Proof of the continuous part at B -hypersymmetric points

Chapter VI: The continuous part

6.2 Proof of the continuous part at B -hypersymmetric points

Notations as in Theorem 6.0.1. The key idea in proving Theorem 6.0.1 is to study the action of the local stabilizer group at x₀ and show that the formal completion H^/x0 ⊆C^/x0 in fact coincides withC^/x0.

Definition 6.2.1. [Cha06, Section 6.1] Let x = [(A_x, λ_x, ι_x, η_x)] ∈ S(k) be a geometric point. Let U_x be the unitary group attached to the semisimple algebra with involution(EndO_B(A_x)⊗_ZQ,∗),where∗is the Rosati involution attached to λ_x.We callU_x(Zp)thelocal stabilizer group atx.

By the assumption of Theorem 6.0.1,x₀isB-hypersymmetric, soU_x0(Zp)coincides withI_B(Zp)for the groupI_Bdefined in Lemma 5.2. By deformation theory, there is a natural action ofU_x0(Zp)on the formal completionS^/x0 and hence on its closed formal subschemesDef(1, r)^(λ)_pdiv andH^/x0.

Recall that the maps in the cascade structure ofMDE(X) are given by [Moo04, Proposition 2.1.9] (see also [Moo04, 2.3.6]). These maps give rise to a group action

ofU_x0(Z^p)on eachDef(i, j)_pdivin an inductive way. We describe this action in the next paragraph.

Suppose that for the triple Def(i, j −1)_pdiv ←−^p Def(i, j)_pdiv −→^r Def(i+ 1, j)_pdiv we have an action of U_x0(Zp) on Def(i, j)_pdiv. Define an action of U_x0(Zp) on Def(i, j−1)_pdiv byu·X:=p(u·X)˜ for anyu∈ U(Zp)andX∈Def(i, j −1)_pdiv, where X˜ is any pre-image of X in Def(i, j)_pdiv under p. For any other choice of pre-imageX˜⁰,by the biextension structure onDef(i, j)_pdiv,X˜ andX˜⁰are in the same DE(i+ 1, j −1)_pdiv-orbit in Def(i, j)_pdiv. Hence,u·X˜ and u·X˜⁰ are also in the sameDE(i+ 1, j −1)_pdiv-orbit, which implies that their images underpcoincide.

Therefore this action of U_x0(Zp)on Def(i, j−1)_pdiv is well-defined. One defines the action onDef(i+ 1, j)_pdiv in the obvious analogous way.

In the following, we study howH^/x0 behaves with respect to the cascade structure ofC^/x0 ∼=MDE(X)^λ_pdiv.

For1≤i≤r−1,letHi,i+1denote the image ofH^/x0 insideDE(i, i+ 1)^(λ)_pdivunder the maps in the cascade structure ofC^/x0.

For1≤ i < j−1 ≤r, we use the biextension structure ofDef(i, j)^(λ)_pdiv and define H_i,j as the quotient Def(i, j)^(λ)_pdiv/(H_i,i+1×Hj−1,j). By [Mum69, Section2], there is a non-canonical injectionDE(i, j)^(λ)_pdiv×_Spec(k)Def(i+ 1, j−1),→H_i,j.

Lemma 6.2.2. For 1 ≤ i ≤ r −1, H_i,i+1 is a formal Barsotti-Tate subgroup of DE(i, i+ 1)^(λ)_pdiv.

Proof. By definition, H is stable under all prime-to-p Hecke correspondences.

Then [Cha06, Proposition 6.1] impliesH^/x0 is stable under the action ofU_x0(Zp);

hence, so are theH_i,i+1’s. As explained in Remark 6.2, H is smooth, so H^/x0 is formally smooth and hence irreducible. The action ofU_x0(Zp)onDE(i, i+ 1)^(λ)_pdiv gives a homomorphism from U_x0(Zp) ⊗_Zp Qp to the unitary group attached to End(DE(i, i + 1)_pdiv)^(λ) ⊗_Zp Qp. Applying [Cha08, Theorem 4.3] finishes the proof.

Lemma 6.2.3. For1≤i≤r−1, H_i,i+1 =DE(i, i+ 1)^(λ)_pdiv.

