Mod p points on Shimura varieties of parahoriclevel
Pol van Hoften
King’s College London

January 27, 2021

Structure of the talkStructure of the talkIntroduction to the Langlands-Rapoport conjecture and aquick survey of previous work.

Structure of the talkIntroduction to the Langlands-Rapoport conjecture and aquick survey of previous work.
Statement of the main results.

Structure of the talkIntroduction to the Langlands-Rapoport conjecture and aquick survey of previous work.
Statement of the main results.
Idea of the proof.

The Langlands-Rapoport conjecture I

The Langlands-Rapoport conjecture I

Langlands and Rapoport conjectured the existence of integral
models of Shimura varieties with good properties.

The Langlands-Rapoport conjecture I

Langlands and Rapoport conjectured the existence of integral
models of Shimura varieties with good properties.
For example the the modular curve Y0 (N) has a smooth
integral model over Z(p) for p - N, using the moduli
interpretation in terms of families of elliptic curves.

The Langlands-Rapoport conjecture I

Langlands and Rapoport conjectured the existence of integral
models of Shimura varieties with good properties.
For example the the modular curve Y0 (N) has a smooth
integral model over Z(p) for p - N, using the moduli
interpretation in terms of families of elliptic curves.
The modular curve Y0 (Np) also has an integral model overZ(p) with p - N, but it is no longer smooth.

The Langlands-Rapoport conjecture I

Langlands and Rapoport conjectured the existence of integral
models of Shimura varieties with good properties.
For example the the modular curve Y0 (N) has a smooth
integral model over Z(p) for p - N, using the moduli
interpretation in terms of families of elliptic curves.
The modular curve Y0 (Np) also has an integral model overZ(p) with p - N, but it is no longer smooth.
Understanding these integral models has interesting
applications, e.g. construction of Galois representations(Deligne, Langlands), Ribet’s proof of the -conjecture.

The Langlands-Rapoport conjecture for the modular curve I

The Langlands-Rapoport conjecture for the modular curve I

The Langlands-Rapoport conjecture gives a description of
Y0 (Np)(Fp )The Langlands-Rapoport conjecture for the modular curve I

The Langlands-Rapoport conjecture gives a description of
Y0 (Np)(Fp ) in two steps.
First we divide elliptic curves into isogeny classes Iφ ,The Langlands-Rapoport conjecture for the modular curve I

The Langlands-Rapoport conjecture gives a description of
Y0 (Np)(Fp ) in two steps.
First we divide elliptic curves into isogeny classes Iφ , and use
Honda-Tate theory to classify the isogeny classes φ.

The Langlands-Rapoport conjecture for the modular curve I

The Langlands-Rapoport conjecture gives a description of
Y0 (Np)(Fp ) in two steps.
First we divide elliptic curves into isogeny classes Iφ , and use
Honda-Tate theory to classify the isogeny classes φ.
Then, we describe elliptic curves lying in a single isogeny class
Iφ using (semi) linear algebra.
If we fix E0 ∈ Iφ and E ∼ E0 via a prime-to-p isogeny, then Eb p -lattice in T p E0 ⊗ Ap . If E ∼ E0 via a
corresponds to a Zfp-power isogeny, then E corresponds to a Dieudonné-lattice inthe p-adic Dieudonné-module of E0 .

The Langlands-Rapoport conjecture for the modular curve II

The Langlands-Rapoport conjecture for the modular curve II

DefineX p (φ) = {Lattices in T p E0 }Xp (φ) = {Dieudonné lattices in Tp E0 }.
Then Iφ is almost equal to their product. Precisely
Iφ ' Iφ (Q)\Xp (φ) × X p (φ)/K p ,where Iφ is the automorphism group of E0 , up to isogeny.

(1)The Langlands-Rapoport conjecture for the modular curve II

DefineX p (φ) = {Lattices in T p E0 }Xp (φ) = {Dieudonné lattices in Tp E0 }.

The Langlands-Rapoport conjecture for the modular curve II

DefineX p (φ) = {Lattices in T p E0 }Xp (φ) = {Dieudonné lattices in Tp E0 }.
Then Iφ is almost equal to their product.

