Λ-ADIC p-divisible GROUPS, I. Contents 1. Introduction 1 2. Construction over Q 3 3. Construction over Z (p) [µ p ] 5 References 7

Transcription

1 Λ-ADIC p-divisible GROUPS, I HARUZO HIDA Contents 1. Intoduction 1 2. Constuction ove Q 3 3. Constuction ove Z (p) [µ p ] 5 Refeences 7 1. Intoduction Let p be a pime. Let R 0 be a discete valuation ing of mixed chaacteistic unamified ove Z (p) (with finite esidue field F of chaacteistic p). Let R = n R n fo R n = R 0 [µ p n]; so, R n is a discete valuation ing with esidue field F. Wite K n fo the quotient field of R n and K fo an algebaic closue of K. A Λ-adic BT goup G /R0 (a Λ-BT goup) is by definition an inductive limit of Basotti Tate goups G n/k0 defined ove K 0 such that G n K0 K n is the geneic fibe of a Basotti Tate goup G n/rn defined ove R n with an action of the Iwasawa algeba Λ = W [[x]] as endomophisms ove K 0 ; so, Λ acts on G n/rn (esp. G n/k0 ) as endomophisms of Basotti Tate gousp ove R n (esp. ove K 0 ). Hee W is a discete valuation ing finite flat ove Z p. We impose the following conditions: (RT) The geneic fibe G n is defined ove K 0 (as an étale Basotti Tate goup ove K 0 ) and the action of Λ on G n/k0 is also defined ove K 0 ; (CT) Witing γ = 1 + x, we have G n = Ke(γ pn 1 : G G) (closed immesion); (DV) G(K ) = Λ fo Λ := Hom cont (Λ, Q p /Z p ) (Pontyagin dual); (DL) We have a Catie self duality G n [p m ] G n [p m ] µ p m ove K n which, afte taking the limit, gives the duality T G G(K ) µ p ove K. Hee T G = lim T G n (K ) (fo T G n = lim G n [p m ](K )) with espect to the map n m G n+1 G n dual to G n G n+1. The fist talk of the two lectues at CRM (Montéal) in Septembe 2005 while the autho was a Clay eseach schola, and the note was evised on Novembe 3, 2009; the autho is suppoted patially by NSF gant: DMS and DMS

2 Λ-ADIC p-divisible GROUPS, I 2 (OD) The connected component of G n ove each stict henselization of R n is a multiplicative goup (i.e., isomophic to a poduct of copies of µ p ) fo all n (so, G n is odinay); (U) On the special fibe, we have the Fobenius map F and its dual V. Thus we have a splitting G F = G G et so that G = Ke(e F ) and G et = Ke(e V ) fo e F = lim n F n! and e V = lim n V n!. Then we have an automophism U of G such that U commutes with F and V and U on G et lifts F G et. Thus a Λ-adic BT goup is give by data (G n/rn ) with the above compatibility conditions. If G /R0 and H /R0 ae Λ-adic BT goups, a mophism f : G H of Λ-adic BT goups is given by a Λ linea map f : G H of abelian fppf sheaves ove R such that the induced mophism f n : G n H n of Basotti Tate goups ove R descends to a mophism G n/rn H n/rn of Basotti Tate goups and it futhe commutes with the action of Gal(K /K 0 ) on the geneic fibes. We wite Hom Λ-BT/R0 (G, H) fo the Λ-module of such mophisms (defined genetically ove K 0 ). If L End Λ-BT/R0 (T ) is a linea opeato, we can think of the p-divisible pat G[L] div of Ke(L : G G). By the classification of Λ-modules, if det(l) 0, G[L] div has finite coank, and it is a classical Basotti Tate goup ove R. Of couse, stating with a self-dual Basotti Tate goup H with a lift U, T H Zp Λ gives a constant Λ-adic BT-goup. We heeafte suppose that all Λ-adic BT-goups we conside ae non-constant. This could be said that the epesentation of Gal(K /K 0 ) on T G is a non-constant defomation of T G 1 in the sense of Mazu. A p-odinay Basotti Tate goup H ove a discete valuation ing B /Z(p) with quotient field F is called a GL(2g)-type if it is self dual with a local ing A End BT/B (H) such that T H = A 2g. We call H minimal if A is geneated by T(σ) A fo all σ Gal(F /F ). Fo a Λ-adic BT goup G /R0, if we have a local Λ[U]-algeba T inside End Λ-BT/R0 (G) such that T G = T 2g and T is self-adjoint unde the duality, we call G a GL(2g)-type ove T. In this Λ-adic case, we call G minimal if T is geneated by T(σ) and U topologically. Supposing the existence of such G (that we will see today), we can ask a lot of simple questions. (Q1) If we ae given G ove a finite field F of chaacteistic p, can one lift it to chaacteistic 0? (Defomation question). (Q2) Is thee any systematic way of constucting such G ove a given R? If it exists, does it ceate all such G ove R of GL(2g)-type? (Constuction). (Q3) If G is nonconstant, can det(u) T be algebaic ove W? (Non-constancy) (Q4) Let us give ouselves a Weil numbe α Q W with α = p of degee 2g. Supposing α odinay (that is, the minimal polynomial of α modulo p can only divisible by X g not moe), does G[U α] div descend to a discete valuation ing? (Descent).

