aa r X i v : . [ m a t h . N T ] D ec A µ -ORDINARY HASSE INVARIANT WUSHI GOLDRING, MARC-HUBERT NICOLE
Abstract.
We construct a generalization of the Hasse invariant for certainunitary Shimura varieties of PEL type whose vanishing locus is the comple-ment of the so-called µ -ordinary locus. We show that the µ -ordinary locus ofthose varieties is affine. As an application, we strengthen a special case of atheorem of one of us (W.G.) on the association of Galois representations toautomorphic representations of unitary groups whose archimedean componentis a holomorphic limit of discrete series. Introduction
Starting with the cornerstone work of Deligne-Serre [DS74] on classical weightone modular forms, the Hasse invariant has been a fundamental tool for constructingcongruences between automorphic forms. In turn, the congruences that arise fromthe Hasse invariant have been used to construct automorphic Galois representationsthat do not appear directly in the ´etale cohomology of Shimura varieties ([Tay91],[Gol12]). One limitation of the Hasse invariant is that there exist many Shimuravarieties for which the Hasse invariant is identically zero. This happens preciselyfor those Shimura varieties whose ordinary locus is empty.The µ -ordinary locus introduced by Rapoport and Richartz [RR96] can be viewedas a substitute to the ordinary locus when the latter is empty. Indeed a theoremof Wedhorn states that, for a prime of good reduction and hyperspecial level, the µ -ordinary locus is dense [Wed99, Th.1.6.2]. It is therefore natural to seek a gen-eralization of the Hasse invariant whose vanishing locus is the complement of the µ -ordinary locus. We construct such an invariant for Shimura varieties Sh( G , X )of PEL-type such that G ( R ) is isomorphic to a unitary similitude group GU( a, b )for some positive integers a, b . This class includes Picard modular varieties.1.1. Main Result.
Suppose U = ( B, V, ∗ <, >, ˜ h ) is a Kottwitz datum, with as-sociated Shimura variety Sh( G , X ) (see [Gol12, § Q -algebra B is an imaginary quadratic field F . Let ℓ be a prime of goodreduction for U (see loc. cit. § K ( ℓ ) ⊂ G ( A ℓf ) an open compact subgroup.Let Sh K ( ℓ ) ,ℓ be the Kottwitz integral model of Sh ( G , X ) at level K ( ℓ ) (see loc. cit. W. G. Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000,USAM.-H. N. Institut de math´ematiques de Luminy, Universit´e d’Aix-Marseille, campus deLuminy, case 907, 13288 Marseille cedex 9, France
E-mail address : [email protected], [email protected] . Date : November 8, 2018.2010
Mathematics Subject Classification.
Primary 14G35 ; Secondary 11F33, 11F55.
Key words and phrases.
Hasse invariant, Galois representations, Shimura varieties, µ -ordinarylocus. µ -ORDINARY HASSE INVARIANT § E = E ( G , X ) be the reflex field of Sh( G , X ) and λ a prime of E lyingabove ℓ . Denote by sh K ( ℓ ) ,λ the special fiber of Sh K ( ℓ ) ,ℓ at λ . Theorem 1.1 ( µ -ordinary Hasse invariant) . There exists an automorphic line bun-dle µ ω K ( ℓ ) , and a section µ H ∈ H ( sh K ( ℓ ) ,λ , µ ω ⊗ ( ℓ − K ( ℓ ) ) such that: ( µ -Ha1) The non-vanishing locus of µ H is the µ -ordinary locus of sh K ( ℓ ) ,λ . ( µ -Ha2) There exists an integer m ∈ N such that ( µ H ) m lifts to characteristic zero. ( µ -Ha3) The construction of µ H is compatible with varying the level K ( ℓ ) . Corollary 1.2.
The µ -ordinary locus sh µ − ord , min K ( ℓ ) ,λ in the minimal compactification sh min K ( ℓ ) ,λ is affine. Remark 1.3.
We do not know the minimal value of m in ( µ -Ha2). The Hasseinvariant of Siegel varieties lifts i.e., m = 1, see [BN07].1.2. Application.
