Example: Take ( G, X ) = (GSp , S ± ) , defined by ( V, ψ ) as above. Let V Z ⊂ V be a Z -lattice, and K p the stabilizer of V Z p = V Z ⊗ Z Z p ⊂ V ⊗ Q p . K p is hyperspecial if and only if a scalar multiple of ψ induces a perfect, Z p -valued pairing on V Z p . Then K p = GSp( V Z p , ψ )( Z p ) . The choice of V Z makes Sh K (GSp , S ± ) as a moduli space for polarized abelian varieties, which leads to a model S K (GSp , S ± ) over O ( λ ) .
Example: Take ( G, X ) = (GSp , S ± ) , defined by ( V, ψ ) as above. Let V Z ⊂ V be a Z -lattice, and K p the stabilizer of V Z p = V Z ⊗ Z Z p ⊂ V ⊗ Q p . K p is hyperspecial if and only if a scalar multiple of ψ induces a perfect, Z p -valued pairing on V Z p . Then K p = GSp( V Z p , ψ )( Z p ) . The choice of V Z makes Sh K (GSp , S ± ) as a moduli space for polarized abelian varieties, which leads to a model S K (GSp , S ± ) over O ( λ ) . The S K (GSp , S ± ) are smooth over O ( λ ) if and only if the degree of the polarization in the moduli problem is prime to p. This corresponds to the condition that ψ induces a perfect pairing on V Z p .
Let O ⊂ E = E ( G, X ) be the ring of integers. For a prime λ | p of E let O ( λ ) be the localization of O at λ.
Let O ⊂ E = E ( G, X ) be the ring of integers. For a prime λ | p of E let O ( λ ) be the localization of O at λ. Conjecture. (Langlands-Milne) Suppose that K = K p K p ⊂ G ( A f ) is open compact and K p is hyperspecial. Then for λ | p, The tower Sh K p ( G, X ) = lim ← K p Sh K p K p ( G, X ) has a G ( A p f ) -equivariant extension to a smooth O ( λ ) -scheme satisfying a certain extension property.
Let O ⊂ E = E ( G, X ) be the ring of integers. For a prime λ | p of E let O ( λ ) be the localization of O at λ. Conjecture. (Langlands-Milne) Suppose that K = K p K p ⊂ G ( A f ) is open compact and K p is hyperspecial. Then for λ | p, The tower Sh K p ( G, X ) = lim ← K p Sh K p K p ( G, X ) has a G ( A p f ) -equivariant extension to a smooth O ( λ ) -scheme satisfying a certain extension property. Theorem. If p > 2 , K p hyperspecial and ( G, X ) is of abelian type, then Sh K p ( G, X ) admits a smooth integral model S K p ( G, X ) .
Let O ⊂ E = E ( G, X ) be the ring of integers. For a prime λ | p of E let O ( λ ) be the localization of O at λ. Conjecture. (Langlands-Milne) Suppose that K = K p K p ⊂ G ( A f ) is open compact and K p is hyperspecial. Then for λ | p, The tower Sh K p ( G, X ) = lim ← K p Sh K p K p ( G, X ) has a G ( A p f ) -equivariant extension to a smooth O ( λ ) -scheme satisfying a certain extension property. Theorem. If p > 2 , K p hyperspecial and ( G, X ) is of abelian type, then Sh K p ( G, X ) admits a smooth integral model S K p ( G, X ) . In the case of Hodge type, S K p ( G, X ) is given by taking the normal- ization of the closure of p (GSp , S ± ) ֒ p (GSp , S ± ) Sh K p ( G, X ) ֒ → Sh K ′ → S K ′ into a suitable moduli space of polarized abelian varieties.
Mod p points - the Langlands-Rapoport conjecture: We continue to assume K p is hyperspecial.
Mod p points - the Langlands-Rapoport conjecture: We continue to assume K p is hyperspecial. Conjecture. (Langlands-Rapoport) There exists a bijection ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ which is compatible with the action of G ( A p f ) and Frobenius on both sides.
