University . BONORA - IRTESTE FEST : Sissa 2015 July 1.2 , - - PowerPoint PPT Presentation

SLIDE 1 ARITHMETIC ASPECTS OF LIOUVILLE ettore Alotrovamoh ' Florida . State

University

BONORA FEST : Sissa

• IRTESTE

, July 1.2 , 2015

SLIDE 2 THE LIOUVILLE

Equation

÷ *

.

• tzet

SLIDE 3 THE LIOUVILLE EQUATION U C . ¢ , ye

C- (

U ,R )

J÷

t.se

• tzet

Z=x

thy

complex

coordinate

SLIDE 4 THE LIOUVILLE EQUATION

Conformal

metrics

:

p=et/del3

(

• ver

UCE

)

J÷

±* .

tee

" ⇐ >

ks=

• 1

¥ uhmcwwatwu

Ks=

• zeosiy
• JE

SLIDE 5 THE LIOUVILLE Equation

÷⇐

,

• tee

" ⇐ >

ks=

. 1

¥ uhmcwwatwu

Ks=

• zi

41 , g=e41dzP 22 JE f- e CTU ,R )

t.FI

,=.$( et 's )=o

SLIDE 6 THE LIOUVILLE EQUAT

:

£÷€ . tae " ⇐ >

¥=

• 1

Salon Curvature Ks=

• zi

41 , g=e41dzP 22 JE f- e CTU ,R )

t.FI

,=.$( et 's )=o

$4

)=r÷fuae^5p+±s

fuiedzndz

SLIDE 7 InvarianceProperties = r÷fuae^5p+ Tt

fuiedzndz

e " dz role = e¢ dz ' A

• tz

' , z '

• f ( z )

Area true : OK

SLIDE 8 Invariance Properties

$4

) = r÷fuae^5p+ Tt Sue .

dzndze

" dz role = e¢ dz ' A

• tz

' , z ' = f ( z ) Area true : OK

24^54-29^59

' = d (something ) Ill

• defined
• n the

nose .

SLIDE 9 The Lionville Functional

( for

real . . . ) X : Riemann Surface , g=g(X) 22 .

(

X = Ha , where * is a Smooth surface

• ver

an arithmetic ring A , eg . A =@ , { 1,5 } , Fa ) ) f : Tx

• Ext ( sheaf of positive

real . valued functions) S : CH ×

• R

K conformal factors ( relative to the fixed conformal structure )

54 )

= £ ( s ) + Sxvols Quadratic Term 9 [ Areatuu

V

SLIDE 10 The Lionville Functional ( for real . . . )

µ

54 )

= £ ( s ) + Sxvds

,

Cup square : SH=#x,sD U It ,s )] Hermit an Deligme Cohomology . Determinant

• f cohomology

. Houuitianhobmomy

• f

a 2- gerke f. . that never was ... )

SLIDE 11 The Lionville Functional ( for real . . . )

µ

54 )

= £ ( s ) + Sxvds

,

Cup square :

5Gt

It ,sD U It ,s )] Hermit an Deligme Cohomology . Determinant

• f cohomology

. Houuitianhobmomy

• f

a 2- gerbe f. . that never was ... ) Regularized Volume

• f

N3 , where JN3=X

Result

• f

a transgression map

( Conjectural

, in part )

SLIDE 12 Metrized lime Bundles . X : algebraic variety / Coeuflex manifold

L

: lime bundle my Invertible sheaf Ucx

• La

xlu ) . module f : L

• Ex

,+ hermit ion fiber

metrics

i

• g ( s )

positive , smooth , R

• valued

X = U ;

Ui

: Sj = si Yij ,

gijgjpegik

, 9 (sj) = f ( si ) / of;jP

PI

( × ) = { [ L , p ] } Arithmetic Picard Grp mm Lemont Hhs ) ] e IBK,

HYAPK

, z→O×→% )

|

• =

At ' ( × , 6x*

• Ex

, + )

SLIDE 13

PRODUCT (

ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) c- weuwyhic . , Pick )

×PIk

) µa(x , aoioxbrx . 1in 1in to

;±!g¥ie

" ) x rhsmooth , (L ,p ) , (Mst )

• 4Th

Imaginary ttterwitiom 2 . Gerke

SLIDE 14

PRODUCT (

ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) c- weuwyhic . , Pick )

×PIk

) *p(× , tnsoxsrx . 1in Inn

t

;±!g¥ie

" ) x 1L ,p ) , (Mst )