Proof. The natural action ofU_x0(Zp)onH_i,i+1restricts to an action onM(Y_i)⊗_WL. Since x₀ is B-hypersymmetric, we have U_x0(Zp) = I_B(Zp). ThusU_x0(Zp)acting onM(Y_i)⊗_W Lis isomorphic to the standard representationStdofGL_dimYi.

If the polarization λ := λ_x0 is nontrivial on DE(i, i+ 1)_pdiv, we obtain a dual- ity pairing between M(Y_i) and M(Y_i+1). In this case, the action of U_x0(Z^p) on M(Y_i+1)⊗_W L is isomorphic to the dual ofStd. Thus, the action of U_x0(Zp) on Hom^λ_W(M(Y_i), M(Y_i+1))⊗_W Lis isomorphic to Sym²Std, which is irreducible.

By Proposition 6.1.2(2), the action of U_x0(Zp) on M(DE(i, i+ 1)^λ_pdiv) ⊗_W L is isomorphic to an irreducible representation.

On the other hand, if the polarization is trivial on DE(i, i+ 1)_pdiv, the action of Ux0(Z^p)onHomW(M(Yi), M(Yi+1))⊗W Lis the tensor product representation of the standard representations ofGL_dimYi and GL_dimYi+1, which is irreducible. By Proposition 6.1.2(1), the action of U_x0(Zp)onM(DE(i, i+ 1)_pdiv)⊗_W Lis again isomorphic to an irreducible representation.

By Lemma 6.2.2, it makes sense to consider the Cartier moduleM(H_i,i+1). We have M(H_i,i+1)⊗_WLas a non-trivial sub-representation ofM(DE(i, i+ 1)_pdiv)^(λ)⊗_WL which is irreducible, so we obtain the desired equalities.

Proof of Theorem 6.0.1. We show inductively that Lemma 6.2.3 implies H^x0 = C^/x0.

Whenr = 2, C^/x0 =DE(1,2)^λ_pdiv and there is nothing to prove.

Whenr = 3, C^/x0 = Def(1,3)^λ_pdiv is a biextension of (DE(1,2)_pdiv,DE(2,3)_pdiv) byDE(1,3)^λ_pdiv.The equalities in Lemma 6.5 induces an isomorphism realized via H_1,3 =Def(1,3)^λ_pdiv/(H_1,2×H_2,3) = Def(1,3)^λ_pdiv/(DE(1,2)_pdiv×DE(2,3)_pdiv)∼= DE(1,3)^λ_pdiv. Thus, the only candidate for the subscheme H^/x0 of Def(1,3)^λ_pdiv is Def(1,3)^λ_pdiv itself.

When r ≥ 3, let H^i,j denote the image of H^/x0 in Def(i, j)^λ_pdiv.By induction we have equalities H^i,j = Def(i, j)^(λ)_pdiv for all (i, j) 6= (1, r). In particular, H^1,r−1 = Def(1, r−1)_pdiv andH^2,r =Def(2, r)_pdiv together imply

H_1,r =Def(1, r)^λ_pdiv/(DE(1, r−1)_pdiv×DE(2, r)_pdiv), so we obtainH^/x0 =Def(1, r)^λ_pdiv =C^/x0.

C h a p t e r 7

PROOF OF THE MAIN THEOREM

The main theorem of our paper is the following.

Theorem 7.0.1. LetD= (B,O_B,∗, V,h·,·i, h)be a Shimura datum of PEL type A or C, for whichpis an unramified prime of good reduction. LetF be the center of B andF₀ its maximal totally real subfield. LetS denote the reduction modulo p of the Shimura variety associated toD of levelK^p ⊆ G(A^pf). LetN be a Newton stratum onS. Assume

1. N contains a hypersymmetric pointx₀,

2. either (i)pis totally split inF/F₀ and the Newton polygon ofN satisfies the condition (*) in Definition 3.2.1; or (ii)pis inert inF/F₀.

WriteN⁰ for the irreducible component ofN containingx₀. ThenH^p(x)is dense inC(x)∩ N⁰ for every x ∈ N⁰(k). Moreover, ifN is not the basic stratum, then C(x)∩ N⁰is irreducible.

We observe that Theorem 5.0.1 implies that for anyx∈ N⁰,C(x)∩N⁰is irreducible and hence coincides with the irreducible component of C(x) containing x. We denote this component by C⁰(x). Moreover, Theorem 5.0.1 and Theorem 6.0.1 together imply that the Hecke orbit conjecture holds for any B-hypersymmetric point. Then the key in deriving Theorem 7.0.1 from Theorem 5.0.1 and Theorem 6.0.1 lies in showing the existence of aB-hypersymmetric point inH^p(x)∩C⁰(x) for everyx∈ N⁰.

