Traverso’s Isogeny Conjecture for <span class="mathjax-formula">$p$</span>-Divisible Groups

Traverso's Isogeny Conjecture for p-Divisible Groups.

MARC-HUBERTNICOLE(*) - ADRIANVASIU(**)

To Carlo Traverso, for his 60th birthday

ABSTRACT - Let k be an algebraically closed field of characteristic p>0. Let c;d2N. Letbc;d1be the smallest integer such that for any twop-divisible groupsHandH⁰overkof codimensioncand dimensiondthe following assertion holds: IfH[p^bc;d] andH⁰[p^bc;d] are isomorphic, thenHandH⁰are isogenous. We show thatbc;dˆl cd

c‡d

m. This proves Traverso's isogeny conjecture forp-di- visible groups overk.

1. Introduction.

Let p2N be a prime. Letk be an algebraically closed field of char- acteristic p. Let c;d2N. Let H be a p-divisible group over k of codi- mensioncand dimensiond; its height isc‡d. Itis well-known (see [Ma], [Tr1, Thm. 3], [Tr2, Thm. 1], [Va, Cor. 1.3], and [Oo3, Cor. 1.7]) that there exists a minimal numbern_H2Nsuch thatHis uniquely determined up to isomorphism byH[p^nH] (i.e., ifH⁰is a p-divisible group overksuch that H⁰[p^nH] is isomorphic toH[p^nH], thenH⁰is isomorphic toH). For instance, we have n_Hcd‡1 (cf. [Tr1, Thm. 3]). This implies that there exists a minimal numberbH2Nsuch that theisogeny classofHis uniquely de- termined by H[p^bH]. We call bH the isogeny cutoff of H. We have 1b_Hn_H. Traverso speculated the following (cf. [Tr3, § 40, Conj. 5]):

(*) Indirizzo dell'A.: University of Tokyo, Department of Mathematical Sciences, Komaba, 153-8914, Tokyo, Japan.

E-mail: nicole@ms.u-tokyo.ac.jp

(**) Indirizzo dell'A.: University of Arizona, Department of Mathematics, 617 N. Santa Rita Ave., P.O. Box 210089, Tucson, AZ 85721-0089, U.S.A.

E-mail: adrian@math.arizona.edu

CONJECTURE 1.1. The isogeny cutoff bH is bounded from above by l cd

c‡d

mi.e., we havebHl cd c‡d

m.

The goal of this paper is to prove an optimal variant of the Conjecture:

THEOREM1.2. Let c;d2Nand let bc;d1be the smallest integer such that for any two p-divisible groups H and H⁰over k of codimension c and dimension d the following assertion holds:If H[p^bc;d] and H⁰[p^bc;d] are isomorphic, then H and H⁰are isogenous. Then b_c;dˆl cd

c‡d

mand there- fore we have b_Hl cd

c‡d m.

In Section 2, we introduce notations and basic invariants which pertain top-divisible groups overkand which allow us to obtain a practical upper bound ofbH(see Corollary 2.13). In Section 3, we prove Theorem 1.2; see Example 2.10 for a simpler proof in the particular case whenHis isosimple.

If thea-number ofHis at most 1, then the inequalityb_Hl cd c‡d

mis in essence due to Traverso (cf. [Tr3, Thm. of § 21, and § 40]; see Theorem 3.1).

We refer to Theorem 2.2 and Example 3.2 for concrete examples with bHˆ1 and withbHˆl cd

c‡d

m(respectively).

2. Estimates of the isogeny cutoffb_H. Letr:ˆc‡dandj:ˆlcd

r

m2N. LetW(k) be the ring of Witt vectors with coefficients ink. Lets be the Frobenius automorphism of W(k) in- duced fromk. Let(M;f) be the (contravariant) Dieudonne module ofH.

ThusMis a freeW(k)-module of rankrandf:M!Mis as-linear en- domorphism such that we havepMf(M). We havecˆdim_k(f(M)=pM) anddˆdim_k(M=f(M)). Let w:M!M be the s ¹-linear Verschiebung map off; we havefwˆwfˆp1_M.

