.

...

### 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

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 *isomorph*of *A* ^{def}*⇔*an object which is isomorphic to *A*
*R*_{+}: the underlying additive module of a ring *R*

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 2 / 23

*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*^{▷}*F**v*

*⇒*deg*q** _{E}* (= [F :Q]

^{−}^{1}log(∏

*♯(O**F**v**/q*_{v}*O**F**v*))) (*≈*6*·*ht* _{E}*)
.The Szpiro Conjecture for Elliptic Curves over Number Fields
..

...A certain upper bound of ht* _{E}*, i.e., deg

*q*

_{E}Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 3 / 23

Suppose that the following(*∗*) holds:

. ...

(*∗*): *∃N* *≥* 2, *∃C* *≥* 0 s.t. deg*q*_{E}^{N}*≤* deg*q** _{E}* +

*C*Then since deg

*q*

_{E}*=*

^{N}*N*

*·*deg

*q*

*, one may conclude that*

_{E}deg*q*_{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*

*(in*

_{E}*S).*

^{‡}Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 4 / 23

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}*⇒*deg

*q*

_{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.: Q*p*, *π*^{´}_{1}^{et}(P^{1}_{Q}_{p}*\ {*0,1,*∞}*): hol.; (Q*p*)_{+},Q^{×}* _{p}*,

*G*

_{Q}

*: mono-an.)*

_{p}Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 5 / 23

*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 that*F/F*_{mod} and *K/F*_{mod} are *Galois.*

*S*^{def}= [

Spec*O**K**/Gal(K/F*mod)]

(the stack-theoretic quotient)

*⇒* The arith. div. on*O**F* determined by *q** _{E}* can be descended to an
arith. div. on

*S, i.e., by considering the arith. div. on*

*S*det’d by

q ^{def}= (q_{v}^{def}= *q*_{v}_{|}_{F}*∈ O**F*^{▷}_{v|}

*F* *⊆ O**K*^{▷}*v*)_{v}_{∈V}*∈* ∏

*v**∈V*

*O**K*^{▷}*v**.*

Note that degq^{def}= [F_{mod} :Q]* ^{−1}*log(∏

*v**∈V**♯(O**K**v**/q*_{v}*O**K**v*)) = deg*q** _{E}*.

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 6 / 23

Recall An arithmetic line bundle on *O**K*

= a certain pair of a l.b. *L* on *O**K* and a metric on *L ×*ZC
1*→ µ(K)→K*

^{×}^{ADiv}

*→*⊕

*w**∈V*(K)(K_{w}^{×}*/O*^{×}_{K}* _{w}*)

*→*APic

*O*

*K*

*→*1 Categories of Arithmetic Line Bundles on

*S*

*F*_{mod}^{⊛} : the Frobenioid of arithmetic line bundles on*S*
.Module-theoretic Description

..

...

*F*mod^{⊛} : the Frobenioid of collections *{a**v**O**K**v**}**v**∈V* s.t.

*a*_{v}*∈K*_{v}* ^{×}*,

*a*

_{v}*∈ O*

_{K}

^{×}*for almost*

_{v}*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}

^{×}*/O*

*K*

^{×}*v*

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 7 / 23

*⇒*The holomorphic structure of *F*_{mod} determines
*F*mod^{⊛}

*−→ F**∼* mod^{⊛}

*−→**∼* *F*MOD^{⊛}

*{a**v**O**K**v**}* *7→* (F_{mod}^{×}*,* *{*ord*v*(a*v*)*}*)
*F*mod^{⊛R},*F*mod^{⊛R}, *F*MOD^{⊛R} : the resp. realifications of *F*mod^{⊛} , *F*mod^{⊛} ,*F*MOD^{⊛} ,
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

*F*mod

^{⊛R}

*→ F**∼* mod^{⊛R}

*→ F**∼* MOD^{⊛R} )

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 8 / 23

The multiplication by 1/N on ⊕

*v*((K_{v}^{×}*/O*^{×}_{K}* _{v}*)

*⊗*R) determines Θ

_{naive}:

^{†}*F*MOD

^{⊛R}

*−→**∼* ^{‡}*F*MOD^{⊛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).