The Langlands-Rapoport conjecture for the modular curve II

DefineX p (φ) = {Lattices in T p E0 }Xp (φ) = {Dieudonné lattices in Tp E0 }.
Then Iφ is almost equal to their product. Precisely
Iφ ' Iφ (Q)\Xp (φ) × X p (φ)/K p ,(1)

The Langlands-Rapoport conjecture for the modular curve II

DefineX p (φ) = {Lattices in T p E0 }Xp (φ) = {Dieudonné lattices in Tp E0 }.
Then Iφ is almost equal to their product. Precisely
Iφ ' Iφ (Q)\Xp (φ) × X p (φ)/K p ,where Iφ is the automorphism group of E0 , up to isogeny.

(1)The Langlands-Rapoport conjecture II

The Langlands-Rapoport conjecture II
Let (G , X ) be a Shimura datum, i.e., G is a reductive groupover Q and X is a Hermitian symmetric domain with an actionof G (R).

The Langlands-Rapoport conjecture II
Let (G , X ) be a Shimura datum, i.e., G is a reductive groupover Q and X is a Hermitian symmetric domain with an actionof G (R).
Let p be a prime number, K p ⊂ G (Apf ) be a compact open
subgroup and Kp ⊂ G (Qp ) a parahoric subgroup and let
K = K p Kp ⊂ G (Af ).

The Langlands-Rapoport conjecture II
Let (G , X ) be a Shimura datum, i.e., G is a reductive groupover Q and X is a Hermitian symmetric domain with an actionof G (R).
Let p be a prime number, K p ⊂ G (Apf ) be a compact open
subgroup and Kp ⊂ G (Qp ) a parahoric subgroup and let
K = K p Kp ⊂ G (Af ).
Let ShK (G , X ) be the associated Shimura variety, which is analgebraic variety over a number field E , the reflex field.

The Langlands-Rapoport conjecture II
Let (G , X ) be a Shimura datum, i.e., G is a reductive groupover Q and X is a Hermitian symmetric domain with an actionof G (R).
Let p be a prime number, K p ⊂ G (Apf ) be a compact open
subgroup and Kp ⊂ G (Qp ) a parahoric subgroup and let
K = K p Kp ⊂ G (Af ).
Let ShK (G , X ) be the associated Shimura variety, which is analgebraic variety over a number field E , the reflex field.
If v | p is a place of E , then the conjecture predicts that there
should be a ‘canonical’ integral model SK (G , X ) over OE(v ) .

The Langlands-Rapoport conjecture II
Let (G , X ) be a Shimura datum, i.e., G is a reductive groupover Q and X is a Hermitian symmetric domain with an actionof G (R).
Let p be a prime number, K p ⊂ G (Apf ) be a compact open
subgroup and Kp ⊂ G (Qp ) a parahoric subgroup and let
K = K p Kp ⊂ G (Af ).
Let ShK (G , X ) be the associated Shimura variety, which is analgebraic variety over a number field E , the reflex field.
If v | p is a place of E , then the conjecture predicts that there
should be a ‘canonical’ integral model SK (G , X ) over OE(v ) .
For example G = GL2 , X = H± and Kp = GL2 (Zp ) or
Kp = Γ0 (p), then E = Q and the integral models from theprevious slide are ‘canonical’.

The Langlands-Rapoport conjecture III

The Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’The Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’a
SK (G , X )(Fp ) 'Iφ ,φThe Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’a
SK (G , X )(Fp ) 'Iφ ,φcompatible with the action of prime to p Hecke operators.

The Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’a
SK (G , X )(Fp ) 'Iφ ,φcompatible with the action of prime to p Hecke operators.
Moreover, the isogeny classes have the following description(Rapoport-Zink uniformisation)

The Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’a
SK (G , X )(Fp ) 'Iφ ,φcompatible with the action of prime to p Hecke operators.
Moreover, the isogeny classes have the following description(Rapoport-Zink uniformisation)
Iφ ' Iφ (Q)\Xp (φ) × X p (φ)/K p

The Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’a
SK (G , X )(Fp ) 'Iφ ,φcompatible with the action of prime to p Hecke operators.
Moreover, the isogeny classes have the following description(Rapoport-Zink uniformisation)
Iφ ' Iφ (Q)\Xp (φ) × X p (φ)/K p

Here Xp (φ) parametrises p-power isogenies,

The Langlands-Rapoport conjecture III
The conjecture then predicts that there is a partition into
‘isogeny classes’a
SK (G , X )(Fp ) 'Iφ ,φcompatible with the action of prime to p Hecke operators.
Moreover, the isogeny classes have the following description(Rapoport-Zink uniformisation)
Iφ ' Iφ (Q)\Xp (φ) × X p (φ)/K p

Here Xp (φ) parametrises p-power isogenies, X p (φ)
parametrises prime-to-p isogenies and Iφ (Q) is the group of
self quasi-isogenies.