The DieudonneÂ-Manin classification ofF-isocrystals over k(see [Di, Thms. 1 and 2], [Ma, Ch. II, §4], [De, Ch. IV]) states that:

there exists m2N such that we have a direct sum decomposition Mh1

p i;f

ˆ ^m

iˆ1(Mi;f) into simpleF-isocrystals overk, and

there exist numbers c_i;d_i2N[ f0g which satisfy the inequality r_i:ˆd_i‡c_i>0, which are relative prime (i.e., g:c:d:(c_i;d_i)ˆ1), and for which there exists aB(k)-basis forM_iformed by elements fixed byp ^dif^ri.

The unique slope of (Mi;f) is ai:ˆd_i

r_i2[0;1]\Q. To fix ideas, we assume thata₁a₂ a_m. Eachi2 f1;. . .;mggives a slopea_iwith multiplicityr_i. The Newton polygonN_Hof His a continuous, piecewise linear, upward convex function N_H:[0;r]![0;d] which for all i2 f1;. . .;mghas slopea_ion the interval ⁱP¹

`ˆ1r<sub>`</sub>;Pⁱ

`ˆ1r<sub>`</sub>

. As t he fieldkis algebraically closed, the isogeny class of His uniquely determined by the Newton polygonN_H. The finite setN_c;dof Newton polygons we thus obtain by varyingH, can be partially ordered as follows: we say thatN1

lies above (resp. strictly above) N2 if for all t2[0;r] we have N1(t) N2(t) (resp. if for all t2[0;r] we have N1(t) N2(t) and moreoverN₁6ˆ N₂). This partial order is convenient when studying the variation of the Newton polygon in families. We will use the notationN to denote the Newton polygon of, whereis ap-divisible group over a field that contains k.

Let H_ci;di be the p-divisible group over kwhose Dieudonne module (M_ci;di;f_ci;di) has the property that there exists aW(k)-basisfe₀;. . .;e_{ri 1}g for M_ci;di such that we have f_ci;di(e_l)ˆe_di‡l if l2 f0;. . .;c_i 1g and f_ci;di(e_l)ˆpe_di‡lifl2 fc_i;. . .;r_i 1g(here the lower right indices ofeare taken modulo ri). Itis well-known thatHci;di has the following two properties (see [dJO, Lem. 5.4]): (i) it has codimension c_i, dimension d_i, and unique Newton polygon slope a_i, and (ii) its endomorphism ring End(H_ci;di) is the maximal order in the simple Q_p-algebra End(H_ci;di)

Zp

Q_pof invarianta_i2Q=Z. In addition, these two properties determine H_ci;di up to isomorphism (see [Ma] and [Oo4, § 1.1]). Let H₀:ˆQ^m

iˆ1H_ci;di; we have N_H0 ˆ N_H. The Dieudonne module of H₀ is (M0;f₀):ˆQ^m

iˆ1(Mci;di;f_ci;di).

DEFINITION 2.1 (Oort). We say that a p-divisible group H over k is minimal if it is isomorphic to the p-divisible group H₀ determined un- iquely from the Newton polygonN_Hof H.

Thep-divisible groupH₀is theunique(up to isomorphism) minimalp- divisible overkthat is isogenous toH.

THEOREM2.2. A minimal p-divisible group H over k is determined up to isomorphism by H[p](thus we have b_Hˆn_Hˆ1).

PROOF. For the isoclinic case (i.e., when the pairs (ci;di) do notdepend oni) we refer to [Va1, Exa. 3.3.6]. For the general case we refer to either

[Oo4, Thm. 1.2] or [Va2, Thm. 1.8]. p

DEFINITION2.3. By the minimal height q_Hof a p-divisible group H over k we mean the smallest non-negative integer q_Hsuch that there exists an isogenyc₀ :H!H0to a minimal p-divisible group, whose kernelKer(c₀) is annihilated by p^qH.

DEFINITION2.4. By the a-number a_Hof a p-divisible group H over k we mean the number dim_k(a_p;H[p])ˆdim_k(M=f(M)‡w(M))2N[ f0g, wherea_pis the local-local group scheme of order p.

See [Oo2, Prop. 2.8] for a proof of the following specialization trick.

