STRONG CM LIFTING PROBLEM II

Transcription

1 STRONG CM LIFTING PROBLEM II TAISONG JING VERSION: /9/4 ABSTRACT. Let R be a complete dcrete valuaton rng of mxed charactertc, X be a p-dvble group over R, and X be the cloed fber of X. We ay a ubgroup G of X potentally lftable, f after a fnte bae change G lft to a fnte locally free ubgroup cheme of X R, where R/R a fnte extenon. In th artcle, we compute the complete lt of potentally lftable ubgroup n a frt non-trval example, where X a CM p-dvble group wth heght 4 and dmenon. A an applcaton, we obtan a new type of counterexample to the queton of trong CM lftng.. INTRODUCTION Th a contnued work of [5] on trong CM lftng problem (CML). The queton (CML) for abelan varete ak whether every g-dmenonal abelan varety over a fnte feld F q wth an acton by the whole rng of nteger n a CM feld L of degree g admt an L-lnear CM lftng to charactertc. Th problem can be reduced to a queton on lftng ubgroup of CM p-dvble group. Namely, f F a p-adc local feld and X an O F -lnear CM p-dvble group over a complete dcrete valuaton rng R of charactertc wth redue feld F p, whch O F -table ubgroup of the cloed fber X Fp lftable nto a fnte locally free ubgroup cheme of X? In 6 of [5] we gave a condton on p-adc CM type of X uch that every O F -table ubgroup of X Fp lftable nto a fnte locally free ubgroup cheme of the bae change of X to a fnte extenon of R. In thee example, the p-adc CM type nduced from an unramfed extenon of Q p. A a corollary we proved that the anwer to queton (CML) for abelan varete affrmatve when every place v above p n the maxmal totally real ubfeld L nert n L. A complete anwer to the queton on lftng O F -table ubgroup of the cloed fber X Fp to charactertc requre u to conder all fnte ubgroup of the geometrc generc fber of X, and compute the reducton over F p of ther cheme-theoretc cloure. We do not know any uch attempt n the pat except for ome very pecal cae, e.g., when dm X or codm X. In th artcle, we wll tudy an example of O F -lnear CM p-dvble group X wth dmenon and heght 4 over a complete dcrete valuaton rng wth redue feld F p, where F a p-adc local feld of degree 4. The anwer urprng to u: whether a fnte ubgroup of the geometrc generc fber of X ha an O F -table reducton completely determned by t order. Namely, f the order p n, then the reducton equal to X Fp [π n ],.e., the kernel of multplcaton on X F p by π n, where π a unformzer of O F ; f the order p n+, then the reducton a certan ubgroup G between X Fp [π n ] and X Fp [π n+ ], and we have a decrpton on t embeddng n X Fp [π n+ ]; ee (Theorem.3). Th reult ndcate that the ubgroup of the geometrc generc fber of X eem to try very hard to have an O F -table reducton, though n charactertc they may be far from beng O F -table. Baed on th obervaton, we can ak the followng queton. Let Φ be a prmtve p-adc CM type for F. I there a general condton on the p-adc CM type Φ, uch that there ext an nteger d(φ) (equal to n the example above) whch only depend on Φ, atfyng that for any fnte locally free ubgroup cheme G of an O F -lnear CM p-dvble group wth p-adc CM type Φ over a complete dcrete valuaton rng n mxed charactertc, the cloed fber of G contan an O F -table ubgroup wth ndex unformly bounded by p d(φ)?

2 TAISONG JING VERSION: /9/4 Th true when #Φ = or [F : Q p ]. In thee cae, an O F -lnear CM p-dvble group wth p-adc CM type Φ n mxed charactertc ha dmenon or codmenon, and all fnte locally free ubgroup cheme have O F -table reducton; n other word, d(φ) = n thee cae. The man example we tudy n th artcle the frt example that doe not belong to thee cae. We do not know any further example or neceary condton on Φ o far. We ue the theory of Kn module from ntegral p-adc Hodge theory a the man tool. A Kn module a W(κ)[[u]]-module atfyng certan addtonal condton, where κ a perfect feld of charactertc p and W(κ) the rng of Wtt vector over κ. There a p-dvble group or a fnte locally free ubgroup cheme n mxed charactertc aocated to a Kn module, and roughly peakng the Deudonne module of the cloed fber the quotent module by modulo u ; for a prece tatement, ee [] (B.4.7) or [5] (3..). The localzed W(κ)((u))-module of the Kn module carre the nformaton on the generc fber. In [5] (5..7), va the theory of Lubn-Tate formal group law we have computed element n Kn module uch that they correpond to the toron pont on the geometrc generc fber of the p-dvble group. Such element generate the W(κ)((u))- module attached to a fnte locally free ubgroup cheme. After that, the computaton of the cloed fber of a fnte locally free ubgroup cheme reduced to a computaton of the W(κ)[[u]]-module before nvertng u; ee (3.7.) and (3.7.). Th computaton poble becaue of our knowledge on the toron pont, baed on the explct nformaton on ther coordnate gven by the Lubn-Tate theory; ee (3.6.) and (3.6.3). In the example of the O F -lnear CM p-dvble group X wth dmenon and heght 4, a a corollary of the computaton on the reducton of t fnte locally free ubgroup cheme, we obtan a complete lt of the cloed fber of all F-lnear CM p-dvble group n mxed charactertc wth the ame p-adc CM type a X. Th lead to a counterexample of (CML). In and 4 of [5], we tuded the counterexample of (CML) comng from an extra ymmetry on the Le type of the cloed fber. That ymmetry caued by mall redue feld of the reflex feld of the p-adc CM type n the ene of [5] (4.). In the new counterexample n th artcle, however, the reflex feld equal to F and hence t redue feld not mall. Therefore th counterexample doe not fall n the framework of 4 of [5]. Bede that, we alo tudy the F-lnear CM lftng for O F -lnear CM p-dvble group over F p, uch that the p-adc CM type Φ of the lftng cannot be nduced from an unramfed extenon of Q p. Let F be the mnmal ubfeld of F uch that Φ nduced by a p-adc CM type for F. In the cae when the ramfcaton ndex of F mall, we can prove a property mlar to the one we proved n 6 of [5]: f X an O F -lnear CM p-dvble group wth p-adc CM type Φ over a complete dcrete valuaton rng R of charactertc wth redue feld F p, then every O F -table ubgroup of X Fp lft to a fnte locally free ubgroup cheme of the bae change of X to a fnte extenon of R. Th property on lftng O F -table ubgroup of X Fp tronger than ayng that every O F -lnear CM p-dvble group ogeneou to X Fp admt an F-lnear CM lftng wth p-adc CM type Φ; for the prece tatement and an explanaton on the property, ee (3.). A a corollary, we prove that the anwer to queton (CML) affrmatve under the followng broader condton on the CM feld L: for every place v above p n the maxmal totally real ubfeld L, ether v nert n L, or v plt n L and the ramfcaton ndex e(v) < p ; ee (.3.). On the other hand, th trong lftng property on O F -table ubgroup of X Fp may fal to hold f the ramfcaton ndex of F hgh; ee Example (3..). Th obervaton ndcate the ubtlety n F-lnear CM lftng wth p-adc CM type Φ that cannot be nduced from an unramfed extenon of Q p. Acknowledgement. Th work part of the author doctoral dertaton. Many thank to C.-L. Cha for ntroducng the queton to me and for many dcuon and encouragement. I alo thank F. Oort for h