Let U be as in § π is a cuspidal automorphicrepresentation of G ( A ) with p -adic component π p for every (rational) prime p .Given a (rational) prime ℓ , let P ( ℓ ) be the set of (rational) primes p different from ℓ such that π p is unramified and G is unramified at p . Let P ( ℓ ) be the set of primesof F that are split and lie over some p ∈ P ( ℓ ) .Assume ℘ ∈ P ( ℓ ) . One has a decomposition G ( Q p ) ∼ = Q × p × GL( n, F ℘ ), where n is given by n = dim F End B V . Write π p ∼ = χ p ⊗ π ℘ , with χ p a character of Q × p and π ℘ a representation of GL( n, F ℘ ).Our result on Galois representations is: Theorem 1.4.
Suppose π is a cuspidal automorphic representation of G ( A ) whosearchimedean component π ∞ is an X -holomorphic limit of discrete series represen-tation of G ( R ) (see [Gol12, § ). Assume ℓ is a prime (of Q ) of good reductionfor U . Then there exists a unique semisimple Galois representation (1) R ℓ,ι ( π ) : Gal( F /F ) −→ GL( n, Q ℓ ) satisfying the following two conditions: (Gal1) If p ∈ P ( ℓ ) and ℘ is a prime of F dividing p then R ℓ,ι ( π ) is unramified at ℘ . In particular R ℓ,ι ( π ) is unramified at all but finitely many places. (Gal2) If ℘ ∈ P ( ℓ ) then there is an isomorphism of Weil-Deligne representations (2) ( R ℓ,ι ( π ) | W F℘ ) ss ∼ = ι − rec( π ℘ ⊗ | · | − n2 ℘ ) , where W F ℘ is the Weil group of F ℘ , the superscript ss denotes semi-simplificationand rec is the Local Langlands Correspondence, normalized as in Harris-Taylor [HT01] . Remark 1.5.
Comparison with [Gol12, Th.1.2.1]. The improvement in Th. 1.4with respect to loc. cit. is the removal of the assumption that some prime λ of E above ℓ is split in E .2. Construction of the µ -ordinary Hasse invariant Assume henceforth, without loss of generality, that a ≤ b . The assumption that ℓ is a prime of good reduction for U implies that ℓ is unramified in E . µ -ORDINARY HASSE INVARIANT 3 If λ is split in E , Th. 1.1 is well-known (see e.g., [Gol12, § a = b then E = Q , so λ is necessarily split in E . Hence we assume from now on that a < b and that λ is inert in E .As in [Gol12, § K ( ℓ ) decomposes as(3) Ω K ( ℓ ) = Ω ⊕ r K ( ℓ ) ,a ⊕ Ω ⊕ r K ( ℓ ) ,b , where Ω K ( ℓ ) ,a (resp. Ω K ( ℓ ) ,b ) has rank a (resp. b ) and r is the rank of B over F .Let ω K ( ℓ ) ,a (resp. ω K ( ℓ ) ,b ) be the determinant of Ω K ( ℓ ) ,a (resp. Ω K ( ℓ ) ,b ).Let A be an abelian scheme representing the universal isogeny class above sh K ( ℓ ) ,λ As in (4.6) of loc. cit. , the Verschiebung Ver : A ( ℓ ) → A induces a map(4) Ver ∗ : Ω K ( ℓ ) → Ω ( ℓ ) K ( ℓ ) . Since λ is inert, the restrictions of Ver ∗ to Ω K ( ℓ ) ,a (resp. Ω K ( ℓ ) ,b ) have the form(5) Ver ∗| Ω K ( ℓ ) ,a : Ω K ( ℓ ) ,a −→ Ω ℓ K ( ℓ ) ,b and Ver ∗| Ω K ( ℓ ) ,b : Ω K ( ℓ ) ,b −→ Ω ℓ K ( ℓ ) ,a . Therefore, if (Ver ∗ ) denotes the composite of Ver ∗ with itself, then we have(6) (Ver ∗ ) | Ω K ( ℓ ) ,a : Ω K ,a −→ Ω ( ℓ ) K ,a . Let(7) µ h ( A ) : ω K ( ℓ ) ,a −→ ω ⊗ ( ℓ ) K ( ℓ ) ,a , be the top exterior power of that map, where we have used that ω ( ℓ ) K ( ℓ ) ,a = ω ⊗ ( ℓ ) K ( ℓ ) ,a since ω K ( ℓ ) ,a is a line bundle. The map µ h ( A ) induces a global section(8) µ H ( A ) ∈ H ( sh K ( ℓ ) ,λ , ω ⊗ ( ℓ − K ( ℓ ) ,a ) . If B is another representative of the universal isogeny class above sh K ( ℓ ) ,λ and ϕ : A → B is an isogeny compatible with the endomorphism actions of A , B , thenas in [Gol12, § loc. cit. ) implies that ϕ ∗ ( µ H ( B )) = µ H ( A ). Hence we may omit reference to therepresentatives A or B and we have a section µ H ∈ H ( sh K ( ℓ ) ,λ , ω ⊗ ( ℓ − K ( ℓ ) ,a ), whichwe call the µ -ordinary Hasse invariant.3. Proofs.