PROPOSITION2.5. For everyp-divisible groupHoverk, there exists ap- divisible groupHoverk[[x]]that has the following two properties:

(i) its fibre over k is H;

(ii) if k((x))is an algebraic closure of k((x)), thenH_k((x))has the same Newton polygon as H and its a-number is at most1.

DEFINITION2.6. By the weak isogeny cutoff of a p-divisible group H over k we mean the smallest number~b_H2Nsuch that the following two properties hold:

(i) if H⁰ is a p-divisible group over k such that H⁰[p^~bH] is iso- morphic to H[p^~bH], then its Newton polygonN_H⁰is not strictly aboveN_H; (ii) there exists a p-divisible groupHover k[[x]]that has the fol- lowing three properties:

(ii.a) its fibre over k is H;

(ii.b) the fibreH_k((x))over k((x))has the same Newton polygon as H;

(ii.c) the isogeny cutoff of H_k((x)) is at most~b_H and the a-number of H_k((x))is at most1.

Note that property (i) holds for any level greater or equal tob_H. Thus the existence of the number~b_H2Nis implied by Proposition 2.5.

FACT2.7. If H^tis the Cartier dual of H, then qHˆq_H^t, bHˆb_H^t, and

~b_Hˆ~b_H^t.

PROOF. This follows from Cartier duality (the Cartier dual of Hci;diis

Hdi;ci). p

LEMMA2.8. The isogeny cutoff b_His the smallest natural number such that for every element g2GL_M(W(k)) congruent to 1_M modulo p^bH, the Dieudonne module(M;gf)is isogenous to(M;f).

PROOF. Lett2N. The arguments proving either [Va1, Cor. 3.2.3] or [NV, Thm. 2.2 (a)] show that: (i) ifg2GL_M(W(k)) is congruentto 1_Mmodulo p^t, then every p-divisible group H_g over k whose Dieudonne module is isomorphic to (M;gf), has the property thatH_g[p^t] is isomorphic toH[p^t], and (ii) if ap-divisible groupH⁰overkhas the property thatH⁰[p^t] is iso- morphic toH[p^t], then there existsg2GL_M(W(k)) congruentto 1_Mmodulo p^tand such that the Dieudonne module ofH⁰is isomorphic to (M;gf). The Lemma follows from these two statements and the very definition of the

isogeny cutoffb_H. p

LEMMA2.9. For every p-divisible group H over k, the isogeny cutoff bHis bounded from above by the minimal height q_H plus 1 i.e., we have b_Hq_H‡1.

PROOF. Letc:H!H0 be an isogeny whose kernel is annihilated by p^qH. Let(M0;f₀),!(M;f) be the monomorphism of Dieudonne modules associated to the isogenyc; we will identifyM₀ with its image under this monomorphism. We have p^qHMM₀M. Let g2GL_M(W(k)) be con- gruentto 1_Mmodulop^qH‡1. We writegˆ1_M‡p^qH‡1e, wheree2End(M).

We have p^qH‡1e(M0)p^qH‡1e(M)p^qH‡1MpM0. Thus we have g2GLM0(W(k)) and moreovergis congruentto 1M0 modulop. Therefore the reductions modulopof the two triples (M0;f;w) and (M0;gf;wg ¹) co- incide. As the Dieudonne module (M₀;f) is minimal (being isomorphic to (M₀;f₀)), from Theorem 2.2 we get that it is isomorphic to (M₀;gf). The isogeny class of (M;gf) (resp. of (M;f)) is the same as of (M0;gf) (resp. of (M0;f)). From the last two sentences we get that the Dieudonne modules (M;gf) and (M;f) are isogenous. From this and Lemma 2.8, we get that

b_Hq_H‡1. p

EXAMPLE2.10. Suppose thatmˆ1 i.e.,His an isosimplep-divisible group. Let u0:H0!H0 be an endomorphism such that End(H0)ˆ

ˆW(Fp^r)[u0] and u^r₀ˆp, cf. [Ma, Ch. III, § 4, 1] and [dJO, Lem. 5.4].

From [dJO, 5.8] we get that there exist inclusions (M₀;f₀),!(M;f),!