Mod p points - the Langlands-Rapoport conjecture: We continue to assume K p is hyperspecial. Conjecture. (Langlands-Rapoport) There exists a bijection ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ which is compatible with the action of G ( A p f ) and Frobenius on both sides. This requires some explanation but, heuristically, the ϕ parameterize ” G - isogeny classes”, while the S ( ϕ ) parameterize the points in a given isogeny class.
Mod p points - the Langlands-Rapoport conjecture: We continue to assume K p is hyperspecial. Conjecture. (Langlands-Rapoport) There exists a bijection ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ which is compatible with the action of G ( A p f ) and Frobenius on both sides. This requires some explanation but, heuristically, the ϕ parameterize ” G - isogeny classes”, while the S ( ϕ ) parameterize the points in a given isogeny class. We will indicate the definition of the ϕ and then explain the definition of S ( ϕ ) .
Mod p points - the Langlands-Rapoport conjecture: We continue to assume K p is hyperspecial. Conjecture. (Langlands-Rapoport) There exists a bijection ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ which is compatible with the action of G ( A p f ) and Frobenius on both sides. This requires some explanation but, heuristically, the ϕ parameterize ” G - isogeny classes”, while the S ( ϕ ) parameterize the points in a given isogeny class. We will indicate the definition of the ϕ and then explain the definition of S ( ϕ ) . The precise definition of the ϕ involves the fundamental groupoid P of the category of motives over ¯ F p . Then P ¯ Q is a pro-torus. The ϕ run over representations ϕ : P → G satisfying certain conditions.
It is easier to explain some invariants which can be attached to each ϕ. The rough idea is that one can attach to each isogeny class the conjugacy class of Frobenius.
It is easier to explain some invariants which can be attached to each ϕ. The rough idea is that one can attach to each isogeny class the conjugacy class of Frobenius. Fix an integer r >> 0 . Then attached to ϕ is a triple ( γ 0 , b, ( γ l ) l � = p ) where
It is easier to explain some invariants which can be attached to each ϕ. The rough idea is that one can attach to each isogeny class the conjugacy class of Frobenius. Fix an integer r >> 0 . Then attached to ϕ is a triple ( γ 0 , b, ( γ l ) l � = p ) where • γ 0 ∈ G ( Q ) is semi-simple, defined up to conjugacy in G ( ¯ Q ) (stable conjugacy).
It is easier to explain some invariants which can be attached to each ϕ. The rough idea is that one can attach to each isogeny class the conjugacy class of Frobenius. Fix an integer r >> 0 . Then attached to ϕ is a triple ( γ 0 , b, ( γ l ) l � = p ) where • γ 0 ∈ G ( Q ) is semi-simple, defined up to conjugacy in G ( ¯ Q ) (stable conjugacy). • For l � = p, γ l ∈ G ( Q l ) is a semi-simple conjugacy class, stably conjugate to γ 0 ∈ G ( ¯ Q l ) .
It is easier to explain some invariants which can be attached to each ϕ. The rough idea is that one can attach to each isogeny class the conjugacy class of Frobenius. Fix an integer r >> 0 . Then attached to ϕ is a triple ( γ 0 , b, ( γ l ) l � = p ) where • γ 0 ∈ G ( Q ) is semi-simple, defined up to conjugacy in G ( ¯ Q ) (stable conjugacy). • For l � = p, γ l ∈ G ( Q l ) is a semi-simple conjugacy class, stably conjugate to γ 0 ∈ G ( ¯ Q l ) . • b ∈ G (Fr W ( F p r )) is an element defined up to Frobenius conjugacy ( b �→ g − 1 bσ ( g ), σ abs. Frobenius) such that Nb = bσ ( b ) . . . σ r − 1 ( b ) is stably conjugate to γ 0 .
