if - math.columbia.edu
TRANSCRIPT
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