7.1 The case ofB =F.

We consider first the situation whereB =F is a totally real field. In this case, our PEL datum is of type C. The algebraic groupGis symplectic, and the Hilbert trick still applies. We use the Hilbert trick to embed a Hilbert modular variety where every central leaf in the Newton strata corresponding to N containsB-hypersymmetric point.

For the remainder of this section, we fix a point x = [(A_x, λ_x, ι_x, η_x)] ∈ N(Fp).

There is a decompositionAx ∼F-isogA^e₁¹× · · · ×A^e_nⁿintoF-simple abelian varieties

A_i such that A_i and A_j are not F-isogenous whenever i 6= j. For each i, let E_i ⊆ End_F(A_i)⊗_Z Q be the maximal totally real subalgebra fixed under the involution given by the polarization of A_i induced by λ_x, then E_i/F is a totally real field extension of degreedim(A_i)/[F : Q].DefineE :=Qn

i=1E_i,thenE is a subalgebra ofEnd_F(A_x)⊗_ZQof dimensiondim(A).

This construction relies on the assumption thatxis defined overFp,so the method in this section only applies whenk =Fp.However, by [Poo17, Theorem 2.2.3], to prove the Hecke orbit conjecture over any algebraically closed fieldk of characteristicp, it suffices to prove it overFp.

Lemma 7.1.1. Let F be a totally real number field and d a positive integer. Let u₁,· · · , u_nbe the places ofF abovep.Suppose that for eachi, there is a finite sep- arable extensionK_i/F_ui of degreed.Then there exists a totally real field extension LofF of degreedsuch that all theu_i’s are inert inL/F andL_wi ∼=K_ioverF_ui for w_iofLaboveu_i.

Proof. For a place u_i of F above p, we have K_i ∼= F_ui(α_i), where α_i is a root of some irreducible separable monic polynomial f_i⁰ ∈ F_ui[X] of degree d. By Krasner’s lemma, we can approximate f_i⁰ by some irreducible separable monic polynomialf_i ∈F[X]such thatF_ui[X]/(f_i)∼=F_ui[X]/(f_i⁰)∼=K_i.Letv₁,· · · , v_m denote the archimedean places ofF and letg_i =Qd

j=1(X−β_ij)for distinctβ_ij ∈F, soF_vi[X]/(g_i)∼=R[X]/(g_i)∼=R^dsinceF is totally real. By weak approximation, there exists some monicf ∈F[X]that isui-close tofifor eachiandvi-close togi

for eachi.In particular,f = f_i for somei,sof is irreducible inF_ui[X]and hence irreducible inF.

LetL =F[X]/(f),thenL/F is separable of degreed. Moreover, for eachui, we have

Y

w|ui

L_w ∼=F_ui⊗L∼=F_ui[X]/(f)∼=K_i.

This implies that there is a unique placewiofEaboveui andLwi ∼=Ki.Similarly, for each archimedean placevi, we haveQ

w|v_iLw ∼=Fvi⊗L∼=R^d.Thus,Lis totally real.

Lemma 7.1.2. The closure of the prime-to-pHecke orbit ofxinS contains a super- singular pointz = [(A_z, λ_z, ι_z, η_z)].Moreover, there exists a product of totally real fieldsL=Q

i,jL_i,jand an injective ring homomorphismα :L→End_F(A_z)⊗_ZQ such that

• for everyi, j,F ⊆L_i,j andL_i,j/F is inert at every prime ofF abovep;

• dim_QL= dimA_z;and

• the Rosati involution induced byλ_zacts trivially on the imageα(L).

Proof. First of all, the same argument as in the last paragraph of the proof of Proposition 4.0.4 shows H^p(x)contains a basic pointz = [(A_z, λ_z, ι_z, η_z)].Since B = F is a totally real number field, z is supersingular. Indeed, letd = [F : Q], thenF =Q(x)wherexis the root of some degreedmonic irreducible polynomial f ∈ Q.Then endomorphism ring of a d-dimensional supersingular abelian variety is isomorphic to Matd(D_p,∞),where D_p,∞ denotes the quaternion algebra over Q ramified exactly atpand∞,and Matd(D_p,∞)clearly contains the companion matrix off.