It is easier to explain some invariants which can be attached to each ϕ. The rough idea is that one can attach to each isogeny class the conjugacy class of Frobenius. Fix an integer r >> 0 . Then attached to ϕ is a triple ( γ 0 , b, ( γ l ) l � = p ) where • γ 0 ∈ G ( Q ) is semi-simple, defined up to conjugacy in G ( ¯ Q ) (stable conjugacy). • For l � = p, γ l ∈ G ( Q l ) is a semi-simple conjugacy class, stably conjugate to γ 0 ∈ G ( ¯ Q l ) . • b ∈ G (Fr W ( F p r )) is an element defined up to Frobenius conjugacy ( b �→ g − 1 bσ ( g ), σ abs. Frobenius) such that Nb = bσ ( b ) . . . σ r − 1 ( b ) is stably conjugate to γ 0 . The data is required to satisfy certain conditions (corresponding to those on the ϕ ).
Then we can define S ( ϕ ) . ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Recall that S ( ϕ ) is meant to parameterize points in a fixed isogeny class.
Then we can define S ( ϕ ) . ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Recall that S ( ϕ ) is meant to parameterize points in a fixed isogeny class. X p ( ϕ ) ← → p -power isogenies
Then we can define S ( ϕ ) . ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Recall that S ( ϕ ) is meant to parameterize points in a fixed isogeny class. X p ( ϕ ) ← → p -power isogenies G ( A p f ) ← → prime to p -isogenies
Then we can define S ( ϕ ) . ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Recall that S ( ϕ ) is meant to parameterize points in a fixed isogeny class. X p ( ϕ ) ← → p -power isogenies G ( A p f ) ← → prime to p -isogenies I ( Q ) ← → automorphisms of the AV+extra structure and I is a compact (mod center) form of the centralizer G γ 0 .
← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Let O L = W (¯ F p ) and L = Fr O L . Then we have g − 1 bσ ( g ) ∈ G ( O L ) µ σ ( p − 1 ) G ( O L ) } X p ( ϕ ) = { g ∈ G ( L ) /G ( O L ) :
← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Let O L = W (¯ F p ) and L = Fr O L . Then we have g − 1 bσ ( g ) ∈ G ( O L ) µ σ ( p − 1 ) G ( O L ) } X p ( ϕ ) = { g ∈ G ( L ) /G ( O L ) : The condition in the definition of X p ( ϕ ) corresponds is a group theoretic version of the usual condition on the shape of Frobenius on the Dieudonn´ e module of a p -divisible group.
← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Let O L = W (¯ F p ) and L = Fr O L . Then we have g − 1 bσ ( g ) ∈ G ( O L ) µ σ ( p − 1 ) G ( O L ) } X p ( ϕ ) = { g ∈ G ( L ) /G ( O L ) : The condition in the definition of X p ( ϕ ) corresponds is a group theoretic version of the usual condition on the shape of Frobenius on the Dieudonn´ e module of a p -divisible group. Here µ is a cocharacter of G conjugate to the cocharacter µ h corresponding to h h µ h : C → Res C / R G m ( C ) = C × C → G ( C ) .
← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Let O L = W (¯ F p ) and L = Fr O L . Then we have g − 1 bσ ( g ) ∈ G ( O L ) µ σ ( p − 1 ) G ( O L ) } X p ( ϕ ) = { g ∈ G ( L ) /G ( O L ) : The condition in the definition of X p ( ϕ ) corresponds is a group theoretic version of the usual condition on the shape of Frobenius on the Dieudonn´ e module of a p -divisible group. Here µ is a cocharacter of G conjugate to the cocharacter µ h corresponding to h h µ h : C → Res C / R G m ( C ) = C × C → G ( C ) . If the conjugacy class of µ is fixed by σ s , then the p s -Frobenius acts on X p ( ϕ ) by Φ s ( g ) = ( bσ ) s ( g ) = bσ ( b ) . . . σ s − 1 ( b ) σ s ( g )
Conjecture. ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim
Conjecture. ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Remarks: (1) Implicit in the statement is a generalization of Tate’s theorem on the Tate conjecture for endomorphisms of AV’s over finite fields, since one seeks to classify isogeny classes in terms of Frobenius conjugacy classes.