*F*mod^{⊛} depending on hol. str. suited to deg. estimates
*F*MOD^{⊛} only multiplicative str. not suited to deg. estimates

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 9 / 23

.Goal ..

...

Relative to a link such as Θ_{naive}, the computation of deg* ^{†}*q

*is, up to mild indeterminacies,*

^{N}*compatible*with the comp. of deg

*q.*

^{‡}Note that ∄a ring automorphism of *K** _{v}* s.t.

*q*

_{v}

^{N}*7→q*

*(if*

_{v}*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.)

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 10 / 23

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.

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 11 / 23

Multiradial Algorithm Suppose that we are given

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

an “underlying” object*C* 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 space*R^{⊕}^{2}
*R: the field* Q*p* *C: the underlying additive module* (Q*p*)_{+}
*R: the ´*etale fundamental gp *π*^{´}_{1}^{et}(V) of a hyperbolic curve *V /*Q*p*

*C: the absolute Galois groupG*_{Q}* _{p}* of Q

*p*

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 12 / 23

Roughly speaking, we shall say that the algorithm Φis:

*coric* if Φdepends only on *C*

*multiradial*ifΦ(is rel’d to *R* but) may be described in terms of*C*
*uniradial* if Φ is not multiradial, i.e., essentially depends on*R*
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 is*unaﬀected* by alterations in a“spoke”

*· · ·* *R*1 *· · ·*

y

*R*_{2} *−−−→* *C* *←−−−* *R*_{3}

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 13 / 23

.(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 GL_{2}(R)-orbit of the hol. str. on *R* *⇒* multiradial
incompatible H.S. on * ^{†}*C compatible H.S. of

*C*

^{†}

y ^{GL}^{2}^{(}^{R}^{)}y^{↷}
H.S. of * ^{‡}*C

*−−−→*R

^{⊕}^{2}H.S. of

*C*

^{‡}*−−−−→*

^{GL}

^{2}

^{(}

^{R}

^{)}

↷ R^{⊕}^{2}

uniradial multiradial

(cf.GL_{2}(R)/C* ^{×}* =H+

*⊔*H

*−*)

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 14 / 23

Summary

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

Θnaive *cannot be compatible*w/ 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.

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 15 / 23

.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

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

*v**∈V* (*O**K**v*)_{∗}*F*_{mod} ↷ ∏

*v**∈V*

((K* _{v}*)

*“via (*

_{∗}*O*

*K*

*v*)

*”)*

_{∗}Moreover, this algorithm is *compatible* with

Θ_{naive}: ^{†}*F*_{MOD}^{⊛R} *→*^{∼}^{‡}*F*_{MOD}^{⊛R} ; * ^{†}*q

^{N}*7→*

*q.*

^{‡}Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 16 / 23

Main Theorem of IUT*⇒* Szpiro Conjecture
Θ_{naive}: ^{†}*F*_{MOD}^{⊛R} *→*^{∼}^{‡}*F*_{MOD}^{⊛R} ; * ^{†}*q

^{N}*7→*

*q*

^{‡}Th’m*⇒*

*{*(^{†}*O**K**v*)_{∗}*}**v**∈V*

*†*q* ^{N}* ↷∏

(^{†}*O**K**v*)_{∗}

*†**F*_{mod} ↷∏

(^{†}*K** _{v}*)

_{∗}

ind.↷

*−→**∼*

*{*(^{‡}*O**K**v*)_{∗}*}**v**∈V*

*‡*q* ^{N}* ↷∏

(^{‡}*O**K**v*)_{∗}

*‡**F*_{mod} ↷∏

(^{‡}*K** _{v}*)

_{∗}

*⇒*

*c*mod: ^{†}*F*mod^{⊛}
ind.↷

*→**∼* ^{‡}*F*mod^{⊛} which maps *{*^{†}*q*_{v}^{N}^{†}*O**K**v**} 7→ {*^{‡}*q*^{N}_{v}^{‡}*O**K**v**}*
*c*_{□}: ^{□}*F*mod^{⊛}

*→**∼* ^{□}*F*_{MOD}^{⊛} which maps *{*^{□}*q*^{(N}*v* ^{)}□*O**K**v**} 7→*^{□}q^{(N)}
which are *compatible* with Θ_{naive}, i.e.,