By assumption,N admits anF-hypersymmetric point, so its Newton polygonζ is F-symmetric, i.e. ζ is the amalgamation of disjoint F-balanced Newton polygons ζ₁,· · · , ζ_afor somea.By [Zon08, Definition 4.4.1], for anyj, ζ_j either has no slope 1/2, or only has slope1/2.Hence, there exist positive integersm_j, h_j such that at every primeuofF abovep, ζ_j has exactlyh_j symmetric components with at most 2slopes, and each component has multiplicitym_j.

Write X = A_x[p^∞], then the above decomposition of ζ gives a decomposition X=La

j=1Xj,whereXj is the Barsotti-Tate group corresponding toζ_j.Further, for each fixedj,the numerical properties ofζj mentioned above gives a decomposition

X^j =M

u|p hj

M

i=1

X^uj,i

into Barsotti-Tate groups of at most2slopes.

Hence,

End_F(X)⊗_ZpQp ∼=Y

u|p a

Y

j=1 hj

Y

i=1

End_Fu(X^uj,i).

Its subalgebraE⊗_QQp similarly admits a decomposition E⊗_QQ^p =Y

u|p a

Y

j=1 hj

Y

i=1

E_j,i^u

into local fields. Note that this has to coincide with the decompositionE ⊗Qp ∼= Qn

i=1Ei⊗Q^p =Qn i=1

Q

w|pEw.

Now we regroup the local data{E_j,i^u /F_u}_j,i,u to construct totally real extension of F.Notice that for a fixedj,the numerical conditions onζ_j implies that for any fixed i,

• #{E_j,i^u}_u|p =h_j,and

• [E_j,i^u :F_u] = [E_j,i^u0 :F_u⁰] = dim(X^uj,i)/[F :Q]for any primesu, u⁰ofF above p.

By Lemma 7.1.1, for a fixed pair(j, i), the data{E_j,i^u}u|p gives rise to a totally real field extensionL_i,j/F with[L_i,j :Q] = dim(Xj,i)such that all primes ofF above pare inert and{(Li,j)w}w|p ={E_i,j^u }u|p as multi-sets.

TakingL=Q

i,jL_i,j,it is easy to see thatdim_QL⁰ = dimA_z.

In general, O_E ∩ End_F(A_z) ⊆ O_E is of finite index. Following the argument in [Cha05b, Theorem 11.3], up to an isogeny correspondence, we may assume End_F(A_z) contains O_E. Similarly, we may and do assume O_L ⊆ End_F(A_z).

By construction, we then have O_E ⊗_Z Zp ∼= O_L ⊗_Z Zp as maximal orders of EndF(Az)⊗_Z Z^p, so the Noether-Skolem theorem implies that E = γLγ⁻¹ for someγ in the local stabilizer gorup ofz.Thenα:=Ad(γ)satisfies the property in the statement of this lemma.

Proposition 7.1.3. Theorem 7.0.1 holds whenB =F is a totally real field.

Proof. If x is (isogenous to) a F-hypersymmetric point, the desired statement follows immediately from Theorems 5.0.1 and 6.0.1, so we assume x is not F- hypersymmetric.

We use an idea analogous to that of [Cha05b, Section 10] and show thatH^p(x)∩C⁰ contains anF-hypersymmetric pointt. Then we haveH^p(t)∩C⁰ ⊆ H^p(x)∩C⁰, butH^p(t)∩C⁰ =C⁰sincetisF-hypersymmetric, and we are done.

LetE be as given in the beginning of this section, and let M_E denote the Hilbert modular variety attached to E.In general, E ∩End_Fp(A_x) is an order of the ring OE.However, up to an isogeny correspondence, we may assumeOE ⊆End_Fp(Ax).

Then there exists a finite morphism M_E → S passing through x, compatible with the prime-to-pHecke correspondence on M_E andS and such that for each geometric point ofM_E,the map induced byfon the strict henselizations is a closed

embedding (for details, see [Cha05b, Proposition 9.2] and the proof of [Cha05b, Theorem 11.3]).