Conjecture. ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Remarks: (1) Implicit in the statement is a generalization of Tate’s theorem on the Tate conjecture for endomorphisms of AV’s over finite fields, since one seeks to classify isogeny classes in terms of Frobenius conjugacy classes. (2) The LR conjecture is related to the conjecture that every isogeny class contains a CM lifting. In fact Langlands-Rapoport showed that the first conjecture implies the second. In practice one ends up proving the second conjecture on the way to proving the first.
Conjecture. ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim Remarks: (1) Implicit in the statement is a generalization of Tate’s theorem on the Tate conjecture for endomorphisms of AV’s over finite fields, since one seeks to classify isogeny classes in terms of Frobenius conjugacy classes. (2) The LR conjecture is related to the conjecture that every isogeny class contains a CM lifting. In fact Langlands-Rapoport showed that the first conjecture implies the second. In practice one ends up proving the second conjecture on the way to proving the first. (3) In the case of PEL type (A,C) this is due to Kottwitz and Zink. A subtle point is that Kottwitz doesn’t quite construct a canonical bijection, and neither do we. (This seems to me to require a new idea.)
Conjecture. ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim (4) The conjecture leads, via the Lefschetz trace formula, to an expres- sions for Sh K p ( G, X )( F q ) in terms of (twisted) orbital integrals involving ( γ 0 , ( γ l ) l � = p , δ ) .
Conjecture. ∼ → S K p ( G, X )(¯ � S ( ϕ ) − F p ) ϕ ← K p I ( Q ) \ ( X p ( ϕ ) × G ( A p f )) /K p S ( ϕ ) = lim (4) The conjecture leads, via the Lefschetz trace formula, to an expres- sions for Sh K p ( G, X )( F q ) in terms of (twisted) orbital integrals involving ( γ 0 , ( γ l ) l � = p , δ ) . Kottwitz has explained how one can use the Fundamental Lemma to sta- bilize this expression, and compare it with the stabilized geometric side of the trace formula. This should allow one to express the zeta function of Sh K p ( G, X ) in terms of automorphic L -functions.
The LR conjecture for Abelian type Shimura varieties:
The LR conjecture for Abelian type Shimura varieties: Theorem. Suppose p > 2 , and ( G, X ) is of Abelian (e.g Hodge) type with K p hyperspecial.
The LR conjecture for Abelian type Shimura varieties: Theorem. Suppose p > 2 , and ( G, X ) is of Abelian (e.g Hodge) type with K p hyperspecial. Then the LR conjecture holds for S K p ( G, X ) .
The LR conjecture for Abelian type Shimura varieties: Theorem. Suppose p > 2 , and ( G, X ) is of Abelian (e.g Hodge) type with K p hyperspecial. Then the LR conjecture holds for S K p ( G, X ) . Moreover, every mod p isogeny class contains a point which lifts to a CM point.
The LR conjecture for Abelian type Shimura varieties: Theorem. Suppose p > 2 , and ( G, X ) is of Abelian (e.g Hodge) type with K p hyperspecial. Then the LR conjecture holds for S K p ( G, X ) . Moreover, every mod p isogeny class contains a point which lifts to a CM point. First consider ( G, X ) of Hodge type. What are the difficulties compared to the PEL case ?
The LR conjecture for Abelian type Shimura varieties: Theorem. Suppose p > 2 , and ( G, X ) is of Abelian (e.g Hodge) type with K p hyperspecial. Then the LR conjecture holds for S K p ( G, X ) . Moreover, every mod p isogeny class contains a point which lifts to a CM point. First consider ( G, X ) of Hodge type. What are the difficulties compared to the PEL case ? 1) Suppose x ∈ S K p ( G, X )(¯ F p ) so that x ❀ A x an AV. If g − 1 bσ ( g ) ∈ G ( O L ) µ σ ( p − 1 ) G ( O L ) } g ∈ X p ( b ) = { g ∈ G ( L ) /G ( O L ) : then gx ❀ A gx .