Previous WorkPrevious Work
Kottwitz (1992) proved closely related results for PEL type Shimuravarieties of type A and C, at primes p > 2 with Kp hyperspecial.

Previous Work
Kottwitz (1992) proved closely related results for PEL type Shimuravarieties of type A and C, at primes p > 2 with Kp hyperspecial.

Theorem (Kisin, 2008 and 2013)
Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is unramified and that Kp is hyperspecial. Thenthe Langlands-Rapoport conjecture∗ holds for (G , X , p).

Previous Work
Kottwitz (1992) proved closely related results for PEL type Shimuravarieties of type A and C, at primes p > 2 with Kp hyperspecial.

Theorem (Kisin, 2008 and 2013)
Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is unramified and that Kp is hyperspecial. Thenthe Langlands-Rapoport conjecture∗ holds for (G , X , p).

Theorem (Kisin-Pappas, 2015)
Canonical integral models exist for (G , X ) of abelian type and Kp
parahoric.

Previous Work
Kottwitz (1992) proved closely related results for PEL type Shimuravarieties of type A and C, at primes p > 2 with Kp hyperspecial.

Theorem (Kisin, 2008 and 2013)
Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is unramified and that Kp is hyperspecial. Thenthe Langlands-Rapoport conjecture∗ holds for (G , X , p).

Theorem (Kisin-Pappas, 2015)
Canonical integral models exist for (G , X ) of abelian type and Kp
parahoric.

Theorem (Zhou, 2017)
Let (G , X ) be a Shimura datum of Hodge type, let p > 2 andsuppose that GQp is residually split, then isogeny classes have
Rapoport-Zink uniformisation for arbitrary parahorics Kp .

Main Results I

Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is quasi-split and unramified.

Main Results I

Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is quasi-split and unramified. Let K p ⊂ G (Apf )be compact open and let Kp ⊂ G (Qp ) be a parahoric subgroup.

Main Results I

Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is quasi-split and unramified. Let K p ⊂ G (Apf )be compact open and let Kp ⊂ G (Qp ) be a parahoric subgroup.

Theorem 1 (-)
The Langlands-Rapoport conjecture∗ holds for the Kisin-Pappas
integral models of ShK (G , X ).

Main Results I

Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is quasi-split and unramified. Let K p ⊂ G (Apf )be compact open and let Kp ⊂ G (Qp ) be a parahoric subgroup.

Theorem 1 (-)
The Langlands-Rapoport conjecture∗ holds for the Kisin-Pappas
integral models of ShK (G , X ).

RemarkMain Results I

Let (G , X ) be a Shimura datum of abelian type, let p > 2 andsuppose that GQp is quasi-split and unramified. Let K p ⊂ G (Apf )be compact open and let Kp ⊂ G (Qp ) be a parahoric subgroup.

Theorem 1 (-)
The Langlands-Rapoport conjecture∗ holds for the Kisin-Pappas
integral models of ShK (G , X ).

Remark
We can also prove Theorem 1 when GQp is quasi-split and tamely
ramified, under the assumption that ShK (G , X ) is proper.

Idea of the proof I

Idea of the proof I
Since we know the results at hyperspecial level, it suffices tounderstand the fibers of the forgetful map.

Idea of the proof I
Since we know the results at hyperspecial level, it suffices tounderstand the fibers of the forgetful map. When G = GL2 , thenthe forgetful map has the following description:
Y0 (Np){(E , αN , H ⊂ E [p]}(2)

Y0 (N){(E , αN }

Idea of the proof I
Since we know the results at hyperspecial level, it suffices tounderstand the fibers of the forgetful map. When G = GL2 , thenthe forgetful map has the following description:
Y0 (Np){(E , αN , H ⊂ E [p]}(2)

Y0 (N){(E , αN }Here E is an elliptic curve, αN is a Γ0 (N) level structure andH ⊂ E [p] is a subgroup of order p.