,!(u₀^{(c 1)(d 1)}M0;f₀) between Dieudonne modules over k. Let c₀:H!H0 be the isogeny defined by the first inclusion (M0;f₀),! ,!(M;f). Its kernel is annihilated by u^(c₀ ^{1)(d 1)} ˆp^{(c 1)(dr 1)} and therefore we haveqHl(c 1)(d 1)

r

mˆj 1. ThusbHj, cf. Lemma 2.9.

REMARK2.11. The functionf :(0;1)(0;1)!(0;1) defined by the rulef(x;y)ˆ xy

x‡yis subadditive i.e., for allx1;x2;y1;y22(0;1), we have an inequality f(x1;y1)‡f(x2;y2)f(x1‡x2;y1‡y2). But the function g(x;y):ˆ df(x;y)eis not subadditive. Due to this and the plus 1 part of Lemma 2.9, our attempts to use Lemma 2.9 in order to prove Theorem 1.2 by induction onm2N, failed. On the other hand, in most cases Lemma 2.9 provides better upper bounds ofbHthan those provided by the following Proposition.

PROPOSITION2.12. For everyp-divisible groupH overk, the isogeny cutoffb_His bounded from above by the weak isogeny cutoffb~_Hi.e., we have b_H b~_H.

PROOF. Letk((x)) be an algebraic closure ofk((x)). LetHbe ap-divisible group overk[[x]] of constant Newton polygon such that its fibre overkisH and the isogeny cutoffbofH_k((x))is atmost~b_H, cf. property (ii) of Definition 2.6. We recall that the statement thatHhas constant Newton polygon means that the Newton polygons ofHandH_k((x))coincide i.e.,NHˆ NH_k((x)). For the proof of this Proposition it is irrelevant what thea-number ofH_k((x))is.

LetH⁰be ap-divisible group overksuch thatH⁰[p^~bH]ˆH[p^~bH]. Based on [Il, Thm. 4.4. f)] we getthatfor alln2N, there exists ap-divisible group H⁰_n over Spec(k[[x]]=(xⁿ)) that lifts H⁰ and such that H⁰_n[p^~bH]ˆ

ˆ H[p^~bH]k[[x]]k[[x]]=(xⁿ). Based on loc. cit., we can assume that H⁰_n‡1 restricted to Spec(k[[x]]=(xⁿ)) isH⁰_n. Therefore theH⁰_n's glue together to define ap-divisible groupH^0fover the formal scheme Spf(k[[x]]). We recall that the categories ofp-divisible groups over Spf(k[[x]]) and Spec(k[[x]]) are naturally equivalent, cf. [dJ, Lem. 2.4.4]. Thus letH⁰be thep-divisible group over Spec(k[[x]]) that corresponds naturally toH^0f.

Thep-divisible groupH⁰liftsH⁰and we have

H⁰[p^~bH]ˆproj:lim:_n!1H⁰_n[p^~bH]ˆ H[p^~bH]:

AsH⁰[p^~bH]ˆ H[p^~bH] andb~bH, from the very definition ofb, we getthat

H⁰_k((x))has the same Newton polygon asH_k((x)); thusN_H⁰

k((x))ˆ NH. As t he Newton polygons go up under specialization (see [De, Ch. IV, §7, Thm.]), we conclude that the Newton polygon N_H⁰ of H⁰ is above the Newton polygon NH of H. But NH⁰ is notstrictly above NH, cf. property (i) of Definition 2.6. From the last two sentences we get that NH⁰ ˆ NH. This

implies thatb_H~b_H. p

From Lemma 2.10 and Proposition 2.12 we get:

COROLLARY2.13. For everyp-divisible group H overk, we have the following inequalitybHminfb~H; qH‡1g.

3. The proof of Theorem 1.2.

In this Section we prove Theorem 1.2. We begin by proving the fol- lowing particular case of Conjecture 1.1 which in essence is due to Traverso (cf. [Tr3, Thm. of § 21, and § 40]).

THEOREM3.1. Suppose that a_H1. Then we have b_Hj:

PROOF. We first recall how to compute the Newton polygonNHofH(cf.

[Ma], [De, Ch. IV, Lem. 2, pp. 82-84], [Tr3, §21, Thm.], and [Oo1, Lem. 2.6]).

