Towards a higher Siegel Weil formula for unitary groups function field Joint with Tony Feng Wei Zhang a classical S W formula unitary F IF quadratic extra global 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 IA F Weil repr TE E SC n F function OC h Cwc h F n
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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