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

LIOUVILLE ASPECTS ARITHMETIC OF Alotrovamoh ' ettore Florida State University . BONORA - IRTESTE FEST : Sissa 2015 July 1.2 , ,

THE LIOUVILLE Equation tzet ÷ * - .

THE EQUATION LIOUVILLE U C- C ¢ U ,R ) ( . ye , J÷ tzet - t.se Z=x coordinate thy complex

THE EQUATION LIOUVILLE metrics Conformal p=et/del3 : ( over UCE ) " J÷ . tee ⇐ > -1 ks= ±* ¥ uhmcwwatwu - zeosiy Ks= JE -

THE LIOUVILLE Equation " ÷ ⇐ - tee ⇐ > 1 ks= , . ¥ uhmcwwatwu 41 Ks= zi g=e41dzP - , 22 JE e CTU ,R ) f- t.FI ,=.$( et 's )=o

THE EQUAT LIOUVILLE : " £÷ € . tae ⇐ > -1 ¥= Curvature Salon 41 Ks= zi g=e41dzP - , 22 JE e CTU ,R ) f- t.FI ,=.$( et 's )=o $4 fuiedzndz )=r÷fuae^5p+±s

Invariance Properties = r÷fuae^5p+ fuiedzndz Tt " ' A ' = e¢ ' - f ( z ) dz dz otz role e z , Area true OK :

Invariance Properties . = r÷fuae^5p+ $4 Sue dzndze Tt ) " ' A ' = e¢ ' = f ( z ) dz dz otz role z , Area true OK : ' d ( something ) 24^54-29^59 = on the - defined Ill nose .

The real Lionville . ) Functional ( for . . where * X Riemann ( is Smooth : , g=g(X ) 22 a Surface X Ha . = , arithmetic ring A surface over an , Fa ) ) A { 1,5 } =@ eg , . , : Tx valued functions ) real Ext ( sheaf of positive f - . R S CH × : - structure ) the fixed conformal factors ( relative to K conformal = £ ( s ) 54 ) + Sxvols 9 [ Areatuu Term Quadratic V

The Lionville real Functional . ) ( for . . µ = £ ( s ) 54 ) + Sxvds , SH=#x,sD Cup square : U It ,s ) ] Hermit an Deligme Cohomology . Determinant of cohomology . . that ... ) Houuitianhobmomy 2- gerke f. was never of a

The Lionville real Functional . ) ( for . . µ = £ ( s ) 54 ) + Sxvds , 5Gt Cup square : ,sD It U It ,s ) ] Hermit an Deligme Cohomology . Determinant of cohomology . . that ... ) Houuitianhobmomy 2- gerbe f. was never of a Volume N3 where Regularized of JN3=X , in part ) Result ( Conjectural of transgression a map ,

Bundles lime Metrized . variety / X Coeuflex manifold : algebraic - La L bundle Invertible sheaf lime : Ucx my module xlu ) . hermit L ion : Ex fiber f - ,+ - valued positive , smooth R g ( s ) i - metrics , U X : = Yij = si = f ( si ) / of ;jP Sj 9 ( sj ) Ui , ; , = { [ L Picard p ] } Arithmetic ( × ) Grp , PI gijgjpegik mm HYAPK e IBK , Lemont Hhs ) ] | z → O × → % ) , - ' ( × At = + ) 6x* Ex - , ,

PRODUCT ( ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) weuwyhic c- . , Pick ) × PIk , aoioxbrx µ a(x ) " ) ;±!g¥ie 1in 1in . to x rhsmooth , ( L 4Th ,p ) , ( Mst ) - Imaginary ttterwitiom . Gerke 2

PRODUCT ( ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) weuwyhic c- . , Pick ) × PIk , tnsoxsrx *p( × ) " ) ;±!g¥ie 1in Inn . t x 1 L 4Th ,p ) , ( Mst ) rlsmooth - , < Herwitiom Imaginary . Gerbe 2 GEOMETR.tn x