The LR conjecture for Abelian type Shimura varieties: Theorem. Suppose p > 2 , and ( G, X ) is of Abelian (e.g Hodge) type with K p hyperspecial. Then the LR conjecture holds for S K p ( G, X ) . Moreover, every mod p isogeny class contains a point which lifts to a CM point. First consider ( G, X ) of Hodge type. What are the difficulties compared to the PEL case ? 1) Suppose x ∈ S K p ( G, X )(¯ F p ) so that x ❀ A x an AV. If g − 1 bσ ( g ) ∈ G ( O L ) µ σ ( p − 1 ) G ( O L ) } g ∈ X p ( b ) = { g ∈ G ( L ) /G ( O L ) : then gx ❀ A gx . But it is not clear that gx ∈ S K p ( G, X )(¯ F p ) because this is defined as a closure and has no easy moduli theoretic description. So it isn’t clear there is a map X p → S K p ( G, X ); g ❀ A gx .
2) x ❀ ( b, ( γ l ) l � = p ) but the existence of γ 0 is not clear: it isn’t clear that the γ l are stably conjugate.
2) x ❀ ( b, ( γ l ) l � = p ) but the existence of γ 0 is not clear: it isn’t clear that the γ l are stably conjugate. 3) Even once one has a map X p × G ( A p f ) → S K p ( G, X ); g ❀ A gx It isn’t clear that the stabilizer of a point is a compact form of G γ 0 .
2) x ❀ ( b, ( γ l ) l � = p ) but the existence of γ 0 is not clear: it isn’t clear that the γ l are stably conjugate. 3) Even once one has a map X p × G ( A p f ) → S K p ( G, X ); g ❀ A gx It isn’t clear that the stabilizer of a point is a compact form of G γ 0 . i.e we need to know that Aut( A x , ( s α )) is big enough. Here A x is thought of as an abelian variety up to isogeny, and the s α are certain cohomology classes (coming from the Hodge cycles).
2) x ❀ ( b, ( γ l ) l � = p ) but the existence of γ 0 is not clear: it isn’t clear that the γ l are stably conjugate. 3) Even once one has a map X p × G ( A p f ) → S K p ( G, X ); g ❀ A gx It isn’t clear that the stabilizer of a point is a compact form of G γ 0 . i.e we need to know that Aut( A x , ( s α )) is big enough. Here A x is thought of as an abelian variety up to isogeny, and the s α are certain cohomology classes (coming from the Hodge cycles). In the PEL case one can deduce this from Tate’s theorem.
Some ideas:
Some ideas: To overcome 1) let x ∈ S K p ( G, X )(¯ x ∈ S K p ( G, X )( ¯ F p ) , and choose ˜ Q p ) lifting x. Then one gets a map G ( Q p ) → S K p ( G, X )( ¯ Q p ); x �→ g ˜ ˜ x
Some ideas: To overcome 1) let x ∈ S K p ( G, X )(¯ x ∈ S K p ( G, X )( ¯ F p ) , and choose ˜ Q p ) lifting x. Then one gets a map G ( Q p ) → S K p ( G, X )( ¯ Q p ); x �→ g ˜ ˜ x Reduction of isogenies mod p gives a map G ( Q p ) → X p ( ϕ ); g �→ g 0 .
Some ideas: To overcome 1) let x ∈ S K p ( G, X )(¯ x ∈ S K p ( G, X )( ¯ F p ) , and choose ˜ Q p ) lifting x. Then one gets a map G ( Q p ) → S K p ( G, X )( ¯ Q p ); x �→ g ˜ ˜ x Reduction of isogenies mod p gives a map G ( Q p ) → X p ( ϕ ); g �→ g 0 . For g ∈ G ( Q p ) , we can define the map X p ( ϕ ) → S K p ( G, X )(¯ F p ) at g 0 by sending g 0 to the reduction of g ˜ x.