3 Λ-ADIC p-divisible GROUPS, I 3 (Q5) Is it possible to embed the p-divisible pat G[U α] div of G[U α] = Ke(U α) into an abelian scheme defined ove a finite extension of R? (Relation to abelian vaieties). (Q6) Fo a given minimal G 1 of GL(2g)-type with ieducible T G 1 Z Q, is thee a univesal G? (Univesality). Hee the univesality is defined as follows. If we have a minimal p-divisible goup H of GL(2g)-type with a mophism i : G 1 H with finite kenel (so, i T(σ G1 ) = T(σ H ) i), we have a mophism i H : H G with finite kenel making the following diagam commutes: G 1 i H i H G G. The Λ-adic BT goup cannot descend to (an inductive limit of) G n/r ove a finite extension R R of R 0 (independent of n) unless it is a constant. Hee is a eason fo it. Suppose that G is minimal of GL(2)-type and suppose that G extends as a BT-goup to the integal closue of Z[ 1 ] in R. If G is defined ove a discete valuation N ing R = Z p o Z (p). Then by Raynaud s classification of p-odinay divisible goups [R] 4.2, the deteminant of the Galois epesentation on T G has to be the p-adic cyclotomic chaacte χ. Thus T G is a defomation of T G 1 which is p-odinay and of deteminant χ. If T G 1 is modula whose esidual epesentation is ieducible ove Q[ p ] (p = ( 1) (p 1)/2 p), by Wiles (see [W]), the univesal Galois defomation ing fo p-odinay defomations unamified outside N p with fixed deteminant χ is of finite ank ove Z p. Thus T G has to be constant; so, G has to be constant. Thus if such a G exists, at least R contains the p-adic valuation ing of the cyclotomic Z p -extension Q /Q. Questions elated to the above have been studied in [H86b], [MW1], [Ti] and [Oh1]. Today I will give an automophic way of constucting such G ove Z (p). By the solution of Galois defomation poblems (of odinay type) by Mazu and Wiles Taylo, this gives almost all such Λ-adic BT-goups, basically solving (Q2) and (Q6) fo GL(2)- type goups. 2. Constuction ove Q Fix a pime p 5 and a positive intege N pime to p. We conside the modula cuve X 1 (Np ) which classify elliptic cuves E with an embedding µ Np E[Np ] = Ke(Np : E E). Suppose N 4 so that X 1 (Np ) gives a fine moduli of the poblem. Let J = Pic 0 X 1 (Np )/Q be the Jacobian vaiety. Similaly we take Js to be the Jacobian vaiety associated to the modula cuve with the conguence subgoup