3 STRONG CM LIFTING PROBLEM II 3 queton and uggeton. I thank Inttute of Mathematc, Academa Snca for hoptalty durng the frt half of 3, where the man part of th work wa done.. REDUCTIONS OF FINITE LOCALLY FREE SUBGROUP SCHEMES Throughout th artcle, let p be a prme number, q be a power of p, and k := F p. For a perfect feld κ of charactertc p, let W(κ) be the rng of Wtt vector over κ, and let B(κ) := W(κ)[ p ]. Denote the Frobenu automorphm on B(κ) by σ. For a p-adc local feld F, we denote t maxmal unramfed ubextenon of Q p by F ur, and t redue feld by κ F. Defnton.. Let R be a complete dcrete valuaton rng of charactertc and redue charactertc p. Let κ be the redue feld of R. Let X be a p-dvble group over R. A fnte ubgroup G of X κ ad to be potentally lftable, f there ext a fnte extenon R over R wth redue feld κ, and a fnte locally free ubgroup cheme G of X R, uch that G κ = G κ. Let F be a p-adc local feld, Φ be a prmtve p-adc CM type for F, F be the reflex feld. Let X be the (unque) O F -lnear CM p-dvble group over R := O F B(k) wth p-adc CM type Φ. A complete lt of potentally lftable ubgroup of X k would allow u to dentfy whch F-lnear CM p-dvble group admt an F-lnear CM lftng wth p-adc CM type Φ. In [5] (Thm. 6.), for a cla of p-adc CM type Φ, we proved that every O F -table ubgroup of X k potentally lftable. We wll prove the ame property for a broader cla of p-adc CM type n (3.), and gve example of other p-adc CM type uch that not every O F -table ubgroup of X k potentally lftable n (3.). In general to gve a complete lt of potentally lftable ubgroup of X k, we need to let R run over all the fnte extenon of R, and compute the reducton of all fnte locally free ubgroup cheme of X R. When dm X = or codm X =, a we wll explan n (.5. (b)), the computaton trval mply becaue the cloed fber X k doe not have many ubgroup. In th ecton, we wll compute a frt non-trval example... The man theorem. we frt et up the example and make ome defnton to tate the man theorem and t corollare. Let p >, F = B(F p )[π ]/(π ɛ p), where ɛ W(F p ) a Techmuller lft and not a quare. The degree 4 extenon F/Q p Galo, and Gal(F/Q p ) a cyclc group of order 4 generated by the automorphm τ : F F, uch that τ B(Fp ) = σ, and τ(π ) = ɛ p π. Throughout th ecton, we denote ɛ p by λ for mplcty. A prmtve p-adc CM type for F ha the form of {, τ}, where an embeddng of F nto Q p. We dentfy F wth t mage n Q p by when there no danger of confuon. Take an dentfcaton between Hom(F ur, Q p ) and {, } a Gal(F ur /Q p ) Z/-toror uch that F ur =. The reflex feld F of (F, Φ) equal to F. Let X be the O F -lnear CM p-dvble group over R = W(k)[π ]/(π ɛ p). The cloed fber X := X k an O F -lnear CM p-dvble group over k. The Grothendeck group R k (O F ) of the category of fntely generated O F Z k-module omorphc to R k (O F OF ur, k) R k (O F OF ur, k) Z Z. The Le type of X defned to be [Le(X)] = (, ) n R k (O F ). Defne a Deudonne module M a follow: (a) M = W(k)[π]/(π ɛ p)e W(k)[π]/(π ɛ σ p)e ; (b) there an O F -acton on M defned by: α e = αe, α e = α σ e for α W(F p ), and π e = πe ; (c) the O F -lnear Frobenu and Verchebung map on M are defned by: Fe = ɛ λ πe, Fe = ɛ πe, Ve = πe, Ve = λ σ πe The p-dvble group attached to M O F -lnear wth Le type (, ), hence O F -lnearly omorphc to X. Therefore M O F -lnearly omorphc to the Deudonne module attached to X. We ay an O F -ba e, e of M

4 4 TAISONG JING VERSION: /9/4 good, f the condton (a), (b), (c) above are atfed. If e, e another good O F-ba of M, then there ext ζ O F uch that e = ζe, e = ζσ e. One can check dm k M/(FM+V M) =, o the a-number of X equal to. The et of α p embedded n X n bjectve correpondence wth P (k),.e., the et of lne n π M/M ke +ke. Defne the followng equvalent relaton on P (k): [a, b ] [a, b ] f and only f there ext c F p uch that [a c, b c p ] = [a, b ] n P (k). Denote the equvalent clae on P (k) by L. The et L can be naturally dentfed wth {, } {k /(F p ) p } by conderng a/b for [a, b] P (k). For each ubgroup G of X wth order p, a an α p embedded n X, we can aocate to G an element δ (G) n L. By our defnton, δ (G) doe not depend on the choce of the good O F -ba e, e n M, o t a well-defned nvarant for ubgroup G of X wth order p. Smlarly, uppoe G a ubgroup of X uch that X[π n ] G for ome nteger n, and [G : X[πn ]] = p, then the Deudonne module N attached to G between π n M/M and π (n+) M/M. Thu we can alo aocate to G a well-defned nvarant δ n (G) L by lookng at the drecton of the k-lne N/(π n Now we are ready to tate the man reult of th ecton: M/M) n π (n+) M/π n M. Theorem.3. Notaton are a above. () Suppoe R a fnte extenon of R and G a fnte locally free ubgroup cheme of X R wth order p t, where t an nteger. Then we have the followng decrpton on the cloed fber G := G k a a ubgroup of X: (a) If t = n even, then G = X[π n ]. (b) If t = n + odd, then X[π n ] contaned n G wth ndex p, and the nvarant δ n(g) equal to ether [] or [λ] n L. () Converely, for each ubgroup H of X uch that X[p n ] H wth ndex p and δ n (H) = [] or [λ], there ext a fnte extenon R of R and a fnte locally free ubgroup cheme H of X R uch that H k = H. In partcular, the cloed fber G O F -table f and only f the order of G an even power of p. Theorem (.3) ha the followng conequence: Corollary.4. Let X be the O F -lnear CM p-dvble group over k wth Le type (, ). If Y an F-lnear CM p-dvble group over k, then Y admt an F-lnear CM lftng wth p-adc CM type compatble wth τ f and only f: ether (a) Y F-lnearly omorphc to X; or (b) Y F-lnearly omorphc to X/G, where G a ubgroup of X wth order p, and δ (G) = [] or [λ]. In partcular, f Y O F -lnear, then Y admt an F-lnear CM lftng wth p-adc CM type compatble wth τ f and only f [Le(Y)] = (, ) n R k (O F ). Proof. Sayng a p-adc CM type Φ for F compatble wth ι equvalent to ayng Φ ha the form {, τ} for ome Hom(F, Q p ). Suffcency follow mmedately from Theorem (.3 ()). For necety, uppoe R a complete dcrete valuaton rng of charactertc and redue feld k, Y an F-lnear CM p-dvble group over R lftng Y wth p-adc CM type Φ compatble wth τ. Then Φ mut be prmtve. Let F be the reflex feld, R := O F B(k), and X be the O F -lnear CM p-dvble group over R wth p-adc CM type Φ. Then Y F-lnearly ogeneou to X, and the necety of the tatement alo follow from Theorem (.3 ()). For the lat tatement, we need to how that f Y O F -lnear and the Le type of Y equal to (, ) or (, ), then Y doe not admt an F-lnear CM lftng wth p-adc CM type compatble wth τ. It eay to check that under uch condton, there ext an F-lnear ogeny X Y uch that the Deudonne module attached to Y equal to π M M or M π M. Therefore Y omorphc to X/G, where G a ubgroup of X wth order p and δ (G) = or. Th G not potentally lftable by (b).

