Yoichi

Klap finite

§| What is Fadic unitornization

Thu Hate ) qek'

, lqkl,

] an elliptic curve # K

st. Eqag

"

I Giilq '2as rigid analytic spaces /k

.

An elliptic curve 1k is isomorphic to such an Eq iff

it has split multiplicative reduction

Remark : Gnpane with the complex uniformization El @I 61A

¥ ¢Yq'2,

where there is no obstruction for elliptic curve

b have such a uniformization.

Defn:

RI =P "k\Pl*( k ),

RI can be given a natural

rigid analytic spare structure fatherwith the

natural action ofPGLZCK )

Rmk : Rieiepi P 'Cep)|P' C k ),

compare with h±=P' (e) IPLR)

Thin ( Mumford ) Let T be a discrete oompait and torsion free

swbgp of PGLZCK),

I projeitie smooth curve #k st .

T\R2k it Xf"

.

Moreover,

a smooth proper curve

X/kWith genus ZIZ is isomorphic fg such an XT iff

X has astrictly semistabk model A/gk st .

all irreducible

lohponet of Ak is P'

.

Rmk : In complex Case, every Curve of genus 72 has

a uuiformization Tlhhtfor TCPGLZLR )

552 Immediate applications of p- adic uniformization

2.1 Arithmetic application

EYEa !/q2'

⇒ Eq[ c H = { xetiyoplxt "=a }

= 4 Lin ( qf )'s

Iasi.je oil } as Gakklk ) - modules

so with ban's Sen,

9¥,

the Galois action has the

torn(, → [ Kla

t.de'

] where K the cyclonic

character and Elgin ) = 9¥ style)

.

Taking limit on n,

we have the same matrix representing

the tadic Tate module TLEQ

Thin o→ al .) → TcEq→Zccd → 0

We can also compute the Etale cohomology of

THEThu ( Schneider - Stahler) -7 an isonorminn

Him,

( Chide , Qc ) ± IT ; ti )

as Glzlk ) .x Gatlklk ) - modules,

where

TL ; is the unique irreducible quotient of Indy#

19

where P.

= [ *

Ff ,Pi= TBGL.

Cor T C PGLZCK) discrete ocompaot and tension free,

HETCLXHE, a ) = ale )

TFEELHAE, a) = atom

put )

µCa → au ) → Heckle,

a) → au )"

→ e

where µ4 ) is the multiplicity at Tlo in Ind ,M↳W1

5$ 2.2 Geometric application

A formal model

Aof Gma

"

can be construed as follows,

Let Y be the blowup of A' on at OEAKCA 'o

.

Yk : P ! UA 'u,

Pkn At =D, cheese a ⇐ Phu p }

and 4- a is a scheme over Ok,

let D= @a)"

,

which

is a formal annuli,

Dn9iCp)= { zeeplkyekl :| )

Let

*= U D

; where Di is a scaled Diez

,

ht . (Dit "9(Clp ) = { zeCp| titekle KEY )

The special fiber of

Ais

Pl

. . . ×¥y . ...

then

*/q "

is aformal model of Eq

"

,

which is

sanistabk f Ok,

we can prove I a projective scheme 1110k

Hi X"

I

fyqa

,

and X is a senistabe medal of Eq.

sin also has a formal medel £ ; 10k st. PGL.CH

Pkk ( Deligarane),

it is strictly senistable 10k

the irreducible components of #dkted is given by

the Bruhat - Tits building of PGL, ( k ),

each component

is P'

.The quotient TIBI is algebrazabk

,so

we have a semistabk model of XT.

§ } Shimura Carves havep

. adil uniformization

B quaternion alg /Q that is ramified at p and unranifredatn

( BQAR ± Mln ) , take a maximal order OB of B and

let OBP = OBQZP,

U C (OB @E)"

Gnpait open' ltupup where

KUP=

odpyp,

WTCBTCAPF )

The Shimura curve

Suis a scheme /q , representing

the Hunter

Su( S ) = { ( A. i. I ) / Als abelian surface

1 : OB → End (A) satisfying certain Kottwitz Condition

T is a

UP. equivalence class at non V : TPA : ftp.hA

IsOB @ Er of OB - modules } µ

where Ship a scheme.

Let B be the quaternion alg 10,

unranified at p ,

and

ramifies at all ramified planes of B other Than p and aa.

Note that PFE Endo.CA) QQ,

fix an isom

B' '

C At ) a B× ( at )

pThin For U small

,

I an isom of formal

schaes1c2p5uIGhcapl@apbiEpirxuPlBcAplByaJwheretheactionofGL.ca

.p) on Esiap $ Fur is

the natural one shop and g→ Frtldets 'on Ifr

Gr Su is semistabk/ *P

Sketch of the proof :

Defn :A special formal

0¥. module over a Zp - alg R

is a formal

gpmwith anda homomorphisn

Dpp- Endlm ) satisfying certain conditions

,it is unique

over Ip up to

0psisogeny

,fix one Io

Nilp = Category of Zp . algebra, qt . p is nilpotat

RENilp

G4BR=

{ ( 4,

X. f) | 4 :[ → R/pR a k -hononnphisn

X a special formal Op . module of height 4 over RD

P : 4*10 →Xpypra quasi .

isogey of height µG is represented by

5i@pEzFuPlBlAp1ByoyI4CA.e.v) |

AIRabelian surface

with OB . action,

E :

'

A' →" X OB . isom,

V :

TEA "→ OB @EP K - level structure }

go RapidIF × UPIBCAPIB'' (a) represents the tumor

R - { ax. e. A. e. HI - - . - }Over

Fpwe bugtrut a map

qfighter×

uplB.cat/BYa))*p-lSu)=p

From (4. Xp ,A

,e,T)y¥pwe Can Construct Canonically ( A

'

,I )

where

AYRan abelian surface with

Ob- action

E'

: AT n→ X Op . isom which make the diagram

Commutative

'

A' a.#

R f- XaEp$k'

it µof g- ×

The level structure T induces a levelstructure T'on A

'

define

I(d.X. C. A. e. i ) ) = ( A

'

,J ")

Using Serre . Tate,

we can show that E is an

iron by showing it is Etak and bijeotive on

EFTP- points

Deformation theoryalso

ensures that I Can be extended

to the whole formal scheme.