Some ideas: To overcome 1) let x ∈ S K p ( G, X )(¯ x ∈ S K p ( G, X )( ¯ F p ) , and choose ˜ Q p ) lifting x. Then one gets a map G ( Q p ) → S K p ( G, X )( ¯ Q p ); x �→ g ˜ ˜ x Reduction of isogenies mod p gives a map G ( Q p ) → X p ( ϕ ); g �→ g 0 . For g ∈ G ( Q p ) , we can define the map X p ( ϕ ) → S K p ( G, X )(¯ F p ) at g 0 by sending g 0 to the reduction of g ˜ x. Of course one can’t expect that every element of X p ( ϕ ) has the form g 0 , but one shows that X p ( ϕ ) has a ”geometric structure” and that ˜ x can be chosen so that the composite map G ( Q p ) → X p ( ϕ ) → π 0 ( X p ( ϕ )) is surjective. This uses joint work with M. Chen and E. Viehmann.
Some ideas: To overcome 1) let x ∈ S K p ( G, X )(¯ x ∈ S K p ( G, X )( ¯ F p ) , and choose ˜ Q p ) lifting x. Then one gets a map G ( Q p ) → S K p ( G, X )( ¯ Q p ); x �→ g ˜ ˜ x Reduction of isogenies mod p gives a map G ( Q p ) → X p ( ϕ ); g �→ g 0 . For g ∈ G ( Q p ) , we can define the map X p ( ϕ ) → S K p ( G, X )(¯ F p ) at g 0 by sending g 0 to the reduction of g ˜ x. Of course one can’t expect that every element of X p ( ϕ ) has the form g 0 , but one shows that X p ( ϕ ) has a ”geometric structure” and that ˜ x can be chosen so that the composite map G ( Q p ) → X p ( ϕ ) → π 0 ( X p ( ϕ )) is surjective. This uses joint work with M. Chen and E. Viehmann.
Finally X p → S K p ( G, X ) . is well defined on a connected component once it is defined at a point. This is a deformation theoretic argument.
Finally X p → S K p ( G, X ) . is well defined on a connected component once it is defined at a point. This is a deformation theoretic argument. To solve 3) (that I is big enough) one uses a reformulation of the proof of Tate’s theorem. (It uses the same key inputs, but is phrased so that it applies to AV’s with Hodge cycles.)
Finally X p → S K p ( G, X ) . is well defined on a connected component once it is defined at a point. This is a deformation theoretic argument. To solve 3) (that I is big enough) one uses a reformulation of the proof of Tate’s theorem. (It uses the same key inputs, but is phrased so that it applies to AV’s with Hodge cycles.) We’ll sketch this in Tate’s original context of principally polarized AV’s.
Tate’s theorem again !:
Tate’s theorem again !: Suppose ( A , ψ ) is a principally polarized abelian variety over F q .
Tate’s theorem again !: Suppose ( A , ψ ) is a principally polarized abelian variety over F q . For l � = p define a group I l over Q l by I l = Aut Frob ( H 1 ( A ¯ F p , Q l ) , ψ ) . (i.e automorphisms compatible with Frobenius and, up to scalar, the po- larization).
Tate’s theorem again !: Suppose ( A , ψ ) is a principally polarized abelian variety over F q . For l � = p define a group I l over Q l by I l = Aut Frob ( H 1 ( A ¯ F p , Q l ) , ψ ) . (i.e automorphisms compatible with Frobenius and, up to scalar, the po- larization). View A as an AV up to isogeny, and define a group I over Q by I = Aut( A , ψ ) (again compatible with ψ up to scalar).