5 STRONG CM LIFTING PROBLEM II 5 Remark.4.. A a corollary, the anwer to queton (CML) relatve to (F, F ur ) for p-dvble group negatve; ee [5] (3..8) for the prece tatement of the queton. Note that the reflex feld F of Φ equal to F, o the redue feld κ F not mall n the ene of [5] (4.). Thu we obtan a new counterexample to queton (CML) that doe not fall n the framework n 4 of [5]. Corollary.5. Suppoe p >, L a CM feld and L t maxmal totally real ubfeld. If there ext a place v of L above p uch that the nerta degree of v and v ramfe n L, then the anwer to queton (CML) for abelan varete negatve. Proof. The completon L,v a degree unramfed extenon over Q p, and L v a degree ramfed extenon over L,v. It an eay exerce n number theory to how that when p >, L v B(F p )[π ]/(π p) or B(F p )[π ]/(π ɛ p), where ɛ a Techmuller lft n W(F p ) and not a quare. Then the tatement follow from (.4.), [5] (4.), and [5] (3..). The mot nteretng phenomenon revealed by Theorem (.3) that, no matter how arbtrary the ubgroup cheme G n charactertc, t reducton G eem to try very hard to be O F -table. It natural to ak the followng queton: Let F be a p-adc local feld, Φ be a prmtve p-adc CM type for F. Let F be the reflex feld of Φ. Let X be the O F -lnear CM p-dvble group wth p-adc CM type Φ over R := O F B(k). I there a general condton on the p-adc CM type Φ, uch that there ext an nteger d(φ) whch only depend on Φ, atfyng that for any fnte extenon R/R and any fnte locally free ubgroup cheme G of X R, the cloed fber G := G k contan an O F -table ubgroup wth ndex unformly bounded by p d(φ)? Remark.5.. (a) If we drop the aumpton that Φ prmtve, we can ealy produce a cla of fnte locally free ubgroup cheme G wth arbtrarly large order, uch that G doe not contan any nontrval O F -table ubgroup. In fact, uppoe Φ nduced from a p-adc CM type Φ for F F. Let X be the O F -lnear CM p-dvble group wth p-adc CM type Φ over R. Then X O F -lnearly omorphc to the Serre tenor contructon X OF O F. For any fnte locally free ubgroup cheme G of X, when we embed t nto X va the natural homomorphm X X, the cloed fber of G doe not contan any O F -table ubgroup of X. (b) When #Φ = or [F : Q p ], we can take d(φ) =. In fact, f G X a ubgroup, take a fltraton = G G G G = G, uch that the ndex of G n G + equal to p for =,,,. The a-number of each X/G equal to nce ether the dmenon or the codmenon equal to. Hence G + /G the unque ubgroup of X/G wth order p and G + /G mut be O F -table. Th prove every ubgroup G of X O F -table. (c) In the example we compute n th ecton, #Φ = and [F : Q p ] = 4. Th a frt nontrval example concernng th queton. A a corollary of Theorem (.3), we can ay d(φ) = n our example. In the ret of the ecton we prove Theorem (.3). The proof organzed a follow. For each potve nteger m, there ext a fnte extenon E m /Frac R uch that the p m -toron pont on X Qp are ratonal over E m. In (.6), we recall the comtructon from 5 of [5] on the Kn module M m attached to X OEm. A m run over all the potve nteger, we compute the cloed fber of the p m -toron fnte locally free ubgroup cheme G of X OEm. The fnte Kn module N attached to G a W(k)[[u]]-module, and the Deudonne module of the cloed fber G k N/(N (up m M m /M m )) (N + up m M m /M m )/(up m M m /M m ), whch we wll denote by N mod u n the future. At the end of ubecton (.6), we reduce the tatement n Theorem (.3) about the

6 6 TAISONG JING VERSION: /9/4 cloed fber G k to the extence of certan pecal element n N; ee (.6)(a), (b), and (c). On the other hand, the generator of the localzaton N := W(k)((u)) W(k)[[u]] N have been computed n [5] (5.). In (.7) we wrte thee generator nto explct form. In order to compute N mod u, we need to fnd a W(k)[[u]]-ba of N before the localzaton. Th can be vewed a an analogy of fndng a lattce n a vector pace. We how everal example n (.8), and then ummarze a general lnear algebra approach n (.9). Th approach uccefully compute the cloed fber G k n the cae when the geometrc generc fber of G generated by at mot two element; ee (.). The remanng eental cae when the geometrc generc fber of G generated by three element. In that cae, t dffcult to apply drectly the lnear algebra approach n (.9); ee the example (..3) at the end of (.). In (.), we explan how the Serre dual of X OEm come to recue for the problem. Fnally n (.) we compute the cloed fber G k va a detour by Serre dual n the cae when the geometrc generc fber of G generated by three element, and complete the proof of Theorem (.3). If M a Kn module (or a fnte Kn module), and x M := W(k)((u)) W(k)[[u]] M, we defne ord u x to be the mallet nteger d uch that u d x M. If ord u (x x ) D, we alo wrte x x mod ord u D..6. The Kn module attached to X and t bae change. Now we prepare to prove Theorem (.3). We frt recall the contructon from 5 of [5] on the Kn module attached to X, and t bae change to fnte extenon of R. Take h(x) = π x + x p. For all potve nteger r, defne h (r) (x) := h h h to be the r-th teraton of h, h r (x) := h(r) (x) h (r ) (x). For all potve nteger m, Let π m be a root of h m (x) n Q p, and defne E m := B(k)(π m ). Let E m (u) be the mnmal Eenten polynomal of π m over B(k); o E m (u) = h m (u)h m (u), where h m (u) the conjugate of h m (u) under π π. One can check the contant term of E m (u) equal to ɛ p. Let M m be the Kn module contructed a n 5 of [5] wth (E m, π m ), and let X m be the aocated p-dvble group over O Em. By [5] (5..7), X m the O F -lnear CM p-dvble group over O Em wth p-adc CM type Φ, and all the p m -toron pont on t geometrc generc fber are ratonal over E m. By [5] (3..5) and (5..7), X m omorphc to X OEm, and the omorphm nduce dentty over the cloed fber. Thu to prove Theorem (.3), t uffce to compute the cloed fber of p m -toron fnte locally free ubgroup cheme of X m when m run over all potve nteger. By [5] (5.), the Kn module M m = W(k)[[u]] Zp O F e wth the natural O F -acton, and the (φ, O F )-lnear endomorphm φ Mm (whch we wll abbrevate a φ m n the future) defned a φ m e = P Φ,πm,B(k) Qp F(u), the charactertc polynomal of the natural acton of π m on the W(k) Qp F-module (E m ) (E m ) τ, where the ndex ndcate the F-tructure. For the convenence of computaton, we dentfy W(k)[[u]] Zp O F e wth W(k),OF ur O F [[u]]e W(k),OF ur O F [[u]]e W(k)[π][[u]]/(π ɛ p)e W(k)[π][[u]]/(π ɛ σ p)e. Under uch an dentfcaton, one can check that a e = ae, a e = a σ e for a O F ur, and π e = πe. The (φ, O F )-lnear endomorphm φ m defned by φ m (e ) = τ (h m (u))e, φ m (e ) = τ (h m (u))e, where τ (rep. τ ) the W(k)[[u]]-omorphm from F B(k)[[u]] = B(k)[π ]/(π ɛ p)[[u]] to W(k)[π]/(π ɛ p)[[u]] (rep. W(k)[π]/(π ɛ σ p)[[u]]) that end π to π (rep. λ π). Let X m be the cloed fber of X m, let M(X m ) be the attached Deudonne module; by [] (B.4) M(X m ) M m /um m. If we tll ue e to tand for the mage of e n M(X m ), one can check that e, e a good O F -ba of M(X m ) (ee the begnnng of the ecton for the defnton of a good O F -ba of M(X m )). Now uppoe G a p m -toron fnte locally free ubgroup cheme of X m, #G = p t. Let N be the attached fnte Kn ubmodule. To prove Theorem (.3()), t uffce to how: (.6.a) When t = n even, there ext w (n) x W(k). N for =,, uch that w (n) x π n e mod u, where