• 4Th

rlsmooth , < Herwitiom 2 . Gerbe Imaginary GEOMETR.tn x

SLIDE 15

PRODUCT (

ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) c- weuwyhic . ' 2

• l

Pick )

×PIk

)

*p(x

, a →

• x

→ six . 1in Inn to

;±!g¥ie

" ) x (L ,p ) , (Mst )

• 4Th

rlsmooth , £ Herwitiom 2 . Gerbe Imaginary GEOMETR.tn

7

×

SLIDE 16

PRODUCT (

ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) c- weuwyhic . ' 2

• l

Pick )

×PIk

)

*p(x

, a →

• x
• six

. 1in 1in

t

;±!g¥ie

" ) x (L ,p ) , (Mst )

• 4Th

rlsmooth , € Herwitiom 2 . Gerbe Imaginary GEOMETR.tn

→

×

SLIDE 17 Batten ,¥ X : Riemann Surface / Smooth Aly . Curve

€iP.nu

%w*r*#

.,

SLIDE 18 Batten ,¥ X : Riemann Surface

HmoothAhy.Cwwee@P.nwNowters.I

., STILL . . . .

SLIDE 19 Batman ,¥ X : Riemann Surface /Smooth Aly . Curve

€fP÷%.

%w*r*#

., STILL . . . . : . →H4x,ei±ei¥ieb ") →

Askin )

• Htolx

, a KDOEK

"Cx )

→ ...

=

RQZIF ,

• →

C L , Me ] " 1

• 1

✓ So , we get an evaluation to H2(× , R )@2#Fi ( Further : extract a number by evaluating against Effx )

SLIDE 20 Back to Lionville

gtlggabote

Metric

(L ,e)=(

the )

= ( Tx ,e9dH2 ) . < (Tx ,e)U( The) , [X] ) =

Eiet

|¥k%B9i

Standard

Quadratic Term

SLIDE 21 Back to Lionville

gtlgpabote

Metric

(L ,e)=(

the )

= ( Tx ,e9dH2 ) . < (Tx , e) U( The) , [X] ) =

Eiet

fats

%B9i r Correction

<

Zia

>€I'¥o%k'

hylgiolg

• bglgytdag

, Tons + [ , ... ,⇐[ , So;µ( Block . Wigner dibgs ) .µ Ttm < it ,e ) ,E×,eD+f×vf well defined ; gtsyetfp )=o⇐>Ke=

• 1

SLIDE 22 Remark : The Determinant

• f Cohomology

, X as before L , M with national sections s , t with ( s ) =D , HE L s , t > : e . torso ; Relations : < fs , t > = f ft ) < s , t > < s , gt > = go , , s , + , ) compatible by Weie Reciprocity . Metized Lime Bundles :( L , e) , A , =) Demote the norms by 11.112 . The e- torso < s ,t > is meted with norm given by exp {

It

, fxotbgllslibg HE +

bgllsli CEt

+ bglltlild ) }

Thnf

Up to a factor =

Show

( p )

( The quadratic

part)

SLIDE 23

PICTORIALHEXPLANATIONYF

¥###÷¥€±0¥

: " ' viii. x# . )

eibersjijii

"

;.i

. " 1

f-

;

Spec

A

SLIDE 24 ONE MORE THING . . . Use an Etoile covering U Twx Shottkyllniformizxtion * .

*¥##÷

• *

. Yi ¥ aenentotsof p , ,#p .

)

TCPSLKE ) he general : T kleimiangroup , 2nd kind Fundamental ( discrete sbgrp ) fD°maim

• f discontinuity off

. domain X=2N3 , N3= Dp , Vd(N9=a ,

%)

= regal (

N3j(

Well known)

SLIDE 25 ONE MORE THING . . . Use an Etoile covering

RISX

Shotthyllniformizxtion * .

#€nDE÷

• *

. Yi ¥ aenentotsof p , ,§y⇒ .

)

TCPSLKE ) he general : T kleimiangroup , 2nd kind Fundamental ( discrete sbgrp ) fD°maim

• f discontinuity off

. domain X=2N3 , N3= Dp , Vd(N9=a ,

%)

= regal (

N3j(

Well known) Compute Scp ) as before using RISX

;

Hyperbolic tetrahedron

sina.is#iitiiiiEtxfE*.jI

"o¥¥

, in TPCBT , R )

SLIDE 26 ONE MORE THING . . . Owe is tempted to formulate the cemjecturt :

%)

is the transgression

• f the

hyperbolic

volume chess

HF

• Br
• Bpsyce

)

lwmotopy

fiber

SLIDE 27

A

FINAL WORD

...

SLIDE 28