We begin with the proof of Th. 1.1. The following two lemmas and their corol-laries will establish that µ H satisfies ( µ -Ha1). Lemma 3.1.
The Newton polygon N ord of the underlying isogeny class of abelianschemes of a µ -ordinary geometric point of sh K ( ℓ ) ,λ has the following slopes: / with multiplicity ar b − a ) r ar Proof.
The case r = 1 follows from [Wed99, 2.3.2]. The case of general r followssubsequently from [Moo04, 1.3.1 and 3.2.9]. (cid:3) Proposition 3.2.
The µ -ordinary locus is the maximal ℓ -rank stratum of sh K ( ℓ ) ,λ . A µ -ORDINARY HASSE INVARIANT Proof.
The key point is that, by [RR96, Prop. 2.4(iv) and Th.4.2], the Newtonpolygon N ord described in Lemma 3.1 is the lowest among the Newton polygons ofthe underlying isogeny classes of abelian schemes corresponding to geometric pointsof sh K ( ℓ ) ,λ . Let A be an abelian scheme with Newton polygon N ( A ). Then N ( A )is symmetric and the ℓ -rank of A is the multiplicity of 0 (=the multiplicity of 1)as a slope of N ( A ). But if the multiplicity of 0 in N ( A ) is at least the multiplicityof 0 in N ord and N ( A ) lies on or above N ord , then by Lemma 3.1 we must have N ( A ) = N ord . (cid:3) Corollary 3.3.
The maximal ℓ -rank stratum of sh K ( ℓ ) ,λ has ℓ -rank ar .Proof. This follows directly from Lemma 3.1 and the proof of Prop. 3.2. (cid:3)
Lemma 3.4.
Suppose A is an abelian scheme which is a representative of theunderlying isogeny class of a geometric point of sh K ,λ . Then µ H ( A ) = 0 if and onlyif the ℓ -rank of A is equal to ar .Proof. One has H ( A, O A ) ∼ = H ( A, Ω A ) and under this isomorphism the actionof Frobenius on H ( A, O A ) corresponds to that of Verschiebung on H ( A, Ω A ).Hence [Mum08, §
15] implies that the ℓ -rank of A equals the semisimple rank of(Ver ∗ ) j : Ω → Ω ( ℓ j ) for all j ∈ N . Since dim A = ( a + b ) r , keeping in mind (3)and using the last corollary of §
14 of loc. cit. , (Ver ∗ ) j is semisimple for j ≥ a + b .Therefore the ℓ -rank of A equals the rank of (Ver ∗ ) j for j ≥ a + b . We take the ⊗ ( a + b ) iterate of the section µ H ( A ), see (7). It is clear that µ H ( A ) = 0 if andonly if µ H ( A ) n = 0 for any n ∈ L N > , in particular for n = a + b .Since a ≤ b , both Ver ∗| Ω K ( ℓ ) ,a and Ver ∗| Ω K ( ℓ ) ,b have rank at most a . So also(Ver ∗ ) j | Ω K ( ℓ ) ,a and (Ver ∗ ) j | Ω K ( ℓ ) ,b each have rank at most a . By (3), (Ver ∗ ) j has rankat most 2 ar .The ℓ -rank of A equals 2 ar if and only if the rank of (Ver ∗ ) j is 2 ar for j ≥ a + b .In turn, the rank of (Ver ∗ ) j is 2 ar if and only if both Ver ∗| Ω K ( ℓ ) ,a and Ver ∗| Ω K ( ℓ ) ,b haverank a . Since Ω K ( ℓ ) ,a and Ω ( ℓ a + b ) ) K ( ℓ ) ,a are rank a vector bundles, the determinant ofa map between them is nonzero if and only if it has rank a . (cid:3) We now conclude the proof of Th. 1.1.