7 STRONG CM LIFTING PROBLEM II 7 (.6.b) When t = n + odd, there ext w N uch that w x π (n+) e + x π (n+) e mod u, where x, x W(k), and x /x (F ) p or λ(f ) p. Here x p p mean the mage of x n k modulo p. Converely, to prove Theorem (.3()) t uffce to how: (.6.c) for every x, x W(k) uch that x /x (F ) p or λ(f ) p, there ext a potve nteger m and p p a p m -toron fnte locally free ubgroup cheme G of X m uch that #G = p n+ and we can fnd an element w n N atfyng w x π (n+) e + x π (n+) e mod u..7. The fnte Kn module attached to fnte locally free ubgroup cheme. To acheve the goal n.6, we need a prece decrpton on the fnte Kn module attached to p m -toron fnte locally free ubgroup cheme of X m. The endomorphm φ on W(k)[[u]] extend to φ : W(k)[π][[u]]/(π ɛ p) W(k)[π][[u]]/(π ɛ σ p), uch that φ W(k) = σ, φ(π) = π, and φ(u) = u p. Smlarly we can defne φ : W(k)[π][[u]]/(π ɛ σ p) W(k)[π][[u]]/(π ɛ p) n the ame way. Accordng to [5] (5..7), f we defne v := τ (h (m ) (u)) φ τ (h (m ) (u))e + τ (h (m ) (u)) φ τ (h (m ) (u))e then all the oluton x p m M m /M m to φ m x = ɛ E m(u)x have the form of η v wth η p m O F /O F. For any ubgroup A of p m O F /O F, let N A := W(k)((u)){η v η p m O F /O F }, and N A := N A p m M m /M m. Let G A be the aocated fnte locally free ubgroup cheme. When A run over ubgroup of p m O F /O F, G A enumerate all p m -toron fnte locally free ubgroup cheme of X m. Denote N A /(N A up m M/M) (N A + up m M m /M m )/(up m M m /M m ) by N A mod u, then N A mod u the Deudonne module of the cloed fber of G A. Now we derve a more prece formula for η v. By the defnton of h (m ) (u), we can wrte h (m ) (u) m π A (u) mod p m, uch that A (u) W(F p )((u)) and ord u A = p (m ). Therefore = v := τ (h (m ) (u)) φ τ (h (m ) (u))e + τ (h (m ) (u)) φ τ (h (m ) (u)) = ( m A n (u)(λ π) n ) φ ( m A n (u)π n )e + ( m A n (u)(λ π) n )( m A n (u)π n ) φ e n= = m n= n= π n ( n A k (u) φ A n k (u)λ kσ )e + m π n ( n A k (u) φ A n k (u)λ (n k) )e k= Recall that λ = ɛ p and ɛ a Techmuller lft, o λ +σ = ɛ p. Becaue ɛ W(F p ) \(W(F p ) ), we deduce ɛ p =. Hence we have λ σ = λ and we can then rewrte the above formula for v a: v = m Defnton.7.. Defne n= n= π n ( n A k (u) φ A n k (u)( λ) k )e + m (λ π) n ( n A k (u) φ A n k (u)λ k )e k= π := π b n := λ A (u) φ A n (u), y j := b j c j, Under the notaton above, v = m n= π n y ne + m n= n= n= k= k= n= π := τπ = λ π c n := λ + A + (u) φ A n (u) z j := b j + c j π n z ne. Now we derve a more prece formula of η v for η p m O F /O F. Let ν be the valuaton on F uch that ν(π) =.

8 8 TAISONG JING VERSION: /9/4 Defnton.7.. Suppoe η p m O F /O F and k the mallet nteger uch that η p k O F /O F. Let α W(F p ) and β W(F p ) be the unque element uch that η = p k (α + π β) (rep. η = p k π (α + π β)) when ν(η) = k (rep. ν(η) = k + ). Defne v[η, r, ] := ɛ k (αy ν(η) r + βy ν(η) r ), v[η, r, ] = ɛ k (λ k+ν(η) α σ z ν(η) r + β σ λ k+ν(η)+ z ν(η) r ) Under uch notaton, one can check η v = m = r= π r v[η, r, ]e ; when r > ν(η) we treat v[η, r, ] a zero. Th formula wll be refered to a the preentaton of η v n the future. Before we dve nto the computaton, let u look nto the defnton of the y, z and v[η, r, ], and derve ome properte of them. Propoton.7.3. Defne d := + p. The followng tatement about b, c, y, z are true: () b, c are both unt n W(k)((u)), and mn{ord u b, ord u c } = p 4m d, max{ord u b, ord u c } = p 4m ( p + p )d. () If mn{ord u b, ord u c } = ord u b (rep. ord u c ), then mn{ord u b +, ord u c + } = ord u c + (rep. ord u b + ). (3) y, z are both unt n W(k)((u)), and ord u y = ord u z = p 4m d. (4) u p4m d y ( ) [ + ] u p4m d z mod u. (5) v[η, r, ] a unt n W(k)((u)), and ord u v[η, r, ] = p 4m +ν(η)+r d; n partcular, t ndependent of and ncreang n r. (6) For any m, y z z y a unt n W(k)((u)), and ord u (y z z y ) = ord u y + ord u z = ord u z + ord u y = d(p 4m + p 4m ). (7) Let, j be dfferent nteger between and m, and uppoe γ W(F p ). Then γy y j ±γ σ λz z j, γz z j ± γ σ λy y j, and γy z j ± γ σ λz y j are all unt n W(k)((u)), and ther order are all equal to d(p 4m + p 4m j ). Proof. () and () are clear by a drect examnaton of each ummand n the defnton of b, c and ung the elementary lemma (.7.4) below. (3) becaue of (), and (4) follow from (). (5) clear by the defnton of v[η, r, ]. To ee (6), note that y z z y = (b c )(b + c ) (b + c )(b c ) = b c b c, then the tatement follow from () and (). To ee (7), when we expand them baed on b, c, b j, c j, the coeffcent of b b j, b c j, c b j, and c c j γ ± γ σ λ. If p γ ± γ σ λ, t mple that λ σ+ γ σ = mod p, contradcton to the fact that λ σ+ = ɛ p =. Moreover, by () there a unque term among b b j, b c j, c b j, and c c j that ha the lowet order, and th order equal to d(p 4m + p 4m j ). Th prove the tatement. Lemma.7.4. Let x = y + z, x, y, z W(k)((u)). If y a unt n W(k)((u)) and ord u z > ord u y, then x alo a unt n W(k)((u)) and ord u x = ord u y..8. Example of reducton of fnte locally free ubgroup cheme. We take th ubecton to compute a few example of N A and N A mod u. Example.8.. Let m, η p O F /O F, and A = η Z/p. Then N A = W(k)((u)){η v} p M/M. In the preentaton η v = π j v[η, j, ]e, we know v[η, j, ] and v[η, j, ] are both unt n W(k)((u)), = j= and ther order are both equal to p 4m +ν(η)+ j d. Let w := u p4m +ν(η)+ jd (η v), then w x, x W(k), and the goal of (.6.b) acheved. = x π e mod u for