Let v_p:W(k)n f0g !N[ f0;1g be the normalized valuation of W(k);

thusv_p(p)ˆ1. Letx2Mbe such that its reduction modulopgenerates the k-vector space M=f(M)‡w(M). Let a0;a1;. . .;ac;b1;. . .;bd2W(k) be such that the map

c:ˆX^c

iˆ0

a_{c i}fⁱ‡X^d

`ˆ1

b<sub>`</sub>w<sup>`</sup>:M!M

annihilates x. It is easy to see that we can choose x such that vp(a0)ˆvp(bd)ˆ0 andfx;f(x);. . .;f^{c 1}(x);w(x);. . .;w^d(x)gis aW(k)-basis for M. For `2 f1;. . .;dg let ac‡`:ˆp<sup>`</sup>b`; thus a_i is well defined for all i2 f0;. . .;rg. Let

Qx(t):ˆt^r‡ X

i2f1;...;rg ai6ˆ0

vp(ai)t^{r i}ˆt^r‡ X

i2f1;...;rg ai6ˆ0

vp(s^d(ai))t^{r i}2Z[t]:

We recall that the Newton polygon ofQ_x(t) (more precisely, of the (r‡1)- tuple (a₀;. . .;a_r)) is the greatest continuous, piecewise linear, upward convex function N_x:[0;r]![0;d] with the property that for alli2 f0;. . .;rgwe

haveNx(i)vp(ai). Then we have

N_Hˆ N_x: (1)

To check Formula (1) we viewMas aW(k)[F]-module, whereFlˆl^sF and whereFacts onMasfdoes. TheW(k)[F]-moduleMis isogenous to the W(k)[F]-moduleM⁰:ˆW(k)[F]=W(k)[F]c⁰, where

c⁰ˆF^dc:ˆX^r

iˆ0

s^d(ai)F^{r i}:

But the Newton polygon of theW(k)[F]-moduleM⁰isN_x, cf. [De, Ch. IV, Lem. 2, pp. 82-84]. Thus Formula (1) holds.

We recall thatjˆlcd r

m. Asc;d>0, we havej1. Letg2GL_M(W(k)) be congruentto 1_Mmodulop^j. LetHgbe ap-divisible group overkwhose Dieudonne module is isomorphic to (M;gf). Asj1,H_g[p] is isomorphic to H[p] and therefore a_Hg ˆa_H1. Based on Lemma 2.8, to prove the Theorem it suffices to show that the Newton polygons NHg and NH

coincide. By replacing (f;w) with (gf;wg ¹), the map c:M!M which annihilatesxgets replaced by another map

c_g:ˆX^c

iˆ0

ag;c ifⁱ‡X^d

`ˆ1

bg;`w<sup>`</sup>:M!M

which annihilatesx, wherea_{g;c i}is congruenttoa_{c i}modulop^jand whereb_g;`

is congruentto b` modulo p<sup>j</sup>. For `2 f1;. . .;dg let ag;c‡`:ˆp<sup>`</sup>bg;`; t hus ag;i2W(k) is well defined for all i2 f0;. . .;rg and itis congruentto ai

modulo pj‡minf0;i cg. The pair (Qx(t);Nx) gets replaced by the pair (Q_g;x(t);N_g;x), whereQ_g;x(t):ˆt^r‡ P

i2f1;...;rg ai6ˆ0

v_p(a_g;i)t^{r i}and whereN_g;xis the Newton polygon of Q_g;x(t). As N_x is upward convex and as N_x(0)ˆ0 and N_x(r)ˆd, we have N_x(t)dt

r for all t2[0;r]. Thus jˆlcd r

m

dNx(c)e Nx(c) Nx(i) for all i2 f0;. . .;cg. Asr>d andjcd r, for i2 fc‡1;. . .;rg we have j‡i cdi

r. From the last two sentences we getthatj‡minf0;i cg di

r Nx(i) for all i2 f0;. . .;rg. From these inequalities and the fact that for all i2 f0;. . .;rg the elementsa_g;iand a_iare congruentmodulopj‡minf0;i cg, we easily getthatN_g;xˆ N_x. From this identity and Formula (1) we get thatN_Hgˆ N_H. p

3.1 ±End of the proof of Theorem1.2.