Proof of
Th. 1.1: Combining Prop. 3.2, Cor. 3.3 and Lemma 3.4 gives ( µ -Ha1). By[LS12b, Prop. 7.14] (or [LS12a] in the compact case), there exists k ∈ N suchthat ω ⊗ ( k ( ℓ − K ( ℓ ) ,a extends to an ample line bundle on the minimal compactification sh min K ( ℓ ) ,λ . Given this ampleness result, the existence of a lift of some power of µ H follows by a well-known cohomological argument coupled with the Koecher principle(cf. [Gol12, Lemma 4.4.1]). Thus ( µ -Ha2) is established. Finally ( µ -Ha3) is provedin the same way as Th.4.2.4 of loc. cit. . (cid:3) Next we note that Cor. 1.2 is an immediate consequence of Th. 1.1:
Proof of
Cor. 1.2: The nonvanishing locus of a section of an ample line bundle ona projective scheme is affine. (cid:3)
Finally we note how Th. 1.4 follows from Th. 1.1. µ -ORDINARY HASSE INVARIANT 5 Proof of
Th. 1.4: The proof is analogous to the proof of [Gol12, Th.1.2]: Let
K ⊂ G ( A f ) be an open compact subgroup such that K = K ( ℓ ) K ( ℓ ) with K ( ℓ ) ⊂ G ( A ℓf )and K ( ℓ ) ⊂ G ( Z ℓ ). Let Sh K ,E be the model of Sh( G , X ) at level K over E asdefined in § loc. cit. . Let Sh K ,ℓ be the normalization of Sh K ( ℓ ) ,ℓ in Sh K ,E .Let π : Sh K ,ℓ → Sh K ( ℓ ) ,ℓ be the natural projection.Using ( µ -Ha2), let µ H lift K ( ℓ ) be a lift of a power µ H and let µ H lift K be the pullbackof µ H lift K ( ℓ ) to Sh K ,ℓ along the projection π .Let H = H P ( ℓ ) ( G , Z ℓ ) be the spherical Hecke algebra of G with values in Z ℓ ,trivial at places outside P ( ℓ ) (see § loc. cit. for a more detailed definition). Theorem 3.5.
Suppose V is an automorphic vector bundle on Sh K ,ℓ and f ∈ H ( Sh K ,ℓ , V ) is nonzero modulo λ . Then for all j ∈ N , the product ( µ H lift K ) ℓ j f ∈ H ( Sh K ,ℓ , ω ⊗ ( ℓ − ℓ j K ,a ⊗ V ) is nonzero modulo λ and satisfies (9) T (( µ H lift K ) ℓ j f ) ≡ ( µ H lift K ) ℓ j T ( f ) (mod λ j +1 ) for all T ∈ H . Proof.
Since the µ -ordinary locus is dense [Wed99, Th.1.6.2], the product ( µ H lift K ) ℓ j f is nonzero modulo λ by ( µ -Ha1). As in the proof of [Gol12, Th.6.2.1], ( µ -Ha3)implies (9). (cid:3) Given Th. 3.5, the remainder of the argument to establish Th. 1.4 is identical to §§ loc. cit. . (cid:3) Remark . A tremendous advantage of our µ -ordinary Hasse invariant is thatit satisfies all key properties of the classical invariant. Its applications will thusfollow the classical blueprint: to Galois representations (as we illustrated brieflyabove), but also immediately the (non-effective) existence of its canonical subgroupthanks to the elementary [Far11, Prop.3], and thus also applications to explicitconstructions of eigenvarieties, etc.4. Acknowledgments
We thank the Max-Planck-Institut f¨ur Mathematik for a year-long membershipin 2011 (M.-H. N.) and also for a short visit in May 2011 (W.G.). In particular,the natural albeit key idea of considering higher powers of Verschiebung occurredto M.-H. N. on his very first Monday at MPIM.W.G. thanks Pierre Deligne, Richard Taylor, Barry Mazur, Elena Mantovan,David Geraghty, Benoˆıt Stroh, Jacques Tilouine and Vincent Pilloni for helpfulconversations. W. G. is happy to acknowledge support from 10 BLAN 114 01 ANRARSHIFO, a Simons Travel Grant and NSF MSPRF.
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