PRODUCT ( ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) weuwyhic c- . ' ol 2 → six Pick ) × PIk → a ox *p(x ) " ) , ;±!g¥ie 1in Inn . to x ( L 4Th ,p ) , ( Mst ) rlsmooth - , £ Herwitiom Imaginary . Gerbe 2 GEOMETR.tn × 7

PRODUCT ( ARITHMEtfERED.tt#x;DxTtbK;DU-tTslk,2 ) weuwyhic c- . ' ol 2 Pick ) × PIk → six a ox - *p(x ) " ) , ;±!g¥ie 1in 1in . t x ( L 4Th ,p ) , ( Mst ) rlsmooth - , € Herwitiom Imaginary . Gerbe 2 GEOMETR.tn → ×

Riemann Surface / Smooth Curve Batten X Aly : ,¥ . . , %w*r*# € iP.nu

Riemann Batten X : Surface ,¥ . , HmoothAhy.Cwwee@P.nwNowters.I STILL . . . .

Riemann Curve Batman X Surface /Smooth Aly : ,¥ . . , %w*r*# € fP÷%. STILL . . . . . → H4x,ei±ei¥ieb a KDOEK " ) Askin ) "Cx ) → Htolx - → : ... , = " → , Me ] C L o 1 - o - - - - RQZIF , 1 ✓ we get So evaluation to H2( × an R )@2#Fi , , number ( Further evaluating against Effx ) extract a : by

Lionville Back to Metric gtlggabote ( L ,e)=( = ( Tx ,e9dH2 ) the ) . < ( Tx ,e)U( The ) [X] ) Eiet = |¥k%B9i , Term Quadratic Standard

Lionville Back to Metric gtlgpabote ( L ,e)=( = ( Tx ,e9dH2 ) the ) . < ( Tx , e) U( The ) [X] ) Eiet %B9i = , < fats r > € I'¥o%k' - bglgytdag Zia hylgiolg , Correction Tons Block dibgs ) [ + . Wigner So ; µ ( , , ... , ⇐ [ . µ Ttm ,e ) ,E × ,eD+f × vf well it defined ; gtsyetfp )=o ⇐ >Ke= < -1

: The Remark Determinant of Cohomology X as before , with national sections L , M with t ( s ) =D HE s , , Relations , t . torso < fs , , t L t e : f ft ) s > : < s > compatible by Weie > = ; + , ) < , gt > s go , = s , Reciprocity . , Demote the L 11.112 Lime Bundles , A Metized :( e) norms by , =) . , with is meted The given by e- torso < s ,t > norm + bglltlild ) } , fxotbgllslibg HE bgllsli C Et exp { + It Show Thnf to ( p ) Up factor ( The quadratic part ) a =

: ' viii. " . ) x# PICTORIALHEXPLANATIONYF f- ¥###÷¥ € ±0¥ " " 1 eibersjijii ;.i . ; A Spec

MORE Etoile covering ONE THING Use Twx U an . . . Shottkyllniformizxtion * . *¥##÷ - * . Yi ¥ ) aenentotsof , ,#p TCPSLKE ) he general p . T : , 2nd kleimiangroup kind Fundamental ( discrete sbgrp ) domain of discontinuity off fD°maim . Well known ) N3= Dp N3j( X=2N3 Vd(N9=a %) regal ( = , , ,

MORE Etoile covering ONE THING Use an . . . Shotthyllniformizxtion * . # € nDE÷ - * . Yi ¥ ) aenentotsof TCPSLKE ) , ,§y ⇒ RISX he general p . T : , 2nd kleimiangroup kind Fundamental ( discrete sbgrp ) domain of discontinuity off fD°maim . Well known ) N3= Dp N3j( X=2N3 Vd(N9=a %) regal ( = , , , RISX Scp ) as before Compute using tetrahedron Hyperbolic ; , "o¥¥ sina.is#iitiiiiEtxfE*.jI in TPCBT R ) ,

MORE THING ONE . . . the cemjecturt is tempted Owe to formulate : is the transgression %) of the volume chess hyperbolic Br ) Bpsyce - HF - lwmotopy fiber

A FINAL WORD ...