4 Γ s = Γ 1 (Np ) Γ 0 (p s ). Note that Γ s\γ s ( p s ) Γ1 (Np ) = Λ-ADIC p-divisible GROUPS, I 4 { ( 1 a 0 p s ) a mod p s = Γ 1 (Np )\Γ 1 (Np ) ( ) p s Γ1 (Np ). ( Witing U s (p s ) : J s J fo the Hecke opeato of Γ 1 0 ) s 0 p s Γ1 (Np ). Then we have the following commutative diagam by the above identity: π J Js u u u J π J s, whee the middle u is given by U s (p s ) and u and u ae U(p s ). Thus if we take the odinay pojecto e = lim n U(p) n! on J[p ] fo J = J, J s, Js, noting U(p m ) = U(p) m, we have Js,od [p ] = J od [p ], whee od indicates the image of e. We now identify J[p ](C) with a subgoup of H 1 (Γ, T p fo T p := Q p /Z p ) fo the conguence subgoup Γ defining the modula cuve whose Jacobian is J. Since Γ s Γ 1 (Np s ), by the inflation estiction sequence, we have the following commutative diagam with exact ows: Γ s H 1 (, T Γ 1 (Np s ) p) H 1 (Γ s, T p ) H 1 (Γ 1 (Np s ), T p ) γp =1 H 2 (, T Γ 1 (Np s ) p)? J s [p ] J s [p ][γ p 1]? By shee computation, we can pove H j od (, T Γ 1 (Np s ) p) = 0 and the all the vetical aows above ae injective, we get the contollability Ke(γ p 1 : J od s Γ s } [p ] Js od [p ]) = J od [p ]. Define J od [p ] = lim J od [p ]. Fo each chaacte ε : Γ/Γ p µ p, by the inflation and estiction technique that J od [p] Z[ε][γ ε(γ)] = J od [p] Z[ε][γ ε(γ)] = J1 od [p]. Thus J od [p ] Z[ε][γ ε(γ)] is a nontivial p-divisible goup. Taking the Pontyagin dual T = J od [p ], we find a sujection π : Λ m T fo m = dim Fp J1 od [p]. Then fo a pime P ε = (γ ε(γ)) Λ, T/P T is the dual of J od [p ] Z[ε][γ ε(γ)] which is Z p -fee of ank m (by Nakayama s lemma). Thus Ke(π) P ε Λ m. Moving aound ε, we find that T = Λ m ; so, J od [p ] is a Λ-adic BT-goup satisfying (CT) and (DV). As fo the duality, the canonical polaization of J gives ise to the selfduality paiing [, ] of J [p ] and J = t J. Let U (p) (esp. T (n)) be the image of U(p) (esp. Hecke opeato T (n)) unde the canonical Rosati involution of J in End(J ). The Weil involution τ associated to ( ) 0 1 satisfies τu(p)τ 1 = U (p) Np 0 Γ s

5 Λ-ADIC p-divisible GROUPS, I 5 and τt (n)τ 1 = U (n) inside End(J /Q[µNp ]) because τ is only defined ove Q[µ Np ]. Thus twisting the paiing by τ and U(p), we get the self-duality paiing, = [, τ U(p) ( )] of J od [p m ]. Witing Rs : J od [p ] Js od [p ] fo the inclusion, and N s = p s j=1 γj : Js od [p ] J od [p ] fo γ = γ p, we can veify by computation Fom this we get (DL) ove Q[µ Np ]. R s(x), y s = x, N s (y). 3. Constuction ove Z (p) [µ p ] We constuct the geneic fibe of a Λ-adic BT goup, and in the following section, we extend it to Z (p) [µ p ]. By the above constuction, the Tate module T = T J od [p ] caies Galois epesentations of Hecke eigenfoms satisfying the following popeties: (1) cusp foms in S 2 (Γ 0 (p) Γ 1 (N)); (2) all cusp foms in S 2 (Γ 1 (Np m )) whose Neben chaacte has p-conducto equal to p m fo m = 1, 2,...,. By a theoem of Langlands (and Caayol), the l-adic Galois epesentation (l p) associated to such a Hecke eigenfom f does not amify at p on Gal(Q/Q[µ p ]) except fo the case (1). In the case (1), it is semi-stable at p. Thus the abelian subvaiety A f attached to f extends a semi-abelian scheme ove Z (p) [µ p ]. Let t G = f as above A f J. Thus we have an inclusion t G J. Let J G be the dual quotient unde the canonical polaization twisted by τ. Fo any abelian subvaiety A of X = J stable unde U(q) fo q Np and T (n) fo n pime to Np, if thee exists an abelian subvaiety B stable unde the same Hecke opeatos such that A + B = X and A B is finite, the abelian subvaiety B is uniquely detemined by A (the multiplicity one theoem). The abelian subvaiety B is called the complement of A in X. By definition, G and t G extend to a semi-abelian scheme ove Z (p) [µ p ]. The acts on J, t G and G by the diamond opeatos. If we define goup µ = µ p 1 Z p t G (0) in t G to be the complement of abelian subvaiety fixed by µ, t G (0) and its dual extend to an abelian scheme ove Z (p) [µ p ]. Anyway, we take the Néon quotient G (0) model of these abelian schemes ove Z (p) [µ p ] and take its p-divisible goups (whose p-powe division goup is at wost quasi-finite flat goups schemes). Theoem 3.1. We have t G od [p ] = J od [p ] = G od [p ] canonically ove Z (p) [µ p ]. To pove the theoem, we fist pove the following lemma. Lemma 3.2. Let R be a henselian discete valuation ing with faction field K. Let G K and G K be eithe both Basotti Tate goups o both abelian schemes ove K with abelian geneic fibe. If G K and G K ae abelian schemes, let G R and G R be the identity connected component of the Néon models ove R of G K and G K, espectively.

