Introduction
Motivation and history
Overview of results
Overview of strategy
Structure of the thesis
Preliminaries
Shimura varieties of PEL type
It follows from the definition that V1 and V2 are BC-submodules of VC. The field defining the isomorphism class of the complex representation V1 of B, denoted by E, is called the reflexive field of D. In this case, the algebraic group Gas associated with the data PEL is the core of the composition.
Newton stratification and foliations
An element[b]∈B(G) is said to be µ-admissible ifνG µandκG([b]) =µ\. The setB(G, µ) of µ-admissible elements ofB(G) inherits the partial order, since the Newton map is injective. A Barsotti-Tate groupX0overkis geometrically isomorphic toX(called X0 ~=g X)ifX0andX becomes isomorphic over an extension of The central leaf associated with toX is defined as.
Hecke symmetries and Hecke orbits
If every slope of the Newton polygon attached to N has the same multiplicity, then N contains a B-hypersymmetric point. We prove the discrete part of the Hecke Orbit conjecture under the assumption that the Newton layer in question is irreducible and contains a B-hypersymmetric point. In this section we prove the discrete part of the Hecke orbit conjecture for B-hypersymmetric points.
Write Hp0(x) for the prime-to-pHecke exinS orbit and C00 for the irreducible component of the central leaf inS0 passing through x. Write Hp0(x) for the exinS orbit from prime to pHecke and C00 for the irreducible component of the central sheet at the x transition. If every slope of the Newton polygon associated with N has the same multiplicity, then the Hecke orbit conjecture holds for every irreducible component of N containing aB-hypersymmetric points.
Hypersymmetric Abelian Varieties
New formulation of the characterization of B -hypersymmetric abelian
In modular Siegel varieties, a Newton layer contains a hypersymmetric point if and only if its Newton polygon is symmetric [CO06, Corollary 4.8]. We note that for the purpose of our paper, the condition that the Newton polygon of a non-empty Newton layer admits a "CM-type partition" (see [Zon08, Definition 4.1.8]) is redundant due to non-emptiness. In the above notation, we use λv, to denote the distinct slopes of a place of F over p, wherev,i is the set of λv,i.
Newton's polygonν, derived from aB-linear polarized abelian variety, is B-balanced if there exist positive integers n, m such that nv = n for all primes from F above and mv, i = m for all primes. In other words, for every B-symmetric Newtonian polygon µ there exists a positive integer and a multisetS such that for allv|pofF,nv =nand{mv,i}ni=1v =Sas is a multiset for allv|pofF. A Newtonian stratumN contains a simple B-hypersymmetric point if and only if its Newtonian polygon is B-balanced.
Hypersymmetricity over a subfield
Let b ∈ B(G, µ) and let ζ be the Newton polygon corresponding tob. Writeζ =⊕u|pζu for admixture|pofF. We say that ζ satisfies the condition (*) if, for any u6=u0ofF lying over the same root of F0zgorajp, the Newtonian polygons ζu and ζu0 have no common slope. Let L/K be a finite extension of number fields such that every prime K over p is inert in L/K. Let A be an L-hypersymmetric abelian variety defined over Fp. Suppose that the Newtonian polygon from A satisfies the condition.
Let π denote the Fpa-Frobenius endomorphism of A0 and let ζ denote the Newtonian polygon of A. Then EndL(A) ⊗ ZQ = EndL (A0) ⊗ ZQ, whose center is identified by L (π). Since A is L-hypersymmetric, there exists a positive integer such that it has n slopes at each p|wofL. By [Zon08, proposal 3.3.1],. Kv with the number of factors equal to tou. Since this is true for any u|pofK, we conclude by [Zon08, Proposition 3.3.1] that A is K-hypersymmetric. In the case of F/F0, where F0 is perfectly real and F is a square imaginary extension of F0, if the conditions in Proposition 3.2.2 are not met, an F-balanced Newtonian polygon may not be F0-balanced.
Hypersymmetric points in the µ -ordinary stratum in PEL type A
An important ingredient in [Cha05b] to prove the discrete part of the Hecke circuit conjecture is a monodromy result for Hecke invariant submanifolds ([Cha05c, Proposition 4.1, Proposition 4.4]; cf. [Cha05b, Theorem 5.1]). Proposition 4.0.4 is a more general version of the main theorem in [Kas12], which holds only if no prime factor of D is of type D. Let Z0 be a connected component of Z with generic pointz. We use z to denote a geometric point on Z above z. Let AZ0 denote the restriction to Z0 of the universal abelian scheme over SKp.
Denote by H the neutral component of the Zariski closure M in G. Denote by m the Lie algebra M as an anl-adic Lie group. 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 conclusion H/x0 ⊆C/x0 actually coincides with C/x0. The pro-pro-Hecke orbit closure of xinS contains the supersingular point z = [(Az, λz, ιz, ηz)]. Furthermore, there exists a product of completely real fields L=Q.
First, the same argument as in the last paragraph of the proof of Theorem 4.0.4 shows Hp(x) contains a basic point z = [(Az, λz, ιz, ηz)]. Since B = F is a totally real number field, z is supersingular. Let E be as given at the beginning of this section, and let ME denote the Hilbert modular manifold attached to E. In general, E ∩EndFp(Ax) is an order of the ring OE. However, we can assume an isogenic correspondenceOE ⊆EndFp(Ax). Based on the discussions in Section 3, we prove the following consequences of the main theorem.
The condition on the Galois group of the normal hull is also a weak approximation property (see [VW13]).