9 STRONG CM LIFTING PROBLEM II 9 Example.8.. Let m, η (p O F /O F )\(p O F O F ), A = η Z/p. Let v := η v = and v := (pη) v = = j= 4 = j= π j v[η, j, ]e, π j v[pη, j, ]e. We want to produce w () and w () by a lnear combnaton of v, v wth coeffcent n W(k)((u)), uch that w () π e mod u. A natural canddate for w () gven by (v[pη,, ]v[η,, ] v[η,, ]v[pη,, ]) (v[pη,, ]v v[η,, ]v ). By the contructon of w () π e equal to (v[pη,, ]v[η,, ] v[η,, ]v[pη,, ]) = r=, w() 4 (v[pη,, ]v[η, r, ] v[η,, ]v[pη, r, ])e. It uffce to: (.8..a) Show v[η,, ]v[pη,, ] v[η,, ]v[pη,, ] a unt n W(k)((u)) and etmate t order (n u); (.8..b) For =, and r >, how ord u (v[pη,, ]v[η, r, ] v[η,, ]v[pη, r, ]) > ord u (v[pη,, ]v[η,, ] v[η,, ]v[pη,, ]) Wrte η = p (α + π β) or p π (α + π β) accordng to ν(η) = 4 or 3, where α W(F p ), β W(F p ). By the defnton of v[η, r, ] and v[pη, r, ], v[pη,, ]v[η,, ] v[η,, ]v[pη,, ] = ɛ 3 (αy ν(η) + βy ν(η) )(α σ λ +ν(pη) z ν(pη) + β σ λ 3+ν(pη) z ν(pη) ) ɛ 3 (α σ λ +ν(η) z ν(η) + β σ λ 3+ν(η) z ν(η) )(αy ν(pη) + βy ν(pη) ) = ɛ 3 αα σ λ +ν(η) (y ν(η) z ν(η) 3 z ν(η) y ν(η) 3 ) + Hgher order term By Propoton (.7.3)(6) and Lemma.7.4, t a unt wth order equal to d(p 4m++ν(η) + p 4m +ν(η) ). Now for =, and r, ord u v[η, r, ] dp 4m +ν(η)+r dp 4m+ν(η), ord u v[pη, r, ] dp 4m +ν(pη)+r dp 4m++ν(η). Hence ord u (v[pη,, ]v[η, r, ] v[η,, ]v[pη, r, ]) d(p 4m++ν(η) + p 4m+ν(η) ). Baed on the etmate above, we deduce that ord u (w () π e ) d(p 4m+ν(η) p 4m +ν(η) ). In partcular we have found w () uch that t reduce to π e modulo u. The dered w () can be contructed mlarly. Thu the goal of (.6.a) acheved. Example.8.3. Let m 3, η (p 3 O F O F )\(p O F O F ), and A = η Z/p 3. Take A := pη Z/p. By Example (.8.), we have contructed w () n N A N A uch that ord u (w () π e ) d(p 4m+µ(pη) p 4m +µ(pη) ). Defne w := u dp4m+ν(η) (η v v[η,, ]w () ), then we have w = 6 u dp4m+ν(η) v[η, r, ]π r e u dp4m+ν(η) = v[η,, ](w () = = r= π e ). The order of the econd term d(p 4m+ν(pη) p 4m +ν(pη) ) dp 4m+ν(η) >. Note that ord u v[η, r, ] ncreang n r and doe not depend on, hence u dp4m+ν(η) v[η,, ] are unt n W(k)[[u]] and ord u u dp4m+ν(η) v[η, r, ] > when r >. Thu we deduce w x W(k). Th acheve the goal of (.6.b). = x π e mod u, where Example.8.4. Let m, η, η p O F /O F, and A = η η Z/p Z/p. Let α W(F p ) and β W(F p ) be the unque element uch that w = p (α + π β ) or p π (α + π β ) dependng on ν(η ) = or. We may further aume that f ν(η ) = ν(η ), then α mod p, α mod p are F p -lnearly ndependent. In fact, f otherwe, there ext γ Z p uch that α γα mod p, then we can replace η wth η γη, to reduce to the tuaton when ν(η ) ν(η ). Wthout of lo of generalty we aume ν(η ) ν(η ).

10 TAISONG JING VERSION: /9/4 Defne w () := (v[η,, ]v[η,, ] v[η,, ]v[η,, ]) (v[η,, ](η v) v[η,, ](η v)). Then w () π e equal to (v[η,, ]v[η,, ] v[η,, ]v[η,, ]) (v[η,, ]v[η,, ] v[η,, ]v[η,, ]) We clam v[η,, ]v[η,, ] v[η,, ]v[η,, ] a unt n W(k)((u)), wth order equal to d(p 4m +ν(η) + p 4m +ν(η) ). To verfy th, we dvde the tuaton nto the cae when ν(η ) < ν(η ) and the cae when ν(η ) = ν(η ). When ν(η ) < ν(η ), then ν(η ) =, ν(η ) =. So v[η,, ]v[η,, ] v[η,, ]v[η,, ] = ɛ (α y + β y )α σ λz ɛ (α σ y + β σ λy )α y = ɛ (α α σ λy z α σ α y z ) + Hgher order term. By Propoton.7.3 (7), we ee the clam true. When ν(η ) = ν(η ), v[η,, ]v[η,, ] v[η,, ]v[η,, ] = ɛ (α y ν(η ) + β y ν(η ) )α σ z ν(η ) ɛ (α σ y ν(η ) + β σ λy ν(η ) )α y ν(η ) = ɛ (α α σ ασ α )y ν(η ) z ν(η ) + Hgher order term. Snce we have aumed α mod p, α mod p are F p -lnearly ndependent, (α α σ ασ α ) a unt n W(F p ), and the clam follow. So for =,, we have ord u (v[η,, ]v[η,, ] v[η,, ]v[η,, ]) ord u (v[η,, ]v[η,, ] v[η,, ]v[η,, ]) d(p 4m+ν(η) + p 4m +ν(η) ) d(p 4m +ν(η) + p 4m +ν(η) ) = d(p 4m+ν(η) p 4m +ν(η) ). In partcular, th mple w () reduce to π e modulo u. Smlarly we can fnd w () that reduce to π e modulo u, and the goal of (.6.a) acheved..9. Lnear algebra lemma. Now we ummarze a lnear algebra approach from the example we computed above. For a quare matrx C, we denote the entry on the -th row, j-th column by C[, j], and t cofactor by C, j. Lemma.9.. Suppoe A p m O F /O F, v, v,, v n are element n N A, and for each n we have a preentaton v = m v,r, π r e, where v,r, W(k)((u)). Defne an n n matrx = r= C := = v,n, v,n, v n,n, v,n, v,n, v n,n, v,n, v,n, v n,n, v,n, v,n, v n,n,. v,, v,, v n,, v,, v,, v n,, Suppoe det C W(k)((u)), and there ext a potve nteger D uch that ord u ( n v l,, j C,l ) ord u det C D for, =,, and j n +. Defne w (n) mod ord u D.. l= := n (det C) C,l v l for =,. Then w (n) l= Proof. By the defnton of C, one can check w (n) from the aumpton on the order of (det C) n C,l v l,, j. l= = π n e + = n j n+ l= N A and w (n) π n e (det C) C,l v l,, j π j e, then t follow

11 STRONG CM LIFTING PROBLEM II To apply Lemma (.9.), the key tep to how det C a unt n W(k)((u)), and etmate ord u det C. Wth th am, now we make ome defnton for matrce of pecal type that wll how up n our computaton, and etablh a few techncal lemma. Let R be a commutatve rng wth, and ord u : R Z be a dcrete valuaton on R; here we are not aumng that R the valuaton rng wth repect to ord u. Let k be a potve nteger, and C be a k k matrx wth entre n R. We denote the et of permutaton on {,,, k} by P k. Defnton.9.. We ay C domnated by the dagonal, f for any permutaton σ P k, k ord u (C[σ( j), j]) k ord u (C[ j, j]); f the nequalty trct, then we ay C trctly domnated by the dagonal. We ay C j= fathfully domnated by the dagonal, f C domnated by the dagonal, and ord u det C = k ord u (C[ j, j]). We ay C n parwe order, f for any par of (, j ), (, j ) wth <, j < j, ord u C[, j ] + ord u C[, j ] ord u C[, j ] + ord u C[, j ]; f the nequalty trct, then we ay C trctly n parwe order. In general, let J J Jt be a partton of {,,, k}, we ay C domnated by the dagonal block (J J J t ), f for any permutaton σ P k, there ext a permutaton τ uch that τ(j ) = J for =,,, t, and k ord u (C[σ( j), j]) k ord u (C[τ( j), j]); f the nequalty trct, then we ay C trctly domnated by j= j= the dagonal block (J J J t ). We ay C n parwe order relatve to partton (J J J t ), f for any par of (, j ), (, j ) uch that, j J r,, j J r wth r < r, we have ord u C[, j ] + ord u C[, j ] ord u C[, j ] + ord u C[, j ]; f the nequalty trct, then we ay C trctly n parwe order relatve to partton (J J J t ). The followng lemma traghtforward by the formula det C = σ P k ( ) gn(σ) j= j= k C[σ( j), j]. Lemma.9.3. Notaton a n Defnton (.9.). Then: (a) If C (trctly) n parwe order, then C (trctly) domnated by the dagonal. (b) If C trctly domnated by the dagonal, then C fathfully domnated by the dagonal. (c) If C (trctly) n parwe order relatve to partton (J J J t ), then C (trctly) domnated by the dagonal block (J J J t ). (d) If C trctly domnated by the dagonal block (J J J t ), and each block that cont of the row and column n J fathfully domnated by the dagonal, then C fathfully domnated by the dagonal... The proof of Theorem (.3) n the pecal cae. Let A be a fnte abelan p-group. Let r(a) be the larget potve nteger r uch that (Z/p) r can be embedded n A; th r(a) called the p-rank of the A. The p-rank of A alo the mallet nteger k uch that A can be generated by k element. Suppoe A a ubgroup of p m O F /O F, then we have r(a) r(p m O F /O F ) = 4. Let G := G A be the aocated p m -toron fnte locally free ubgroup cheme of X m. If r(a) = 4, then p O F /O F A, hence X[p] G. Th mple the ogeny X X/G factor through X p X, and the problem reduced to another fntely locally free ubgroup cheme wth a maller order. So we may aume r(a) 3. In th ubecton we prove Theorem.3 n the cae when G = G A uch that r(a). Suppoe A = η η, η (p m O F /O F )\(p m + O F /O F ) for =,. Suppoe #A = p t, then m + m = t. Wthout lo of generalty we aume ν(η ) ν(η ). Let α W(F p ) and β W(F p ) be the element uch that j=