Based on Proposition 2.12, to prove the inequalityb_Hjitsuffices to show that~b_Hj. We recall thatN_c;dis the set of Newton polygons ofp- divisible groups overkof codimensioncand dimensiond. LetD_H be the class ofp-divisible groups of codimensioncand dimensiondoverkwhose Newton polygons are strictly aboveNH.

We prove the inequality~b_Hjby decreasing induction onN_H2 N_c;d. Thus we can assume that for every 2 D_Hwe have~b j; asb ~b (cf.

Proposition 2.12) we also have bj. To show that~b_Hj, letHbe ap- divisible group over k[[x]] such that the properties (ii.a) to (ii.c) of Defi- nition 2.6 hold. Letbbe the isogeny cutoff of H_k((x)). From the very defi- nition of~b_Hwe getthat

~b_Hmaxfb;bj 2 D_Hg:

Asbj(cf. Theorem 3.1 applied toH_k((x))) and asbjfor all 2 D_H(cf.

the inductive assumption), we have~b_Hj. This ends the induction.

Thus~bHjand thereforebHj. AsHis an arbitraryp-divisible group overkof codimensioncand dimensiond, we getthatb_c;dj. Ifjˆ1, then we obviously have b_c;dˆ1. If j2, then the below Example shows that there existp-divisible groupsHoverkof codimensioncand dimensiond and such that we haveb_Hˆj. This implies that forj2, we haveb_c;dj.

We conclude that for all values ofj2Nwe haveb_c;dˆjˆlcd r

m. p

EXAMPLE3.2. Suppose thatj2. Lets2Nbe the smallest number such thatrdividescd s; we have j 1ˆcd s

r . We assume that there exists an element x2M such that the r-tuple (e1;. . .;er):ˆ

:ˆ(x;f(x);. . .;f^{c 1}(x);w^d(x);. . .;w(x)) is an ordered W(k)-basis for M and we have an equality w^d(x)ˆf^c(x). Thus f(e_i)ˆe_i‡1 if i2 f1;. . .;cg andf(e_i)ˆpe_i‡1ifi2 fc‡1;. . .;rg. We havef^r(e_i)ˆp^de_iand therefore all Newton polygon slopes ofHared

r; thusmˆg:c:d:(c;r). Replacing the equality w^d(x)ˆf^c(x) by the equality w~^d(x)ˆ~f^c(x) p^{j 1}x, we geta p- divisible group H~ over k whose Dieudonne module (M;~f) is such that f(e~ i)ˆei‡1 ifi2 f1;. . .;c 1g,~f(ec)ˆp^{j 1}e1‡ec‡1, and f(e~ i)ˆpei‡1 if i2 fc‡1;. . .;rg. As (~f;w) and (f;~ w) are congruentmodulop^{j 1},H[p~ ^{j 1}] is isomorphic to H[p^{j 1}]. We have NH(t)ˆdt

r butthe piecewise linear functionNH~(t) changes slope atc; more precisely we haveNH~(c)ˆj 1

(cf. Formula (1) applied to H, t o~ x2M, to the function

~

c:ˆ p^{j 1}f~⁰‡~f^c w~^d:M!M, and to the polynomial Q~_x(t):ˆt^r‡

‡(j 1)t^d‡d). Thus the Newton polygon slopes of H~ are j 1 c (with multiplicity c) and 1 j 1

d (with multiplicity d) and therefore they are differentfromd

r. This implies thatbHjand therefore (cf. the inequality bHjproved in the previous paragraph) we havebHˆj.

EXAMPLE 3.3. Suppose that cˆd and that His as in Example 3.2.

Then qHˆc 1

2 (cf. [NV, Rmk. 3.3]) and jˆlc 2

m. If c is odd, then

b_Hˆjˆc‡1

2 ˆq_H‡1; moreover, for H~ :ˆ(Q_p=Z_p)Hm_p1 we also havebH~ ˆjˆc‡1

2 ˆqH~ ‡1. Thus forcˆd, the inequalitybHqH‡1 (see Lemma 2.9) is optimal in general.

Acknowledgments. The first author has been supported by the Japa- nese Society for the Promotion of Science (JSPS PDF) while working on this paper at the University of Tokyo. The second author would like to thank University of Arizona for good conditions in which to write this note.

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Manoscritto pervenuto in redazione il 20 maggio 2006.