Newtonssrataintheweaklyadmissiblelocf.tl )Motivation-Aheadmissibleloc#
p prime X : p - div group over Ff
N =D(X )£p the corear .Diend . crystal . Tim dim N = ht X Ap
KlIp , Ok
p - adic period map
it :{(X , g ) l X a p - div gp . 10K P : X Or- Ip→* Oup a
af→GRIN) (K) C = codim X
(X ,g ) s- Hodge filter . :
DIX) N K
Que . : What is the image ? Lie LXVYXOK
More generally : G : reductive gp lap be a l£p ) , µ a minuscule cochar . of G s . t
. [BT EB la,µ )
(a q ' split→ Pg : the parabolic def . by m ,
FCa,er ) -
- Qpr ) p - adic period map (Rapoport-Zink , Scholze) t : ht ( G
, b , er )ko acap , →FlGp)
local Shim . var . itate morph . of rig . sp .
im e - F la ,g) a
: admissible locus , open in Fla,g)
Ex : G = Din b basic , pi = (t , O , . . . .
, O)
F la ,g) a =I =P"
- ' l U H E Pn -' = Fla,y)H : Dp- rat .
hyperplane
CFargnes.caraiani-schoX.ci/ohpalg. closed complete, Cb its tilt
-o have Fargas - Fontaine armee Xlap (for Cb) a 1 - dim North . regular scheme bhp
and a E X with tales ) = C x.• = Bate C C )
G a reductive group over Op (for this talk : a q
'split )
.
for estate topology The (Fargas ) Have a bijection of pointed sets
BCG) ⇐ {G- led . onX}
BIG1=451 I be aCIP )} Classification :
[BT mo (ai) Kottwitz pet . Kalb) e en la)py [GLn : v(def b ) ) Galois coin .
Cii ) Newton point : Vb E X* IA)on, dom T
choose : A max . unram .
Toons ET- Ca (A1 EB Borel
[Glen : (Ipn , b r) isocrystal -o Newton pal . partial order : [by⇐ [big
C- Andon ] ⇐ Kalb ) = Kalb ' I
µ Vb ' - Vb non - neg .
A - bin comb . of pos .woods
e-⇒ -
let x E GlBar ) (Bdr = BartC) ) n
O gene ME Eb Ix yog and trivial bdl
.
, × On X
Remy : 1) only depends on
×e Grabdr CC) = GlBar ) /AlBote) 2) arBad.CC/Efanpt#afdPmotr)MCtHGlBiirl/aCBdT
Grab,d: (C) z x Caraiani-Schake
, Rapoport : Let [b '] E BIG ) then
there is x-CGra.FR CC) s - t Eb , × ⇐ Eb,
⇐ Eb'T E BIG , n , b)= {Eb'T I
lil Kalb') = Ka (b ) -y # Timing:fafr
Cii) Vb' E Vb (f- 'IIIGalois . average }
3) Get decomposition
t .
" Newton strata "
,µ) , [AT E Bla,y , b) unique basic et .
⇒ Grab.fr, " open , coincides with admissible
locus
Hansen -
5) Them l l V. ) Gra,Bndr, lb'T =U Arab ,pole ,
Cb"] (b fbasic ) [b' 'I> (b'T
[b''TEBIG, m .b)
Idea : • Hansen : LHS is a union of Newton strata for [b
"] with {Eb"} ← {EbI in Buna
- V .
determine topology on Buna .
Theweaklyadmissiblelocus Let P be a parabolic Sgp . of a E a G- bundle on X
Def : A reduction of E to P is a P- bundle
Ep s . t . Ep xp G E E
Have a bij . (x e Grabar (C ) )
{reductions of E to P }←{red . of Ex to P}
⇐
Levi M and
every reduction bn of (b) to M f him -1g brlg-nk.tl LEp ) and
energy. ÷: 'I;q±÷÷÷¥. Remotes e) Have Bialynicki-Binda map
Grind = GlBate )pl E ') GlBate ) KcBatey→tTaI
aIpr> -for a minuscule , this is an isom .
On Fca,er ) have weakly adm . locus FCG
,y , b) wa
¥→ FIG,y , b) Wa
( then - Fargas - Shen)
2) Complement of araid ' wa is a profinite union of translates Innde Js Capt of certain U - orbits U : unip- radical of B
3) Let Grains b: ifor bot C- BIG, µ , b) the unique basic element€ Earp
,her, wa ( " adm
whichotherNewtonbratainterseotarad.fm# [b] basic , ME X* (A)done
'
Tgi subgroup M containing the centralizer of Vb .
-
Graffin b '
"
for which non - emptiness holds
.
"
Idea : (O) " ⇒ "
⇐ " Bla,µ , b) has a unique maximal
÷: inHN- indecamp . element Cbo]My
- Vbo Have (Theme) Graffin b z arBad
,fi bo
ara.BY" wa is open ⇒ enough to consider [b'J-- Cbo]
-
- IC) not w. a . ⇐
.
'
isom . class of @b.x)pXp M
get : Dimension C dim . of Newton stratum .