6 Λ-ADIC p-divisible GROUPS, I 6 If G K and G K ae Basotti Tate goups, we assume to have Basotti Tate goups G R and G R ove R whose geneic fibes ae isomophic to G K and G K, espectively. (1) Suppose we have a sujective mophism f K : G K G K and an endomophism g K : G K G K such that Ke(f K : G K G K ) Ke(g K : G K G K ). Then fo the extensions f : G G and g : G G ove R, Ke(f) is a closed subscheme of Ke(g); (2) Suppose we have an injective mophism f K : G K G K and an endomophism g K : G K G K such that Coke(f K : G K G K) is the sujective image of Coke(g K : G K G K ). Then, fo the extensions f : G G and g : G G ove R, Coke(f) is a quotient goup of Coke(g). Poof. We fist pove the assetion (1). Let A be a K-algeba. By the sujectivity of f K, fo each x G (A), we can find a fppf extension A /A such that thee exists y G(A ) with f K (y) = x. Then g K (y) G(A ) is well defined independently of the choice of y. Then by fppf descent, we conclude that g K (y) G(A). Thus we get a mophism of goup functos G K G K sending x to g K (y). Since a functo mophism gives a unique mophism of schemes (Yoneda s lemma), we get a mophism f K : G K G K such that f K f K = g K. Fist suppose that G K and G K ae abelian schemes. Since G and G ae the connected components of the Néon models of G K and G K, espectively (see [NMD] Poposition 7.4.3), any geneic mophism φ K of these schemes extends to a unique mophism ove R. Then f K and f K extend to mophisms f : G G and f : G G ove R, espectively, and f and f satisfies f f = g, which shows that Ke(f) is a closed subscheme of Ke(g). If G and G ae Basotti Tate goups, we only need to veify that extensions f and f exist. This extension popeties follows fom [T] Theoem 4. The second assetion is the dual of the fist. Now we pove the theoem: Poof. Note that ove Q, by the definition, J od [p ] t G od [p ]. Let B = Ke(J G ) which is the complement of t G. By definition, e kills B[p ]; so, it kills the p- pimay pat of H = B t G. Thus ove Q, we have the identity in the theoem. Since H is finite, H is killed by M U(p) L fo an intege M pime to p and anothe intege L sufficiently lage. We apply the fist statement of the lemma to the pojection f Q : t G Q G Q and g Q = M U(p) L. Thus by the lemma, we have Ke(f) Ke(M U(p) L ); so, we get an injection t G od [p ] G od [p ] which ae p-divisible goup of the same coank; so, the injection is a sujection. Coollay 3.3. The natual mophism i : t G od [p ] t G od s [p ] is a closed immesion fo s >. Poof. We can facto the isomophism ι : t G od [p ] = G od [p ] as ι = t i ι s i. This shows that i is a closed immesion.

