛R,Fmod^⊛R, FMOD^⊛R : the resp. realifications of Fmod^⊛ , Fmod^⊛ ,FMOD^⊛ , i.e., obtained by replacing ⊕

v(K_v^×/O^×_K_v)by ⊕

v((K_v^×/O^×_K_v)⊗R) (⇒ The hol. str. of F_mod determines Fmod^⊛R

→ F∼ mod^⊛R

→ F∼ MOD^⊛R )

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The multiplication by 1/N on ⊕

v((K_v^×/O^×_K_v)⊗R) determines Θ_naive: ^†FMOD^⊛R

−→∼ ^‡FMOD^⊛R which maps ^†q^N 7→^‡q.

.Remark ..

...

Θ_naive may be regarded as a “deformation of value groups”.

The link “Θ_naive” will be eventually established by means of nonarchimedean theta functions (cf. p.21).

Fmod^⊛ depending on hol. str. suited to deg. estimates FMOD^⊛ only multiplicative str. not suited to deg. estimates

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.Goal ..

...

Relative to a link such as Θ_naive, the computation of deg^†q^N is, up to mild indeterminacies,compatible with the comp. of deg^‡q.

Note that ∄a ring automorphism of K_v s.t. q_v^N 7→q_v (if q_v ̸= 1).

Thus,Θ_naive cannot be compatible with the holomorphic structures, i.e.,Θ_naive may be compatible with only certain mono-analytic str.

(For instance,Θ_naive is compatible with the local Galois group G_v ^def= Gal(F_v/K_v)for each finite v ∈V

— cf. Θ_naive“=”a deformation of value groups.)

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On the other hand:

.Remark ..

...

The “degree computation” is, at least a priori, performed by means of the holomorphic structure under consideration.

Thus, in order to obtain a certain compatibility of the degree computations, we have to establish a “multiradial representation”

of the degree computations whose coric data consist of suitable mono-analytic structures.

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Multiradial Algorithm Suppose that we are given

a mathematical object R, i.e., aradial data,

an “underlying” objectC of R, i.e., acoric data, and

a func’l algorithm Φwhose input data is (an isomorph of) R.

.Example ..

...

R: the one-dimensional complex linear space C

C: the underlying two-dimensional real linear spaceR^⊕² R: the field Qp C: the underlying additive module (Qp)₊ R: the ´etale fundamental gp π^´₁^et(V) of a hyperbolic curve V /Qp

C: the absolute Galois groupG_Q_p of Qp

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Roughly speaking, we shall say that the algorithm Φis:

coric if Φdepends only on C

multiradialifΦ(is rel’d to R but) may be described in terms ofC uniradial if Φ is not multiradial, i.e., essentially depends onR If one starts with a coric data “C” and applies the alg’m Φ, then:

uniradial ⇒ the output depends on the choice of a “spoke”

multiradial⇒the output isunaﬀected by alterations in a“spoke”

· · · R1 · · ·

 y

R₂ −−−→ C ←−−− R₃

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.(Tautological) Example ..

...

(R, C)∼= (C,R^⊕²)

Φ(R) = the holomorphic structure on R ⇒ uniradial Φ(R) = the real analytic structure on R ⇒ coric

Φ(R) = the GL₂(R)-orbit of the hol. str. on R ⇒ multiradial incompatible H.S. on ^†C compatible H.S. of ^†C



y ^GL²⁽^R⁾y^↷ H.S. of ^‡C −−−→ R^⊕² H.S. of ^‡C −−−−→^GL²⁽^R⁾

↷ R^⊕²

uniradial multiradial

(cf.GL₂(R)/C^× =H+⊔H−)

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Summary

We want to obtain a certain compatibility of the degree computations relative to a link such asΘnaive.

Θnaive cannot be compatiblew/ the holomorphic str., i.e., Θ_naive is compatible w/ only certain mono-an. str., e.g., G_v. On the other hand, the degree computation is, at least a priori, performed by means of the holomorphic structure.

Thus, we have to establish a multiradial representation of the degree computations whose coric data are suitable mono-an. str.

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.An Approximate Statement of the Main Theorem of IUT (tentative) ..

...

∃A suitable multiradial algorithm whose output data consist of the following three objects ↶mild indeterminacies

{(OKv)_∗}v∈V (∗= + if v is finite, ∗=∅ if v is infinite) q^N ↷ ∏

v∈V (OKv)_∗ F_mod ↷ ∏

v∈V

((K_v)_∗“via (OKv)_∗”)

Moreover, this algorithm is compatible with

Θ_naive: ^†F_MOD^⊛R →^∼ ^‡F_MOD^⊛R ; ^†q^N 7→^‡q.