12 TAISONG JING VERSION: /9/4 η = p m (α + π β ) or p m π (α + π β ), dependng on whether ν(η ) = m or m +. Defne the followng nteger aocated to A: L(A) := 4m + ν(η ) + t + Propoton... Notaton and aumpton a above. Then: (a) If t = n, then for =, and r =,,, n, there ext w (r) N A uch that ord u (w (r) π r e ) d(p L(A)+ p L(A) ). (b) If t = n +, then there ext w N A, uch that w π (n+) α e + ( ) c π (n+) α σ λm +ν(η ) e mod ord u d(p L(A)+ p L(A) ), where c = [ ν(η ) n ]. Before provng Propoton (..), we how that t mple Theorem (.3)() n the cae when G = G A uch that r(a), and alo mple Theorem (.3)(). It uffce to how the goal (.6) (a), (b), and (c) are acheved. (..)(a) obvouly mple (.6)(a). Recall that π = π, π = λ π, o the element w n (..)(b) can be wrtten a w α π (n+) e + ( ) c λ m +ν(η )+n+ α σ π (n+) e. Let x := α, x := ( ) c λ m +ν(η )+n+ α σ. Becaue and λ = ɛ p are both n (F p ) p, t then clear that x /x (F p ) p or λ(f p ) p. Thu (.6) (b) acheved. Concernng (.6) (c), f we let η =, η = p n α or p n π α where n run over non-negatve nteger and α run over W(F p ), then Propoton (..)(b) mple that for each [c, c ] P (k) uch that c /c (F ) p or λ(f ) p, there ext a fnte locally free ubgroup cheme G atfyng δ p p n (G k ) = [c /c ] n L. Therefore Theorem (.3)() proved once we prove Propoton (..). The plan to prove Propoton.. a follow: we apply Lemma (.9.) to prove (a). For (b), we knock out the unwanted entre n the preentaton of η v by ung the contructed lft of π r e, where =, and r =,,, n. Frt uppoe t = n. Defne an order on {p j η v =,, j =,,, m } uch that p j η v p j η v when: (a) ν(p j η ) < ν(p j η ); or (b) ν(p j η ) = ν(p j η ) and <. Let v l = p j l η l v be the l-th element n the et under th order. Then defne a matrx (cf. Lemma.9.) C := v[p j η, n, ] v[p j η, n, ] v[p j n η n, n, ] v[p j η, n, ] v[p j η, n, ] v[p j n η n, n, ] v[p j η, n, ] v[p j η, n, ] v[p j n η n, n, ] v[p j η, n, ] v[p j η, n, ] v[p j n η n, n, ].. v[p j η,, ] v[p j η,, ] v[p j n η n,, ] v[p j η,, ] v[p j η,, ] v[p j n η n,, ] If we delete the frt row of C and add the row of (v[p j η, r, ], v[p j η, r, ],, v[p j n η n, r, ]) on top of the remanng (n ) n matrx for =, and r n +, we denote the new n n matrx by C(, r, ). Smlarly, we can delete the econd row of C and add (v[p j η, r, ], v[p j η, r, ],, v[p j n η n, r, ]) on the top to get a new n n matrx; we denote t by C(, r, ). Propoton... Notaton a above, then the n n matrce C, C(, r, ), C(, r, ) are all fathfully domnated by the dagonal, for =,, r n +. In partcular, ther determnant are all unt n W(k)((u)). Proof. Defne a partton of {,,, n} = J J Jn where J := {, }. By the defnton of the matrce and the etmate on the order of ther entre by Propoton.7.3 (5), one can check all the matrce condered n the Propoton are trctly domnated by the dagonal block (J J J n ). Each

13 STRONG CM LIFTING PROBLEM II 3 dagonal block of C ha the form, or v[p j η, r, ] v[p j η, r, ] v[p j η, r, ] v[p j η, r, ]. By computaton mlar to thoe n Example.8. and Example.8.4, t traghtforward to check that thee block are all fathfully domnated by the dagonal. Therefore by Lemma.9.3 (d), the matrx C fathfully domnated by the dagonal. For C(k, r, ) where k =,, =,, and r n +, all the dagonal block are the ame a thoe of C except for the frt block on the upper left corner, and a drect examnaton of that block wll prove they are fathfully domnated by the dagonal, too. For the lat tatement, note that the matrce are trctly domnated by ther dagonal block (J J J n ), and the determnant of all the block are unt n W(k)((u)), by Lemma.7.4 we deduce that the determnant of the n n matrce C, C(, r, ), C(, r, ) are all unt n W(k)((u)). v[p j η, r, ] v[p j+ η, r, ] v[p j η, r, ] v[p j+ η, r, ], v[p j η, r, ] v[p j η, r, ] v[p j η, r, ] v[p j η, r, ] Now we are ready to prove Propoton... Proof of Propoton..: (a) In th cae m + m = n. Prove by nducton on n. Suppoe we have proved for all the ubgroup A p m O F /O F wth order equal to p n and n < n. Let A := pη pη f m >, and p η f m =. Then #A = p (n ) and L(A ) > L(A). By the nducton hypothe we have already produced w (r) for =, and r =,,, n. Now t uffce to produce w (n). Wth v l = p j l η l v for l =,,, n and matrx C defned before Propoton (..), we have hown det C W(k)((u)), o to apply Lemma (.9.) t reman to prove ord u ( n v[p j l η l, r, ]C k,l ) > ord u det C for k, =, and r n +. But n v[p j l η l, r, ]C k,l equal l= to det C(k, r, ), and by Propoton (..) ord u det C and ord u det C(k, r, ) are equal to the um of the order of ther dagonal entre, repectvely. By ther defnton one can check ord u det C(k, r, ) ord u det C d(p 4m +ν(η)+r p 4m +ν(η)+n ) d(p 4m +ν(η)+n p 4m +ν(η)+n ) = d(p L(A)+ p L(A) ). By Lemma (..), we deduce the extence of w (n) n N A uch that ord u (w (n) π n e ) d(p L(A)+ p L(A) ). (b) In th cae m + m = n +. By our aumpton ν(η ) ν(η ), o m > m. Let A := pη η, then #A = p n, hence by (a) we can produce w (r) w (r) N A N A for =, and r =,,, n, uch that n v[η, r, ]w (r) ), then we π r e mod ord u d(p L(A )+ p L(A ) ). Defne w := u dpl(a) (η v have w = m = r=n+ u dpl(a) v[η, r, ]π r e n = r= u dpl(a) v[η, r, ](w (r) l= = r= π r e ). The order of the econd term d(p L(A)+ p L(A) p L(A) ) d(p L(A)+ p L(A) ) becaue L(A ) L(A) +. In the frt term, note that ord u v[η, r, ] dp L(A)+ when r n +, and v[η, n +, ] α y ν(η ) n mod ord u dp L(A)+, v[η, n +, ] α σ λm +ν(η ) z ν(η ) n mod ord u dp L(A)+. Therefore the propoton follow from Propoton (.7.3)(4). Therefore to complete the proof of Theorem.3(), the remanng tuaton when r(a) = 3. In that cae, we wll meet dffculte f we tll try to apply Lemma (.9.) drectly, nce the crucal propoton (..) may no longer hold. Th phenomenon reflected by the followng example. Example..3. Let m, take α W(F p ) \Z p. Let η = p, η = p α, and η 3 = p π, and take A = η η η 3. The preentaton of η v are: η v = 4 y 4 r π r e + 4 z 4 r π r e, η v = 4 αy 4 r π r e + r= r= r= 4 r= α σ z 4 r π r e, and η 3 v = 3 r= y 3 r π r e + 3 λz 3 r π r e. If we follow the lnear algebra approach n (.9) r=