Tate’s theorem again !: Suppose ( A , ψ ) is a principally polarized abelian variety over F q . For l � = p define a group I l over Q l by I l = Aut Frob ( H 1 ( A ¯ F p , Q l ) , ψ ) . (i.e automorphisms compatible with Frobenius and, up to scalar, the po- larization). View A as an AV up to isogeny, and define a group I over Q by I = Aut( A , ψ ) (again compatible with ψ up to scalar). Theorem. We have ∼ I ⊗ Q Q l − → I l .
Theorem. We have ∼ I ⊗ Q Q l − → I l . Proof. By independence of l it is enough to prove this for one l � = p.
Theorem. We have ∼ I ⊗ Q Q l − → I l . Proof. By independence of l it is enough to prove this for one l � = p. Fix a compact open K l ⊂ GSp( Q l ) . Then we have
Theorem. We have ∼ I ⊗ Q Q l − → I l . Proof. By independence of l it is enough to prove this for one l � = p. Fix a compact open K l ⊂ GSp( Q l ) . Then we have I ( Q ) \ I l ( Q l ) /I ( Q l ) ∩ K l ⊂ I ( Q ) \ GSp( Q l ) /K l and the quotient on the right parameterizes PPAV’s which are l -power isogenous to A .
Theorem. We have ∼ I ⊗ Q Q l − → I l . Proof. By independence of l it is enough to prove this for one l � = p. Fix a compact open K l ⊂ GSp( Q l ) . Then we have I ( Q ) \ I l ( Q l ) /I ( Q l ) ∩ K l ⊂ I ( Q ) \ GSp( Q l ) /K l and the quotient on the right parameterizes PPAV’s which are l -power isogenous to A . These corresponds to (some) points on a quasi-projective variety over F q , but they need not be defined over the same finite field as A so the quotient on the right is not finite.
Theorem. We have ∼ I ⊗ Q Q l − → I l . Proof. By independence of l it is enough to prove this for one l � = p. Fix a compact open K l ⊂ GSp( Q l ) . Then we have I ( Q ) \ I l ( Q l ) /I ( Q l ) ∩ K l ⊂ I ( Q ) \ GSp( Q l ) /K l and the quotient on the right parameterizes PPAV’s which are l -power isogenous to A . These corresponds to (some) points on a quasi-projective variety over F q , but they need not be defined over the same finite field as A so the quotient on the right is not finite. However automorphisms in the definition of I l commute with Frobenius, so the quotient on the left is finite. (First ingredient used by Tate !).
In particular I ( Q l ) \ I l ( Q l ) is compact
In particular I ( Q l ) \ I l ( Q l ) is compact Now we choose l so that I l is a split group (use the compatible system - this choice of l is also made by Tate !).
In particular I ( Q l ) \ I l ( Q l ) is compact Now we choose l so that I l is a split group (use the compatible system - this choice of l is also made by Tate !). The theorem follows from the following
In particular I ( Q l ) \ I l ( Q l ) is compact Now we choose l so that I l is a split group (use the compatible system - this choice of l is also made by Tate !). The theorem follows from the following Lemma. Let I ′ be a connected algebraic group over Q l , whose reductive quotient is split. If I ⊂ I ′ is a closed subgroup such that I ( Q l ) \ I ′ ( Q l ) is compact, then I contains a Borel subgroup of I ′ .
In particular I ( Q l ) \ I l ( Q l ) is compact Now we choose l so that I l is a split group (use the compatible system - this choice of l is also made by Tate !). The theorem follows from the following Lemma. Let I ′ be a connected algebraic group over Q l , whose reductive quotient is split. If I ⊂ I ′ is a closed subgroup such that I ( Q l ) \ I ′ ( Q l ) is compact, then I contains a Borel subgroup of I ′ . By the lemma I l /I Q l is projective. But since I is reductive the quotient is also affine, and connected, hence a point.