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Main Theorem of IUT⇒ Szpiro Conjecture Θ_naive: ^†F_MOD^⊛R →^∼ ^‡F_MOD^⊛R ; ^†q^N 7→^‡q

Th’m⇒







{(^†OKv)_∗}v∈V

†q^N ↷∏

(^†OKv)_∗

†F_mod ↷∏

(^†K_v)_∗







ind.↷

−→∼







{(^‡OKv)_∗}v∈V

‡q^N ↷∏

(^‡OKv)_∗

‡F_mod ↷∏

(^‡K_v)_∗







⇒

cmod: ^†Fmod^⊛ ind.↷

→∼ ^‡Fmod^⊛ which maps {^†q_v^N^†OKv} 7→ {^‡q^N_v ^‡OKv} c_□: ^□Fmod^⊛

→∼ ^□F_MOD^⊛ which maps {^□q^(Nv ⁾□OKv} 7→^□q^(N) which are compatible with Θ_naive, i.e.,

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fit into a diagram that is commutative, up to mild indeterminacies

†Fmod^⊛R cmod

−−−→ ^‡Fmod^⊛R c_†



y y^c^‡

†FMOD^⊛R Θnaive

−−−→ ^‡FMOD^⊛R

⇒







{^†q_v^N^†OKv} ⇒ {^‡q_v^N^‡OKv}, {^‡qv‡OKv}

⇑ ⇓

†q^N ⇐ ^‡q







⇒ {^‡q_v^‡OKv}“^log-vol.⊆ ” ∪

indeterminacies

{^‡q_v^N^‡OKv}

⇒ −degq ≤ −degq^N +C, i.e., (∗) in p.4

⇒ the Szpiro Conjecture

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.An Approximate Statement of the Main Theorem of IUT (tentative) ..

...

∃A suitable multiradial algorithm whose output data consist of the following three objects ↶mild indeterminacies

{(OKv)_∗}v∈V (∗= + if v is finite, ∗=∅ if v is infinite) q^N ↷ ∏

v∈V (OKv)_∗ F_mod ↷ ∏

v∈V

((K_v)_∗“via (OKv)_∗”)

Moreover, this algorithm is compatible with

Θ_naive: ^†F_MOD^⊛R →^∼ ^‡F_MOD^⊛R ; ^†q^N 7→^‡q.

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Recall(v ∈V: fin.;p^def= p_v) (OKv)₊ ∈ output, G_v: coric Unfortunately, it is known that:

̸ ∃a func’l(w.r.t. open injections) alg’mfor rec. “(OKv)₊” from “G_v”.

IKv

def= _2p¹Im(O_K^×_v → O^×_K_v ⊗_ZQ

log_p

→∼ (K_v)₊): thelog-shell of K_v IKv: a finitely generated free Zp-module

(OKv)₊, log_p(O^×_K_v) ⊆ IKv ⊆ IKv ⊗_ZQ = (K_v)₊

[IKv : (OKv)₊] (<∞) can be comp’d by the top. gp str. of G_v G_v ^∃^func’l⇒

algorithm an isomorph of IKv

Thus, “{(OKv)_∗}v∈V” in Th’m should be replaced by {Iv

def= IKv}v∈V

(where the log-shell at an infinite v ∈V ^def= π· OKv).

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Recall q^N, F_mod ∈output

Unfortunately, by various indeterminacies arising from the operation of “passing from holom’c str. to mono-anal’c str.”, it is diﬃcult to obtain multiradial representations of q^N,F_mod themselves directly.

To establish a mul’l alg’m of the desired type, we rep. multiradially a suitable functionwhose special value is q_v^N or an∈F_mod.

“q_v^N” will be represented as a special value of a (multiradially represented)theta function.

“F_mod” will be represented as a set of special values of (multiradially represented) κ-coric functions.

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.An Approximate Statement of the Main Theorem of IUT ..

...

For a “generalE/F”,

∃a suitable multiradial algorithm whose output data consist of the following three objects ↶mild indeterminacies

the collection of log-shells{Iv}v∈V

the theta values (={q^j²^/2l}₁_≤_j_≤_l_⋇^def₌l−1 2

) ↷ ∏

v∈V Iv

Fmod viaκ-coric functions ↷ ∏

v∈V

((Kv)+“via Iv”)

Moreover, this alg’m is compatible w/ the Θ-link (more precisely, Θ^×_LGP^µ -link) “^†F_MOD^⊛R →^∼ ^‡F_MOD^⊛R ”; “^†theta values7→^‡q^1/2l”.

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Fundamental Strategy

□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 linksby means of such Frobenius-like objects.

Take an´etale-like object closely related to □ (e.g., “π₁^temp(X

v)” 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 □”.