Let L = Q

iL_i and γ be as given by Lemma 7.1.2. Again, up to an isogeny correspondence, we may assume O_L ⊆ End

F^p(A_z). Similarly, there is a finite natural morphism g : M_L → S passing through z, such that g is compatible with the Hecke correspondences on either side and at every geometric point ofM_L induces a closed embedding on the strict henzelizations. Moreover, writingH_E^p(x) for the prime-to-p Hecke orbit of x in ME, we have γ^/z(H_E^p(x)) ⊆ g^/z(M^/z_L) andγ^/z(H_E^p(x)) ⊆ γ^/z(H^p(x)^/z).By [Cha06, Proposition 6.1], H^p(x)^/z is stable under the action of the local stabilizer group of z, so γ^/z(H^p(x)^/z) = H^p(x)^/z and we obtain γ^/z(H_E^p(x)^/z) ⊆ g^/z(M^/z_L)∩H^p(x)^/z. Therefore the fiber product M_L×_SF (H^p(x)∩C⁰(x))is nonempty.

Now let y˜be an Fp-point of M_L ×_SF (H^p(x)∩C⁰) with image y ∈ M_L(Fp) and image y ∈ (H^p(x)∩C⁰)(Fp). By definition, we have g(y) = y. Moreover, H^p(x)∩C⁰ contains the image undergof the smallest Hecke-invariant subvariety ofMLpassing throughy,which by the Hecke orbit conjecture for Hilbert modular varieties [Cha05b, Theorem 4.5] is precisely the central leaf C_L(y). Therefore, if we can showC_L(y)contains a point isogenous to a product of L_i-hypersymmetric points, then Proposition 3.2.2 implies that C_L(y) contains a F-hypersymmetric point, and so doesH^p(x)∩C⁰.

To see that C_L(y) contains a point isogenous to a product of L_i-hypersymmetric points, consider the canonical decompositionM_L∼=Q

M_Liand the corresponding Newton stratum decompositionNL ∼=Q

NLi. By the construction ofLi,the Newton polygon ofN_Lieither has exactly two slopes at every prime ofL_iabovep,or only has slope1/2.In either case,N_Liadmits anL_i-hypersymmetric point. By the existence of finite correspondences between central leaves onN_Li (see [YCO, Section 1.3]), each central leaf ofN_Li contains a L_i-hypersymmetric point. This completes our proof.

7.2 Proof of the general case.

In this section, we complete the proof of Theorem 7.0.1.

Proof of Theorem 7.0.1. Again, we only need to prove the statement whenxis not B-hypersymmetric.

Case 1. WhenBis totally real, see Proposition 7.1.3.

Case 2. SupposeB =F is a CM field.

Let D⁰ be the Shimura datum given by replacing B by its maximal totally real subfieldF₀ in the definition ofD.LetS⁰ denote the Shimura variety arising from D⁰. Then there is an embedding S → S⁰ given by sending any [(A, λ, ι, η)]

to [(A, λ, ι|_F0, η)]. Let x = [(A, λ, ι, η)] ∈ N⁰ be any point, and we denote its image inS⁰ also byx. By assumption,N⁰contains aF-hypersymmetric pointx₀. Proposition 3.2.2 impliesx₀is alsoF₀-hypersymmetric. WriteH^p0(x)for the prime- to-pHecke orbit ofxinS andC⁰⁰for the irreducible component of the central leaf inS⁰ passing throughx. Then by Proposition 7.1.3,H^p0(x)∩C⁰⁰ =C⁰⁰.

Now we show thatH^p0(x)∩C⁰ =H^p(x).Letx⁰ = [(A⁰, λ⁰, ι⁰, η⁰)]∈H^p0(x),then there is a prime-to-p F₀-isogenyfbetweenxandx⁰.By definition,f◦ι|_F0 =ι⁰|_F0◦f. SinceF/F₀ is a quadratic imaginary extension,f extends to anF-isogeny between xandx⁰.

Finally, we haveH^p(x)∩C⁰ =H^p0(x)∩C⁰∩C⁰ =H^p0(x)∩C⁰ =H^p0(x)∩C⁰⁰∩ C⁰ =C⁰⁰∩C⁰ =C⁰.

Case 3. Now we are ready to show the statement for a generalB.