7 Λ-ADIC p-divisible GROUPS, I 7 Theoem 3.4. Ove Z (p) [µ p s], the natual inclusion t G od [p ] into t G od s [p ] is a closed immesion whose image is equal to the kenel Ke(γ p 1) on t G od s [p ] fo all s >. Poof. Fo simplicity, we wite G fo G od [p ]. Look at the inclusion i : G G s. Since i is a closed immesion, H := Im(i) is a Basotti Tate subgoup of G s. Since G s [p n ] G s [p n ]/H[p n ] is an epimophism of fppf abelian sheaves, G s [p n ]/H[p n ] = (G/H)[p n ] is a finite flat goup scheme. Thus G s /H is a Basotti Tate goup. Geneically, γ p 1 : G s/ks G s/ks factos though G s /H inducing an isomophism Im(γ p 1) /Ks = (Gs /H) /Ks ; so, by Lemma 3.2, γ p 1 factos though G s /H ove R, getting a mophism π : G s /H Im(γ p 1). Resticting the pojection G s G s /H to Im(γ p 1) G s, we get a mophism: Im(γ p 1) G s /H of fppf abelian sheaves, which is geneically the invese of π, and hence we have Im(γ p 1) = G s /H ove R s. Thus we must have H = Ke(γ p 1) ove R s (as the categoy of fppf abelian sheaves is an abelian categoy), showing Ke(γ p 1) = Im(i : G G s ) as desied. Books Refeences [AME] N. M. Katz and B. Mazu, Aithmetic Moduli of Elliptic Cuves, Annals of Math. Studies 108, Pinceton Univesity Pess, Pinceton, NJ, [MFG] H. Hida, Modula Foms and Galois Cohomology, Cambidge Studies in Advanced Mathematics 69, Cambidge Univesity Pess, Cambidge, England, [MFM] T. Miyake, Modula Foms, Spinge, New Yok-Tokyo, [NMD] S. Bosch, W. Lütkebohmet, and M. Raynaud, Néon Models, Sp-inge, New Yok, Aticles [GS] R. Geenbeg and G. Stevens, p-adic L functions and p adic peiods of modula foms, Inventiones Math. 111 (1993), [H86a] H. Hida, Iwasawa modules attached to conguences of cusp foms, Ann. Sci. Ec. Nom. Sup. 4th seies 19 (1986), [H86b] H. Hida, Galois epesentations into GL 2 (Z p [[X]]) attached to odinay cusp foms, Inventiones Math. 85 (1986), [M] B. Mazu, Rational isogenies of pime degee, Invent. Math. 44 (1978), [MTT] B. Mazu, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectues of Bich and Swinneton-Dye, Inventiones Math. 84 (1986), 1 48 [MW] B. Mazu and A. Wiles, Class fields of abelian extensions of Q. Inventiones Math. 76 (1984), [MW1] B. Mazu and A. Wiles, On p adic analytic families of Galois epesentations, Compositio Math. 59 (1986), [Oh1] M. Ohta, On the p-adic Eichle-Shimua isomophism fo Λ-adic cusp foms, J. eine angew. Math. 463 (1995), [Oh2] M. Ohta, Odinay p adic étale cohomology goups attached to towes of elliptic modula cuves, Compositio Math. 115 (1999),

8 Λ-ADIC p-divisible GROUPS, I 8 [Oh3] M. Ohta, Odinay p adic étale cohomology goups attached to towes of elliptic modula cuves. II, Math. Ann. 318 (2000), [R] M. Raynaud, Schémas en goupes de type (p,..., p). Bull. Soc. Math. Fance 102 (1974), [T] J. Tate, p-divisible goups, Poc. Conf. on local filds, Diebegen 1966, Spinge 1967, [Ti] J. Tilouine, Un sous-goupe p divisible de la jacobienne de X 1 (Np ) comme module su l algèbe de Hecke, Bull. Soc. Math. Fance 115 (1987), [W] A. Wiles, Modula elliptic cuves and Femat s last theoem, Ann. of Math. 141 (1995),