14 4 TAISONG JING VERSION: /9/4 and form the 6 6 matrx: One can check that C = y αy y z α σ z λz y αy y y αy z α σ z λz z α σ z y 3 αy 3 y y αy y z 3 α σ z 3 λz z α σ z λz det C λ(α α σ ) y z (y z z y ) mod ord u > d(p 4m + 4p 4m 3 ) However, y z z y = (b c ) b (b + c ) b = 4b c b ha order equal to d(3p4m + p 4m 4 ), hence ord u (y z (y z z y )) = d(3p4m + p 4m 3 + p 4m 4 ) > d(p 4m + 4p 4m 3 ). So ord u det C > d(p 4m + 4p 4m 3 ) = 6 ord u C[ j, j], n partcular the matrx C not fathfully domnated by the dagonal. j= However, a look nto the Serre dual of X m wll come to recue for th example. Recall that τ Gal(F/Q p ) the nvoluton on F, and the p-adc CM type Φ atfe Φ Φ τ = Hom(F, Q p ). Let ρ : O F End(X m ) be the O F -tructure on X m, f we defne the O F -lnear tructure ρ : O F End(X m) on the Serre dual X m by ρ (x) = ρ(ι(x)), then X m and X m are both O F -lnear wth the ame p-adc CM type. Snce the O F -omorphm cla of O F -lnear CM p-dvble group over R unquely determned by the p-adc CM type (ee [5] (3..3)), we know X m and X m are O F -lnearly omorphc. For a fnte locally free p m -toron ubgroup cheme G of X m, denote the Carter dual (X m [p m ]/G) by G ;m ; t a fnte locally free p m -ubgroup cheme of X m. In our example..3, take m =, let G = G A be the fnte locally free p -toron ubgroup cheme aocated to A, then one can check G ; a cyclc group of order p ; n partcular, t aocated wth a ubgroup A wth p-rank. Hence we can apply Theorem.3 to G ; n X, and deduce that G ; k not O F -table. That mple G k not O F -table, too. Thu, we can take a detour va the Serre dual X and reduce to the olved cae. To prove Theorem.3 n the general cae when G = G A where the p-rank of A equal to 3, we need more explct nformaton on G ;m. Th wll conttute the next ubecton... The Serre dual. Recall from (.6) that the Kn module M m attached to X m W(k)[[u]][π]/(π ɛ p)e W(k)[[u]][π]/(π ɛ σ p)e, and φ m (e ) = τ (h m (u))e, φ m (e ) = τ (h m (u))e, where τ (rep. τ ) the W(k)[[u]]-omorphm from F B(k)[[u]] = B(k)[π ]/(π ɛ p)[[u]] to B(k)[π]/(π ɛ p)[[u]] (rep. B(k)[π]/(π ɛ σ p)[[u]]) by endng π to π (rep. λ π). Let g (u), g (u) be the polynomal n W(k)[u] uch that h m (u) = g (u) + π g (u). Then n matrx form we can wrte g (u) ɛ pg (u) g (u) g (u) φ Mm (e, πe, e, πe ) = (e, πe, e, πe ) g (u) ɛ σ pλ g (u) λ g (u) g (u) The Kn module attached to X M m = Hom W(k)[[u]] (M m, W(k)[[u]]). If we denote the dual ba of {e, πe, e, πe } by {e, (πe ), e, (πe ) }, then φ M m (e, (πe ), e, (πe ) ) gven by g (u) g (u) (e, (πe ), e, (πe ) ɛ pg (u) g (u) ) ɛ g (u) λ g (u) ɛ σ pλ g (u) g (u)

15 STRONG CM LIFTING PROBLEM II 5 Here recall that ɛ p the contant term of the Eenten polynomal E m (u). Take µ W(k) uch that µ σ = ɛ. Defne ê := µ(πe ), πê := µe, ê := µ(πe ), πê := µe then M = W(k)[[u]][π]/(π ɛ p)ê W(k)[[u]][π]/(π ɛ σ p)ê, wth φ M ê = τ (h m (u))ê, φ M ê = τ (h m (u))ê. If we twt the natural O F -tructure on M m by ι,.e., defne a e = ae, a e = a σ e for a W(F p ), and π e = πe for =,, then the mappng that end e to ê an O F -lnear omorphm of Kn module from M m to M m. The natural W(k)[[u]]-blnear parng, : M m M m W(k)[[u]] a perfect parng that compatble wth the O F -tructure,.e., x v, w = v, x w for x O F, v M m, and w M m. The parng, : M m M m W(k)[[u]] naturally extend to (Q Z M m ) (Q Z M m ) B(k)[[u]]. For any potve nteger n, t nduce a parng, n : p n M m /M m p n M m/m m p n W(k)[[u]]/W(k)[[u]], by defnng v, w n := p n v, w. If N the fnte Kn module attached to a fnte locally free p n -toron ubgroup cheme G of X, then t orthogonal complement N ;n the fnte Kn module attached to G ;n (ee the end of Example (..3) for the defnton of G ;n ). The followng lemma allow u to extract the nformaton of N ;n from N, and vce vera. Lemma... Let D be a potve nteger, and l be an nteger between and n. Aumpton and notaton on N and N ;n are a above. (a) If N π l M m /M m mod ord u D, then N ;n π (n l) M m/m m mod ord u D; and vce vera. (b) If N π l M m /M m + W(k)[[u]] π (n l) M m/m m + and vce vera. = = µ π (l+) e mod ord u D wth µ W(k)[[u]], then N ;n µ π (n l) ê mod ord u D, where µ W(k)[[u]] atfy λµ µ + µ µ mod u; Proof. Frt look at (a). Let M m be the Deudonne module attached to X = (X m ) k. The Deudonne module attached to G k π l M m /M m, o the Deudonne module aocated to G ;n k the orthogonal complement of π l M m /M m under the nduced parng p n M m /M m p n Mm/M m p n W(k)/W(k), whch ealy een to be π (n l) Mm/M m. Therefore for π j ê wth =, and j =,,, n l, there ext ther lft n N ;n n the form of v, j = π j ê + n h, j,,r π r ê, where h, j,,r uw(k)[[u]]. Becaue they are orthogonal to = r=n l+ N, for each =, and j =,,, l, the parng π j e, v, j n mod ord u D. Take j =, th mple ord u h, j,,n D. Take j =, 3,, l nductvely, we deduce that ord u h, j,,r D for all, j,, r. Th prove (a). (b) can be proved n the ame way, only to notce that under our defnton of ê and ê, we have π (l+) e, π (n l) ê = λ π (l+) e, π (n l) ê. Propoton (..) ha the followng mmedate corollary: Corollary... If X[π ] contaned n G k wth ndex p, then X[π n ] contaned n G ;n k p, and vce vera. If that the cae, let δ (G k ) and δ n (G ;n k ) be the clae of G k and G ;n δ (G k ) = λδ n (G ;n k wth ndex k n L, then ). In partcular, δ (G k ) = [] or [λ] f and only f δ n (G ;n k ) = [] or [λ]. If we defne ˆv := τ (h (m ) (u)) φ τ (h (m ) (u))ê + τ (h (m ) (u)) φ τ (h (m ) (u))ê, then all the oluton x p m M m/m m to φ M m (x) = ɛ E m(u)x have the form η ˆv, η p m O F /O F. For any ubgroup A of p m O F /O F, defne N A := W(k){η ˆv η A}, and N A := N A p m M m/m m. Let Ĝ A be the aocated fnte locally free ubgroup cheme of X m, then they enumerate all p m -toron fnte locally free ubgroup cheme when A run over ubgroup of p m O F /O F. Now upppoe n m, A a ubgroup of p n O F /O F, and N A, G A are the