Let D⁰ be the Shimura datum given by replacing B by its center F in the defini- tion of D. Let S⁰ denote the Shimura variety arising from D⁰. Then there is an embeddingS →S⁰ given by sending[(A, λ, ι, η)]to[(A, λ, ι|_F, η)].Letx∈ N⁰ be any point, and we denote its image in S⁰ also by x. By assumption, N⁰ contains a B-hypersymmetric pointx₀. By [Zon08, Proposition 3.3.1], x₀ is also F-hypersymmetric. WriteH^p0(x)for the prime-to-pHecke orbit ofxinS andC⁰⁰ for the irreducible component of the central leaf inS⁰passing throughx. Then by the previous two cases,H^p0(x)∩C⁰⁰ =C⁰⁰.

Now we show that H^p0(x)∩C⁰ = H^p(x). Let x⁰ = [(A⁰, λ⁰, ι⁰, η⁰)] be a closed geometric point of S such that there is an prime-to-p F₀-isogeny f from x = [(A, λ, ι, η)]tox⁰. Thenf induces a morphismf : End(A) → End(A⁰)such that f◦ι|F =ι⁰|F◦f onF.By the Skolem-Noether Theorem,f◦ι|F =ι⁰|F◦f extends to an inner automorphismϕ:B →B.Hence,f extends to aB-isogeny betweenx andx⁰.

By an argument analogous to the last part of Case 2, we conclude H^p(x)∩C⁰ = C⁰.

7.3 Special cases of the main theorem.

Based on the discussions in Section 3, we prove the following corollaries of the main theorem.

Corollary 7.3.1. 1. Supposepis inert inF. If every slope of the Newton polygon attached toN has the same multiplicity, then the Hecke orbit conjecture holds for any irreducible component ofN containing aB-hypersymmetric point.

2. Suppose the center ofBis a CM field. Assume that the signature ofS has no definite place, and thatpis a prime of constant degree in the extensionF/Q.

Further assume assumption 2 in Theorem 7.0.1 is satisfied. Then the Hecke orbit conjecture holds for every irreducible component of the µ-ordinary stratum.

Proof. By Corollary 3.3.3, in either of the two cases, the Newton stratumNcontains aB-hypersymmetric point. In the second case,G(A^pf)acts transitively onΠ₀(S), so if the µ-ordinary stratum contains a B-hypersymmetric point, then it contains a B-hypersymmetric point in each of its irreducible components. Moreover, the assumptions satisfy the conditions of Proposition 3.2.2. Hence, we may apply Theorem 7.0.1 and to derive the desired results.

Corollary 7.3.2. Let Lbe a quadratic imaginary field inert at the rational prime p. The Hecke Orbit conjecture holds for the moduli space of principally polarized abelian varieties of dimensionn ≥ 3equipped with an action byO_Lof signature (1, n−1).

Proof. Sincepis inert, a Newton stratum contains aL-hypersymmetric point if its Newton polygon is symmetric. By [BW06, Seciton 3.1], any admissible Newton polygon in this case is given byN(r) + (1/2)^n−2r for some integer0≤ r ≤n/2, where

N(r) =











∅, ifr= 0,

(¹₂ −_2r¹) + (¹₂ +_2r¹), ifr >0is even, (¹₂ −_2r¹)²+ (¹₂ +_2r¹)², ifris odd.

From this description, it follows by easy computation that for each pair(n, r),0≤ r ≤ n/2, the admissible Newton polygon uniquely determined by(n, r)is always L-hypersymmetric. Indeed, whenn = 3randr is even, or whenn = 4randr is odd, the Newton polygon consists of3slopes of the same multiplicity and is hence

L-balanced. Otherwise, the Newton polygon is an amalgamation of one polygon of slope1/2and one polygon of slopes ¹₂ − _2r¹,¹₂ + _2r¹. Clearly, each of these two polygons isL-balanced.

It is well-known that in this case, each isogeny class of Barsotti-Tate groups consists of a unique isomorphism class. Thus, every central leaf coincides with the Newton polygon containing it and therefore admits a L-hypersymmetric point. Moreover, by [Ach14, Theorem 1.1], every Newton stratum is irreducible, which then implies that every central leaf is irreducible. Following the same argument as the proof of Theorem 7.0.1, the irreducibility of central leaves combined with Theorem 6.0.1 yields the desired result.

