Towards a higher Siegel Weil formula

for unitary groups function field

Joint with Tony FengWei Zhang

a classical S W formula unitary

F IF quadratic extraglobal fields

V h Herm space din n IF

G V V h IF

W to Y Zn din't Herm F

Max isotropic

H U W U Cn n

W

G A x H AI Cs S V X IAF

Weil repr

TE E S C n

F function

OC h Cwc h F n

0 g h Z Cg.h Te n

xE xq.X F

J 0 g h dgG

GIFA

A

E C h S E Eisenstein series on H

L L induce from Siegel parab

C HUA P c Hhz.li

s E

stab X

S W formula

O g h dg c E h O E

GI l counting representationsinn of thermmatrix

Kudla Arithmetic version

Kudla Rapoportarithmetic intersectionnumber an unitary

E h o EShimura variety

I conjecture on non singular Fourier coeft

d inter ot n

T

deg K R divisors E'T h o Tewith momentmatrix T Lnxn Herm

Chao Li Wei Zhang proved this formula

Fw Fields

Waldspurger formula S W formula

Arithmetic S W

s eu

Higher G 2 formula Higher S W

Y Zhang

2 Statement

or X F

f v 1 functionfields k Eq

X F q odd

v e'tale double cover

GI s ShtrgG U n

Hermitian vector bundle on X

I am

F v b of rk n on X

Ii

Bung moduli stackof F h

Modifications

Foo ho F h

means a E Xisometry

Jolxiyxi.az Flxnlod.rx7

at x lowers by deg 1

at rod raises by deg l

Fo FI

4EngthadaFb112

A Hermiliansehtuka with legs Cxix's IrKil

Fo ho F h Fuhr Fr hr

o ISI

Fo ho over X SFo h

Fo ko idx Frs Foo ho

Shtor modulistackof vk n Herm shtukas

Iijima leikam sane thTShtra has dim _nr

Smooth too f t DM stack

Sp c6s

E v b on X r k nm

E

Ee iFo h El's EE Cfr hr CEh

line bundle tito

has dim n 1 r

when r L this is a divisor

analogue of K R divisor on Sh

Note Shtra for r odd

E L Lz to to Lm

E l O r m

Z Zg x x ZfmShtra

2

intersecting m cycles of cochin r

expected dim Cn m r

rkE n expected dim 0

Nontrivial define a O cycle C CHO SHIthat is the virtual fundamental

E is not adirect sum cycle of ZEof linebundle

Can define the nonsingular part ofthe O cycle 2E C CHO SHE

E v b on X rk n

E

zr te toE Fasho

Is

IEi

a E EE defined on Xffgdiscrete invariant

zr 11 Zoila

Ze He 2e a

a E often

Eat a

nonsingular parts A is injectivei.e non deg at genericptofX

ZElap E CH ZECA

nonsing

Fait 2Ela is a propmer scheme Eq

dy ft la E Q

E h E It U Cn n

Fourier expansion at E

E E 64k Gln Op

E GLEE

f Ee E U Cn n An xn Herm

Fournierexp along 5h In

rateFourier coeffes Hermitian forms an E

E E EE standard

E E standardaV a

afe eEjta.EEs

rail function in qsto only when a E FEU

Theorem Feng Y Zhang r o

E rh n n Xa E HE nonsing

normalizing factorsadded

degfzer a1 Efta o

log95

RI r o S W

r odd both sides are 0

Us ShtI 0RHS Eth s E El s

Idea of proof

Elementary rep theory

Wd 742dX Sd N Wi xWd i

Sguw s 213

SJnd Wd Sd htt

Xd i Wd 742 Itadding742 coordinates

d

R tH Indy wa i 1 tii 0

graded rep of Wd

Wd iREICH Ind s i sgt 1 t

OEjEiedi j Wj Wd i

graded virtual rep of Wd

Exercise Rift t RED tr

degCzeriaTIIcal

Hermad XYx moduli stack ofCQ h

Q torsion coh.sk on Xof length 2 d

10hTEE.co.ma

funnels

i

YogiSpringer theory on Hermzd XYX

Rep Wd Peru Herm x

u

g 1 Fgmiddle extn fromany opendensesubset

RiftCt Red t

Herm Springer th

Ked'T t KEI Ctas virtual perv she on Hermad XYx

E a Coker a is a torsion sh

with Herm Strai E EE E Hermed HX

bothsides of higher S W formula

only depend on Q Coker a

HITTER K its

left.ITroIQEi2ftfEIIe

degfshtinelaiffuna.tnng7T

changeorderhththfhuitas 1

choTamauchii

of intersection 4Den 5a

degfterlaEela sFe

gE Inca o

iEda's