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 17 / 23

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

*†**F*mod^{⊛R}
*c*mod

*−−−→* ^{‡}*F*mod^{⊛R}
*c*_{†}

y y^{c}^{‡}

*†**F*MOD^{⊛R}
Θnaive

*−−−→* ^{‡}*F*MOD^{⊛R}

*⇒*

*{*^{†}*q*_{v}^{N}^{†}*O**K**v**} ⇒ {*^{‡}*q*_{v}^{N}^{‡}*O**K**v**},* *{*^{‡}*q**v**‡**O**K**v**}*

*⇑* *⇓*

*†*q^{N}*⇐* * ^{‡}*q

*⇒ {*^{‡}*q*_{v}^{‡}*O**K**v**}*“^{log-vol.}*⊆* ” ∪

indeterminacies

*{*^{‡}*q*_{v}^{N}^{‡}*O**K**v**}*

*⇒ −*degq *≤ −*degq* ^{N}* +

*C,*i.e., (

*∗*) in p.4

*⇒* the Szpiro Conjecture

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 18 / 23

.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

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

*v**∈V* (*O**K**v*)_{∗}*F*_{mod} ↷ ∏

*v**∈V*

((K* _{v}*)

*“via (*

_{∗}*O*

*K*

*v*)

*”)*

_{∗}Moreover, this algorithm is *compatible* with

Θ_{naive}: ^{†}*F*_{MOD}^{⊛R} *→*^{∼}^{‡}*F*_{MOD}^{⊛R} ; * ^{†}*q

^{N}*7→*

*q.*

^{‡}Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 19 / 23

Recall(v *∈*V: fin.;*p*^{def}= *p** _{v}*) (

*O*

*K*

*v*)

_{+}

*∈*output,

*G*

*: coric Unfortunately, it is known that:*

_{v}*̸ ∃a func’l*(w.r.t. open injections) *alg’m*for rec. “(*O**K**v*)_{+}” from “G* _{v}*”.

*I**K**v*

def= _{2p}^{1}Im(*O*_{K}^{×}_{v}*→ O*^{×}_{K}_{v}*⊗*_{Z}Q

log_{p}

*→**∼* (K* _{v}*)

_{+}): the

*log-shell*of

*K*

_{v}*I*

*K*

*v*: a finitely generated free Z

*p*-module

(*O**K**v*)_{+}, log* _{p}*(

*O*

^{×}

_{K}*)*

_{v}*⊆ I*

*K*

*v*

*⊆ I*

*K*

*v*

*⊗*

_{Z}Q = (K

*)*

_{v}_{+}

[*I**K**v* : (*O**K**v*)_{+}] (<*∞*) can be comp’d by the top. gp str. of *G*_{v}*G*_{v}^{∃}^{func’l}*⇒*

algorithm an isomorph of *I**K**v*

Thus, “*{*(*O**K**v*)_{∗}*}**v**∈V*” in Th’m should be replaced by *{I**v*

def= *I**K**v**}**v**∈V*

(where the log-shell at an infinite *v* *∈*V ^{def}= *π· O**K**v*).

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 20 / 23

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 function*whose 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.*

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 21 / 23

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

...

For a “general*E/F*”,

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

the collection of log-shells*{I**v**}**v**∈V*

the theta values (=*{*q^{j}^{2}^{/2l}*}*_{1}_{≤}_{j}_{≤}_{l}_{⋇}^{def}_{=}*l**−*1
2

) ↷ ∏

*v**∈V* *I**v*

*F*mod via*κ-coric functions* ↷ ∏

*v**∈V*

((K*v*)+“via *I**v*”)

Moreover, this alg’m is *compatible* w/ the Θ-link (more precisely,
Θ^{×}_{LGP}*^{µ}* -link) “

^{†}*F*

_{MOD}

^{⊛R}

*→*

^{∼}

^{‡}*F*

_{MOD}

^{⊛R}”; “

*theta values*

^{†}*7→*

*q*

^{‡}^{1/2l}”.

Yuichiro Hoshi (RIMS, Kyoto University) Introduction to IUT I December 1, 2015 22 / 23

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 *links*by means of such Frobenius-like objects.

Take an*´etale-like* object closely related to □
(e.g., “π_{1}^{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 □”.

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