Monodromy
The discrete part
The continuous part
The cascade structure on central leaves
It proves that the local deformation space at a generic point on a PEL Shimura-type variety is constructed from Barsotti-Tate groups via a system of fibrations over the Witt ring k. For points outside the generic Newtonian stratum, Chai develops an analogous theory in the case where we restrict ourselves to the central sheet (see [Cha06, Sections 4.3, 4.4]). Let A→C be a restriction on Co of a universal abelian variety and let X be its Barsotti-Tate group with the action OB⊗ZZp.
The central leaf C is homogeneous in the sense that formal completions of two points Cat are non-canonically isomorphic. It is therefore sufficient to study what happens at the point x0 ∈ C. The formal completion C/x0 is contained in the deformation space of the above-mentioned slope filtration, which admits an ar-cascade structure in the sense of [Moo04, Definition 2.2. 1]. If Gi is a Barsotti-Tate group, write M(G) for the Cartier module of G. For a polarization λ, we write G(λ) in the sense of G λ when the induced action of λ on Gi is non-trivial is and vice versa.
Proof of the continuous part at B -hypersymmetric points
We see that Theorem 5.0.1 implies that for anyx∈ N0 C(x)∩N0 is irreducible and thus coincides with the irreducible component of C(x) containing x. Moreover, Theorem 5.0.1 and Theorem 6.0.1 taken together imply that the Hecke orbit conjecture holds for any B hypersymmetric point. Moreover, Hp(x)∩C0 contains the image passing through the smallest Hecke-invariant submanifold of ML, which according to the Hecke orbit conjecture for Hilbert modular variants [Cha05b, Theorem 4.5] is precisely the central leaf CL(y) .
Furthermore, by [Ach14, Theorem 1.1], every Newtonian stratum is irreducible, which then implies that every central leaf is irreducible. By the same argument as the proof of Theorem 7.0.1, the irreducibility of central leaves combined with Theorem 6.0.1 gives the desired result. This appendix reproduces Zong's proof of his main theorem [Zon08, Theorem 5.1], whose Theorem 3.1.2 is reformulated in our simpler terminology.
Proof of the Main Theorem
We use the Hilbert trick to embed a Hilbert modular variety where each central leaf of the Newton layers corresponding to N contains B-hypersymmetric point. Let u1,· · · , unbe the sites of F overp. Assume that for eachi there is a finitely separable extensionKi/Fui of degreed. Then there exists a completely real field extension LofF of degreedsuch that all theui's are inert inL/F andLwi ∼=KioverFui for wiofLaboveui. Now we regroup the local data{Ej,iu /Fu}j,i,u to construct a completely real expansion of F. Note that for a fixedj the numerical conditions onζj imply that for any fixed i,.
By construction, we then have OE ⊗Z Zp ∼= OL ⊗Z Zp as the maximum order of EndF(Az)⊗Z Zp, so the Noether-Skolem theorem implies that E = γLγ−1 for some γ in the local stabilizer group ofz . Thenα:=Ad(γ) satisfies the property in the statement of this lemma. Then there exists a finite morphism ME → S passing through x, compatible with the prim-to-pHecke correspondence in ME and S and such that for each geometric point of ME, the map induced by strict henselizations is closed. Similarly, there exists a finite natural morphism g: ML → S passing through z such that g is compatible with Hecke correspondences on both sides and at every geometric point of ML induces a closed inclusion in strict Henzelizations .
Proof of the general case
Thus, if we can show that CL(y) contains a point that is isogenic to a product of Li hypersymmetric points, Theorem 3.2.2 implies that CL(y) contains an F hypersymmetric point, and so does Hp(x )∩C0. To see that CL(y) contains a point that is isogenic to a product of Li hypersymmetric points, we consider the canonical decomposition ML∼=Q. Due to the existence of finite similarities between central leaves on NLi (see [YCO, Section 1.3]), each central leaf of NLi contains a Li hypersymmetric point.
Let D0 be the Shimura data given by replacing B with its largest fully real subfield F0 in the definition of D. Let S0 denote the Shimura variety derived from D0. Let D0 be the Shimura data given by replacing B with its center F in the definition of D. Thenf induces a morphism : End(A) → End(A0) such that f◦ι|F =ι0|F◦f on F .Po The Skolem-Noether theorem extends f◦ι|F =ι0|F◦f to the inner automorphism ϕ:B →B. Hence f expands to aB-isogeny between x and x0.
Special cases of the main theorem
Each central leaf thus coincides with the Newton polygon it contains and therefore admits an L-hypersymmetric point. Zon08, Definition 4.3.1] LetζB denote the Newton polygon with slope1/2, such that at any place vofF above the multiplicity of1/2 is equal to the order of the class[Qp,∞⊗F]−[B]in the Brauer -groupBr( F), where Qp, ∞ is the quaternion Q-algebra, branched exactly to infinity. Let us be a simple B-linearly polarized abelian manifold overFp. If AisB is hypersymmetric, then the Newton polygonζ orζBorB is balanced.
In particular, if π is completely real, A0 is a power of a supersingular elliptic curve with Newton polygonζB, and the order of[D]inBr(F) is[Qp,∞⊗F]−[B]. Let be a B-balanced Newton polygon, such that the corresponding Newton layer N is not empty. Then, similar to the previous case, there is a hypersymmetric B-simple abelian manifold A whose Newton polygon is ζ.
Appendix: Hypersymmetric Abelian Varieties