C h a p t e r 8

APPENDIX: HYPERSYMMETRIC ABELIAN VARIETIES

This appendix reproduces Zong’s proof of his main theorem [Zon08, Theorem 5.1], of which Theorem 3.1.2 is a rephrase in our simpler terminology. As mentioned in Section 3.3, Theorem 3.1.2 follows from the proof but not the statement of [Zon08, Theorem 5.1]. For completeness, we reproduce his proof to make the present paper self-contained. All the proofs in this appendix are due to Zong. Any mistake is my own.

We remark that some of the conditions of supersingular restriction (see [Zon08, Section 4]) come from the presence of a B-action (see in particular “CM-type partitions” in [Zon08, Proposition 4.9]), or, in other words, from the requirement that the Newton strata under consideration is non-empty. Since we are only interested in non-empty Newton strata, we may omit those conditions to get a simpler formulation.

The proofs in this appendix are essentially the same as those given in [Zon08, §6 and §7], except that the proof of Proposition A.4 is simpler than that of [Zon08, Proposition 7.7] by omitting the discussion of CM type partitions. Theorem 3.2 follows from Proposition A.2 and A.3.

Theorem 3.4. A (non-empty) Newton stratum N on S contains a simple B- hypersymmetric point if and only its Newton polygon isB-balanced.

Notation 8.0.1. [Zon08, Definition 4.3.1] LetζB denote the Newton polygon with slope1/2such that at every placevofF abovep,the multiplicity of1/2is equal to the order of the class[Q^p,∞⊗F]−[B]in the Brauer groupBr(F),whereQ^p,∞is the quaternionQ-algebra ramified exactly atpand infinity.

The following proposition is a rephrase of [Zon08, Proposition 6.1].

Proposition 8.0.2. LetAbe a simpleB-linear polarized abelian variety overFp.If AisB-hypersymmetric, then its Newton polygonζ is eitherζBorB-balanced.

Proof. Let A⁰ be a simple B-linear polarized abelian variety over a finite field F^q such that A⁰ ⊗ F^p ∼= A. We may assume q is sufficiently divisible. A⁰ is

B-simple and is isogenous to A⁰⁰ for some absolutely simple abelian variety A⁰⁰. Let π denote the Frobenius ofA⁰ and let K = F(π). Write D := End⁰_B(A⁰)and L_v :=F_v⊗K(Fq).WriteM⁰for theB-linear isocrystal associated toA⁰.ThenM⁰ decomposes as⊕v|pM_vand one checks that eachM_vis a freeL_v-module. Letf_v(T) be the characteristic polynomial ofπas anL_v-linear transformation ofM_v.SinceA isB-hypersymmetric, by [Zon08, Proposition 3.4],

f_v(T) = Y

w|v

(T −ι_w(π))^nw,

where ι_w : K → F_v denotes the F-embeddings of K into F_v indexed by the place w. By Kottwitz [Kot92], eachM_v has [K : F] isotypic components. Thus f(T) = det(T −π|_M⁰)can be factored as

f(T) =Y

v|p

N ormLv/K(Fq)fv(T) = Y

v|p

Y

w|v

N ormFv/Qp(T −ιw(π))^nw.

Consider triples u = (v, w, τ)wherev is a place ofF abovep, w is a place ofK abovev,andτ is an embeddingF_v → Q_p.Then the set of such triples is in 1-to-1 correspondence with the set of embeddingsι_u :K →Q_p.Thus

f(T) =Y

u

(T −ιu(π))^nw.

By Katz-Messing [KM74], f(T) ∈ Z[T], so n_w is independent of w. Hence, n_w = 2 dim(A)/[K :Q].By [Kot92, Lemma 3.3], the multiplicity of each isotypic component ofM is equal to

n[L_v :K(Fq)]/([B :F]^1/2[F_v :Qp]) = order([D]).

In particular, ifπis totally real,A⁰ is a power of a supersingular elliptic curve with Newton polygonζ_B, and the order of[D]inBr(F) is equal to[Q^p,∞⊗F]−[B].

Finally, ifF =F₀, then[K :F]is even and eachζ_v is symmetric, soζ contains no slope1/2component.

The following proposition is a rephrase of the “if” part of [Zon08, Theorem 5.1]

(see [Zon08, Section 7]).

Proposition 8.0.3. Letζbe aB-balanced Newton polygon such that the correspond- ing Newton stratumN is non-empty. Then there exists a hypersymmetricB-linear polarized simple abelian varietyAdefined overF^pwhose Newton polygon isζ.