Idea of the proof I
Since we know the results at hyperspecial level, it suffices tounderstand the fibers of the forgetful map. When G = GL2 , thenthe forgetful map has the following description:
Y0 (Np){(E , αN , H ⊂ E [p]}(2)

Y0 (N){(E , αN }Here E is an elliptic curve, αN is a Γ0 (N) level structure andH ⊂ E [p] is a subgroup of order p. An elliptic curve over Fp has
either one or two choices for H, depending on whether it issupersingular or ordinary.

Idea of the proof I
Since we know the results at hyperspecial level, it suffices tounderstand the fibers of the forgetful map. When G = GL2 , thenthe forgetful map has the following description:
Y0 (Np){(E , αN , H ⊂ E [p]}(2)

Y0 (N){(E , αN }Here E is an elliptic curve, αN is a Γ0 (N) level structure andH ⊂ E [p] is a subgroup of order p. An elliptic curve over Fp has
either one or two choices for H, depending on whether it issupersingular or ordinary. We observe that the fiber only depends
on the p-divisible group E [p ∞ ]Idea of the proof II

Idea of the proof II

For moduli spaces of abelian varieties with extra structures, thesefibers are more complicated and usually not finite, for example thefibers can be projective lines.

Idea of the proof II

For moduli spaces of abelian varieties with extra structures, thesefibers are more complicated and usually not finite, for example thefibers can be projective lines. However, it is still true that the fibers
only depend on the p-divisible group with extra structures.

Idea of the proof II

For moduli spaces of abelian varieties with extra structures, thesefibers are more complicated and usually not finite, for example thefibers can be projective lines. However, it is still true that the fibers
only depend on the p-divisible group with extra structures. Thismeans that we can use Dieudonné theory to understand the fibers.

Idea of the proof II

For moduli spaces of abelian varieties with extra structures, thesefibers are more complicated and usually not finite, for example thefibers can be projective lines. However, it is still true that the fibers
only depend on the p-divisible group with extra structures. Thismeans that we can use Dieudonné theory to understand the fibers.
For Hodge type Shimura varieties, the integral models do not havea moduli interpretation, which makes it difficult to make the above
strategy work.

Idea of the proof II

For moduli spaces of abelian varieties with extra structures, thesefibers are more complicated and usually not finite, for example thefibers can be projective lines. However, it is still true that the fibers
only depend on the p-divisible group with extra structures. Thismeans that we can use Dieudonné theory to understand the fibers.
For Hodge type Shimura varieties, the integral models do not havea moduli interpretation, which makes it difficult to make the above
strategy work. We can still associate a p-divisible group with extra
structures X to an Fp -point, but it is no longer clear that the fiber
only depends on this X .

Idea of the proof III

Idea of the proof III
Let Kp be a hyperspecial parahoric and Kp0 ⊂ Kp another parahoric.

Idea of the proof III
Let Kp be a hyperspecial parahoric and Kp0 ⊂ Kp another parahoric.
Let SK ,Fp (G , X ) be the special fiber of the Kisin-Pappas integral
model, then it has a morphism to the ‘moduli space of p-divisiblegroups with extra structures’.

Idea of the proof III
Let Kp be a hyperspecial parahoric and Kp0 ⊂ Kp another parahoric.
Let SK ,Fp (G , X ) be the special fiber of the Kisin-Pappas integral
model, then it has a morphism to the ‘moduli space of p-divisiblegroups with extra structures’. This map fits into a commutative
diagram together with its variant for K 0

Idea of the proof III
Let Kp be a hyperspecial parahoric and Kp0 ⊂ Kp another parahoric.
Let SK ,Fp (G , X ) be the special fiber of the Kisin-Pappas integral
model, then it has a morphism to the ‘moduli space of p-divisiblegroups with extra structures’. This map fits into a commutative
diagram together with its variant for K 0
SK 0 ,Fp (G , X )ShtG ,µ,Kp0(3)

SK ,Fp (G , X )ShtG ,µ,Kp .

Idea of the proof III
Let Kp be a hyperspecial parahoric and Kp0 ⊂ Kp another parahoric.
Let SK ,Fp (G , X ) be the special fiber of the Kisin-Pappas integral
model, then it has a morphism to the ‘moduli space of p-divisiblegroups with extra structures’. This map fits into a commutative
diagram together with its variant for K 0
SK 0 ,Fp (G , X )ShtG ,µ,Kp0(3)

SK ,Fp (G , X )ShtG ,µ,Kp .