16 6 TAISONG JING VERSION: /9/4 correpondng p n -toron fnte Kn module and fnte locally free ubgroup cheme of X m. The defnton below provde a drect and concrete way to wrte down the ubgroup p n O F /O F attached to N ;n and G ;n. Defnton..3. Defne a ymmetrc Q p -parng on F a follow: a + bπ, c + dπ := (ad + bc) + (ad + bc) σ, a, b, c, d B(F p ) It nduce a ymmetrc parng p n O F /O F p n O F /O F p n Z/Z: a + bπ, c + dπ n := p n ((ad + bc) + (ad + bc) σ ) For any ubgroup A p n O F /O F, let A ;n be t orthogonal complement. Under the defnton above, when n m one can check (N A ) ;n = N A ;n, and hence G ;n A = Ĝ A ;n. Moreover, the followng propoton llutrate the relaton between the tructure of A and A ;n ; we leave the detal to reader. Defnton..4. Suppoe A p n O F /O F a ubgroup. for all potve nteger, denote the kernel of A π A by A[π ]. For =,,, n, defne R (A) := dm Fp A[π ]/A[π ] Snce dm Fp π O F/π ( ) O F =, we know R (A) can only take value,, or. Propoton..5. Suppoe A a ubgroup of p n O F /O F. Then we have: (a) If A 4 Z/p n wth n n, then A ;n 4 Z/p n n. = (b) R (A ;n ) + R n+ (A) = for all =,,, n. = Now we can prove Theorem.3 for G A n the cae when A Z/p Z/p Z/p j p m O F /O F, where j. In fact, by Propoton..5 we know A ; Z/p Z/p j ha p-rank at mot, hence Theorem (.3) for G A follow from Propoton (..) and Corollary (..). Explore th dea further we wll be able to prove Theorem.3 for G A n the general cae when the p-rank of A equal to 3 n the next ubecton... The proof of Theorem (.3) n the general cae. Suppoe G = G A a fnte locally free p m -toron ubgroup cheme of X, and A = 3 η, where η (p m O F /O F )\(p (m ) O F /O F ) wth m m m 3. = Suppoe #A = p t, o t = m + m + m 3. Let α W(F p ) and β W(F p ) be the element uch that η = p m (α + π β ) or p m π (α + π β ), dependng on whether ν(η ) = m or m +. By the argument at the end of the prevou ubecton, we may aume m > m. We may alo aume that X[π] G, otherwe the ogeny X X/G factor through X π X and we may reduce to a ubgroup cheme wth a maller order. Th aumpton tranlate nto R (A) <. Becaue we have aumed the p-rank of A 3 and the p-rank equal to R (A) + R (A), we deduce that R (A) = and R (A) =. Defnton... Aumpton on A are a above. Defne L(A) := 4m + ν(η ) + t+, and D(A) := d(p L(A) p L(A) ) f ν(η ) = m + and m = m + m 3 d(p L(A)+ p L(A). ) otherwe Propoton... Aumpton on A are a n the begnnng of the ubecton. Then: (a) If t = n, then there ext w (r) N A for =, and r =,,, n, uch that ord u (w (r) π r e ) D(A). (b) If t = n+, then there ext w N A uch that w π (n+) α e +( ) c π (n+) α σ λm +ν(η ) e mod ord u d(p L(A)+ p L(A) ), where c = [ ν(η ) n ].

17 STRONG CM LIFTING PROBLEM II 7 Before we prove Propoton.., we frt explan how to deduce Theorem.3() from t under the aumpton on A a n the begnnng of the ubecton. It uffce to prove (.6)(a) and (b). When m m + m 3, they follow mmedately from Propoton..; ee the argument after Propoton (..). In general we prove by nducton on m 3. When m 3 =, nce we have aumed m > m, we alway have m m + = m + m 3. Suppoe m 3 and we have proved the theorem for maller m 3. We may aume m m + m 3. Then A ;m Z/p m Z/p m m 3 Z/p m m by Propoton..5. But now m m < m 3, hence (.6)(a) and (b) follow from nducton hypothe and Corollary (..), and Theorem.3 proved. In the ret of th ubecton we prove Propoton... Once (a) proved, for (b) one can contruct w by knockng out the unwanted entre n the preentaton of η v by ung the contructed lft of π r e, where =, and r =,,, n. The argument mlar to that n the proof of Propoton.. (b) and left a an exerce. Now we look at Propoton.. (a). We pont out that t uffce to prove the cae when m = m + m 3. In fact, uppoe m m +m 3 and we have proved the clam for (m, m, m 3 ). Let A := p η η η 3, then we have already produced w (r) Defne v := η v v n = r= m = r=n N A N A for =, and r =,,, n by nducton hypothe. v[η, r, ]w (r), v := pη v n v[pη, r, ]w (r), then v, v N A and v[η, r, ]π r e, v = m r=n = r= v[pη, r, ]π r e mod ord u D(A ) Defne w () := (v[pη, n, ]v[η, n, ] v[η, n, ]v[pη, n, ]) (v[pη, n, ]v v[η, n, ]v ). One can check that v[pη, n, ]v[η, n, ] v[η, n, ]v[pη, n, ] a unt n W(k)((u)) and ha order d(p 4m +ν(η)+n + p 4m+ν(η)+n ). Snce ord u v[pη, n, ] ord u v[η, n, ] dp 4m +ν(η)+n, and (d(p 4m +ν(η)+n +p 4m+ν(η)+n ))+dp 4m +ν(η)+n + D(A ) D(A), we deduce that w () (v[pη, n, ]v[η, n, ] v[η, n, ]v[pη, n, ]) (v[pη, n, ] v[η, n, ] = m r=n v[pη, r, ]π r e ) mod ord u D(A) m = r=n v[η, r, ]π r e and t routne to check that the rght hand de further congruent to π n e modulo ord u D(A). Smlarly we can contruct w (n) N A uch that w (n) π n e mod ord u D(A), too. Thu the clam n Propoton.. (a) for (m, m, m 3 ) wll be proved. Therefore now we are reduced to the cae when m = m + m 3. We dvde the tuaton nto the cae when ν(η ) = m and ν(η ) = m +. We frt aume ν(η ) = m. Prove by nducton on m 3. Frt uppoe m 3 =, o m = m +. Defne A := pη η η 3 and A := η η. They are both ubgroup of ndex p n A. We wll produce two vector v and v from N A and N A, repectvely, and then produce the dered w (n) and w (n) by a lnear combnaton of v and v. By Propoton..5 (), A ;m = η η, wth η (p m O F /O F )\(p (m ) O F /O F ) and η (p (m ) O F /O F )\(p (m ) O F /O F ). Moreover, by Propoton..5 () we know R m (A ;m ) = R (A ) =, o ν( η ) = m. Wrte η = p m ( α + π β ), where α W(F p ), β W(F p ). Snce the p-rank of A ;m, by Propoton.., we deduce that N ;m A π (m ) M + W(k)[[u]] (π m α ê +( ) c π m α σê ) mod ord u D(A ;m ), where c = [ ν( η ) (m ) ]. By Lemma.. we deduce there ext v mod u. = x π (m +) e mod ord u D(A ;m ), where x W(k)[[u]] uch that λx α +( ) c x α σ