Above we used ’independence of l ’, and we haven’t proved this yet for arbitrary G. Still the above argument shows I = Aut( A x , ( s α )) has the same rank as G. One can use this to construct enough special points
Above we used ’independence of l ’, and we haven’t proved this yet for arbitrary G. Still the above argument shows I = Aut( A x , ( s α )) has the same rank as G. One can use this to construct enough special points Theorem. Every isogeny class in S K ( G, X )(¯ F p ) contains a point which admits a special lifting.
Above we used ’independence of l ’, and we haven’t proved this yet for arbitrary G. Still the above argument shows I = Aut( A x , ( s α )) has the same rank as G. One can use this to construct enough special points Theorem. Every isogeny class in S K ( G, X )(¯ F p ) contains a point which admits a special lifting. Using the theorem one can solve the problem 2) about existence of γ 0 and γ l being stably conjugate, and hence get the indepenence of l.
Shimura varieties of Abelian type: Recall that a Shimura datum ( G 2 , X 2 ) is called of Abelian type if there is a Shimura datum of Hodge type ( G, X ) and a central isogeny G der → G der 2 which induces an isomorphism on adjoint Shimura data ∼ ( G ad , X ad ) → ( G ad 2 , X ad − 2 ) .
Shimura varieties of Abelian type: Recall that a Shimura datum ( G 2 , X 2 ) is called of Abelian type if there is a Shimura datum of Hodge type ( G, X ) and a central isogeny G der → G der 2 which induces an isomorphism on adjoint Shimura data ∼ ( G ad , X ad ) → ( G ad 2 , X ad − 2 ) . The pro-scheme Sh( G, X ) = lim ← K G ( Q ) \ X × G ( A f ) /K has a natural conjugation action by G ad ( Q ) + = G ad ( Q ) ∩ G ( R ) + .
Shimura varieties of Abelian type: Recall that a Shimura datum ( G 2 , X 2 ) is called of Abelian type if there is a Shimura datum of Hodge type ( G, X ) and a central isogeny G der → G der 2 which induces an isomorphism on adjoint Shimura data ∼ ( G ad , X ad ) → ( G ad 2 , X ad − 2 ) . The pro-scheme Sh( G, X ) = lim ← K G ( Q ) \ X × G ( A f ) /K has a natural conjugation action by G ad ( Q ) + = G ad ( Q ) ∩ G ( R ) + . If K p = G ( Z p ) is hyperspecial, then this induces an action of G ( Z ( p ) ) + on ← K p G ( Q ) \ X × G ( A f ) /K p K p Sh K p ( G, X ) = lim which extends to an action on S K p ( G, X ) .
The integral model S K p ( G 2 , X 2 ) is constructed from S K p ( G, X ) using this action; the geometrically connected components of the former are quo- tients of those of the latter. This is analogous to Deligne’s construction of canonical models.
The integral model S K p ( G 2 , X 2 ) is constructed from S K p ( G, X ) using this action; the geometrically connected components of the former are quo- tients of those of the latter. This is analogous to Deligne’s construction of canonical models. It turns out that one can construct the analogous structures for � ϕ S ( ϕ ); there is a G ( Z ( p ) ) + -action and a notion of connected components. This bijection ∼ � S K p ( G, X ) − → S ( ϕ ) ϕ can be made compatible with G ( Z ( p ) ) + -actions.
The integral model S K p ( G 2 , X 2 ) is constructed from S K p ( G, X ) using this action; the geometrically connected components of the former are quo- tients of those of the latter. This is analogous to Deligne’s construction of canonical models. It turns out that one can construct the analogous structures for � ϕ S ( ϕ ); there is a G ( Z ( p ) ) + -action and a notion of connected components. This bijection ∼ � S K p ( G, X ) − → S ( ϕ ) ϕ can be made compatible with G ( Z ( p ) ) + -actions. To do this it seems essential to work with the morphisms ϕ and not just the triples ( γ 0 , ( γ l ) l � = p , δ ) .
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