Here ShtG ,µ,Kp is the stack of G -shtukas of type µ and level Kp .
These were introduced by Xiao-Zhu and generalised by
Shen-Yu-Zhang.

Idea of the proof III
Let Kp be a hyperspecial parahoric and Kp0 ⊂ Kp another parahoric.
Let SK ,Fp (G , X ) be the special fiber of the Kisin-Pappas integral
model, then it has a morphism to the ‘moduli space of p-divisiblegroups with extra structures’. This map fits into a commutative
diagram together with its variant for K 0
SK 0 ,Fp (G , X )ShtG ,µ,Kp0(3)

SK ,Fp (G , X )ShtG ,µ,Kp .

Here ShtG ,µ,Kp is the stack of G -shtukas of type µ and level Kp .
These were introduced by Xiao-Zhu and generalised by
Shen-Yu-Zhang.
The LR conjecture holds for the Shimura variety in the top left
corner if and only if the diagram is Cartesian.

Idea of the proof IV

Idea of the proof IV

So let Y be the fiber product of the diagram

Idea of the proof IV

So let Y be the fiber product of the diagram and consider themorphism i : SK 0 ,Fp (G , X ) → Y , we will show it is an isomorphismin three steps:

Idea of the proof IV

So let Y be the fiber product of the diagram and consider themorphism i : SK 0 ,Fp (G , X ) → Y , we will show it is an isomorphismin three steps:
We show that i is a closed immersion.

Idea of the proof IV

So let Y be the fiber product of the diagram and consider themorphism i : SK 0 ,Fp (G , X ) → Y , we will show it is an isomorphismin three steps:
We show that i is a closed immersion.
We show that Y is equidimensional.

Idea of the proof IV

So let Y be the fiber product of the diagram and consider themorphism i : SK 0 ,Fp (G , X ) → Y , we will show it is an isomorphismin three steps:
We show that i is a closed immersion.
We show that Y is equidimensional.
We prove that Y has the same number of irreducible
components as SK 0 ,Fp (G , X ).

Idea of the proof IV

So let Y be the fiber product of the diagram and consider themorphism i : SK 0 ,Fp (G , X ) → Y , we will show it is an isomorphismin three steps:
We show that i is a closed immersion.
We show that Y is equidimensional.
We prove that Y has the same number of irreducible
components as SK 0 ,Fp (G , X ). We do this by showing that Y
has as few irreducible components as possible.

Idea of the proof IV

So let Y be the fiber product of the diagram and consider themorphism i : SK 0 ,Fp (G , X ) → Y , we will show it is an isomorphismin three steps:
We show that i is a closed immersion.
We show that Y is equidimensional.
We prove that Y has the same number of irreducible
components as SK 0 ,Fp (G , X ). We do this by showing that Y
has as few irreducible components as possible.
This last result is new even for SK 0 ,Fp (G , X )!

Idea of the proof V

Idea of the proof V

Both Y and SK 0 ,Fp (G , X ) have a natural stratification, called the
KR stratification.

Idea of the proof V

Both Y and SK 0 ,Fp (G , X ) have a natural stratification, called the
KR stratification.
KR strata are equidimensional, and both Y and SK 0 ,Fp (G , X ) are aunion of closures of strata.

Idea of the proof V

Both Y and SK 0 ,Fp (G , X ) have a natural stratification, called the
KR stratification.
KR strata are equidimensional, and both Y and SK 0 ,Fp (G , X ) are aunion of closures of strata.
If we could show that the Kottwitz-Rapoport strata of Y were
irreducible, then we would be done.

Idea of the proof V

Both Y and SK 0 ,Fp (G , X ) have a natural stratification, called the
KR stratification.
KR strata are equidimensional, and both Y and SK 0 ,Fp (G , X ) are aunion of closures of strata.
If we could show that the Kottwitz-Rapoport strata of Y were
irreducible, then we would be done.
This isn’t entirely true, but we can deal with the other cases (thosecontained in the basic locus) using results of Zhou.

Idea of the proof VI

Idea of the proof VI

This irreducibility argumnent the main part of my proof;

Idea of the proof VI

This irreducibility argumnent the main part of my proof; some
ingredients areIdea of the proof VI

This irreducibility argumnent the main part of my proof; some
ingredients are
An irreducibility argument of Görtz-Yu

Idea of the proof VI

This irreducibility argumnent the main part of my proof; some
ingredients are
An irreducibility argument of Görtz-Yu
An irreducibility argument of He-Zhou

Idea of the proof VI

This irreducibility argumnent the main part of my proof; some
ingredients are
An irreducibility argument of Görtz-Yu
An irreducibility argument of He-Zhou
Strong approximation

Idea of the proof VI

This irreducibility argumnent the main part of my proof; some
ingredients are
An irreducibility argument of Görtz-Yu
An irreducibility argument of He-Zhou
Strong approximation
Monodromy arguments à la Chai

Main Results II

Main Results II
The modular curve Γ0 (N) comes equipped with the Ekedahl-Oort
stratification;

Main Results II
The modular curve Γ0 (N) comes equipped with the Ekedahl-Oort
stratification; the stratum that an Fp point (E , αn ) is in isdetermined by the p-torsion E [p].

Main Results II
The modular curve Γ0 (N) comes equipped with the Ekedahl-Oort
stratification; the stratum that an Fp point (E , αn ) is in isdetermined by the p-torsion E [p]. More generally, this defines a
stratification on the moduli space of abelian varieties.

Main Results II
The modular curve Γ0 (N) comes equipped with the Ekedahl-Oort
stratification; the stratum that an Fp point (E , αn ) is in isdetermined by the p-torsion E [p]. More generally, this defines a
stratification on the moduli space of abelian varieties.
Ekedahl and van der Geer showed that Ekedahl-Oort strata are
irreducible precisely when they are not contained in the
supersingular locus.

Main Results II
The modular curve Γ0 (N) comes equipped with the Ekedahl-Oort
stratification; the stratum that an Fp point (E , αn ) is in isdetermined by the p-torsion E [p]. More generally, this defines a
stratification on the moduli space of abelian varieties.
Ekedahl and van der Geer showed that Ekedahl-Oort strata are
irreducible precisely when they are not contained in the
supersingular locus.
Let (G , X ) be as above, and let Kp be a hyperspecial subgroup.

Main Results II
The modular curve Γ0 (N) comes equipped with the Ekedahl-Oort
stratification; the stratum that an Fp point (E , αn ) is in isdetermined by the p-torsion E [p]. More generally, this defines a
stratification on the moduli space of abelian varieties.
Ekedahl and van der Geer showed that Ekedahl-Oort strata are
irreducible precisely when they are not contained in the
supersingular locus.
Let (G , X ) be as above, and let Kp be a hyperspecial subgroup.

Theorem 2 (-)
Suppose for simplicity that G ad is Q-simple. Then Ekedahl-Oort
strata that are not contained in the basic locus are irreducible.

Main Results III

Main Results III
Using the Mantovan product formula,

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Theorem 3
Suppose for simplicity that G ad is Q-simple.

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Theorem 3
Suppose for simplicity that G ad is Q-simple. If GQp is split, thennon-basic Newton strata are irreducible.

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Theorem 3
Suppose for simplicity that G ad is Q-simple. If GQp is split, thennon-basic Newton strata are irreducible. More generally the numberof irreducible components of a Newton stratum Shb is given by the
representation theoretic constant

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Theorem 3
Suppose for simplicity that G ad is Q-simple. If GQp is split, thennon-basic Newton strata are irreducible. More generally the numberof irreducible components of a Newton stratum Shb is given by the
representation theoretic constant
Dim Vµ (λb )rel ,introduced by Zhou-Zhu.

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Theorem 3
Suppose for simplicity that G ad is Q-simple. If GQp is split, thennon-basic Newton strata are irreducible. More generally the numberof irreducible components of a Newton stratum Shb is given by the
representation theoretic constant
Dim Vµ (λb )rel ,introduced by Zhou-Zhu. Furthermore, central leaves in non-basic
Newton strata are also irreducible,

Main Results III
Using the Mantovan product formula, and results of Zhou-Zhu on
irreducible components of Rapoport-Zink spaces, we can prove thefollowing result:

Theorem 3
Suppose for simplicity that G ad is Q-simple. If GQp is split, thennon-basic Newton strata are irreducible. More generally the numberof irreducible components of a Newton stratum Shb is given by the
representation theoretic constant
Dim Vµ (λb )rel ,introduced by Zhou-Zhu. Furthermore, central leaves in non-basic
Newton strata are also irreducible, this is known as the discrete partof the Hecke-orbit conjecture.

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