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(1)On an isogenous invariant of a p-divisible formal group" Dedicated to Professor Yoshikazu Nakai on his sixtieth birthday .. By. Keisuke TOKI 4. (Received May 30, 1980). Throughout this paper, let k be an algebraically closed fields of positive characteristic P. Let G, M(G) and E(G) be a P-divisible k-formal group, the. Dieudonn6-module of G and the F-space over k in which M(G) is a F-lattice over le respectively in the sense of Demazure [1]. Moreover, according to [1], let G2, Z==s/r (r & s are integers satisfying rl.)s;-})O and (r, s)==1) be the p-divisi-. ble k-formal group whose Dieudonn6-module M(G2) is VV(le).[F, V]/(Fr"s-vs), where VV(le) is the long Witt-vectors ring with coefficients in k and VV(le).[F, V]. is the non-commutative ring consisting of polynomials relative to F, V with. coefficients in VV(k) defined by relations FV==VF =P, Fw=w(P)F, Vw(P)=wV for all w in Hi(k). Then, due to Manin-Demazure (cf. [1], [4]), we known that G is expressed uniquely up to isogeny as the product-type [[[(Gk)M2(or that. R. the F-space E(G) is isomorphic to the direct-sum OE(Gk)M2). Let M,(G) be. 2 the F-lattice OM(G2)M2 in the F`space OE(G2)M2, and Fo be the F-map of. 22. M,(G). By considering the reduction of Mo(G) {L'Mo(G) via the reduction l)V(le) .. .. -k, we construct the p-th power semilinear endomorphism Vo(G)4, V,(G) of a k-vector space Vo(G) whose dimension is equal to the height of G. And moreover we construct the normal form matrix mG of fo by following the construction of the normal form of matrix A in Hasse-Witt [2], g3, Satz 10 & 11, and remark that the matrix mG is an isogenous invariant of G. Especially, for p-divisible F.-formal groups G's obtained by base-changes to F, of p-divisible groups of abelian varieties defined over a finite field of characteristic p, it is shown that the isogeny among these G's is characterized by the .equality among nta's, when simple factors of the product-type up to isogeny for each of these G's are of the same height except factors with height one or two. Here F- . denotes. the algebraic closure of prime field F. of characteristic P. Additionally we shall show that a p-th power semilinear endomorphism (of a positive finjte dimensional k-vector space) equipped with some conditions comes from a pdivisible le-formal group. Finally, as to our normal form matrix in the case of " Appeared in Sci. Rep. Yokohama Nat. Univ. Sec. I, No. 27, Nov., 1980..

(2) 28 K. ToKi. the P-divisible group of the jacobian variety of a complete non-singular curve defined over a finite field of characteristic p, we observe a connection with the. normal form of Hasse-Witt matrix A for that curve in the case of genus at most five.. Sl. P"-th power semilinear endomorphisms and their normal form matrices. Let V be a positive finite dimensional le-vector space, and N be an arbitrary. non-zero integer. Then we call a group-endomorphism f of V satisfying f(RIxX)==2P"f(v) for all 2 in le, all v in V a P"-th Power semilinearendomorPhism. over k of V. For such an endomorphism f, [V]f denotes the subspace of V generated over le by all vectors v satisfying f(v)==v and [V]f".ii denotes the /) are killed by a power of f. subspace of V consisting of all vectors which PRoposlTIoN 1. Let N, V be as above and f be a' P"-th Power semi-linear. endomorPhism over fe of V. VVe Put n==dim V. Then we obtain that the subspace [V]f has a le-basis consisting of vectors fixed by the action off, the direct-. sum decomPosition V=[V]fO[V]f-.ii which is comPatible zvith the action of f holds, and the subsPace [V]f-.ii coincid2s with the kernel of fn: V-V.. PRooF. We consider Fp as the algebraic closure of Fp in k. Let Fpwi be the finite subfield of F, with Pi"i elements. At first we shall verify that, for a finite number vectors vi, ･･･, vt satisfying f(v,･)= vj (7'=1, ･･･, l), the linearly. independency over Fp,M implies the linearly independency over fe, In fact, suppose that the set {vi, -･･vt} is linearly dependent over le. Among the v's, we choose a maximal subset of linearly independent elements over fe. Let, say {vi, ･･･,v.} (m<l) be so. Then the vt is written by vi== $I) Ri,vj (each Rtjciik).. J'--1. Since f(vJ･)=vj (j'=1, ･･･,l), we have jM..,2tjvj= j=M,22.･"v,k Therefore 1,j--R￡." for Vl'. , and hence 2t,(iFpiNi for Vl'. Then the set {vt, vi, ･･･,v.} is linearly depend-. ent over Fpu,Ni. Let [V]f be not zero. Then there exists a non-zero vector vo fixed by the action of f. If [V]f is not generated over fe by vo, then there exists a vector v6 fixed by the action of f such that the $et {vo, v6} is linearly. independent over F,LNi. If [V]f is not generated over k by {vo, v6},then there exists a vector v6' fixed by the action offsuch that {vo, v6, v6'} is linearly independent over F,,Ni. By repeating this procedure, we see that [V]f has a le-basis consisting of vectors fixed by the action of f. So we have also [V]f. A[V]f..ii=={O}. Now we consider V as a left le[T].-module defined by the action T･=f, where le[T]. is the non-commutative ring consisting of polynomials relative to T with coefficients in k defined by relations T2==2P"T for all 2 in. k. Let v be an arbitrary non-zero element of V. We put Iv={r(T)Gle[T]al r(T)･v=O}. Then I. is a principal left ideal of fe[T]. and the orbit k[T].v of v is a non-zero left k-module isomorphic to k[T]./I.. Since k[T].v is a subspace of the finite dimensional fe-vector space V, the ideal J. is not zero. '.

(3) On an isogenous invariant of a P-divisble formal group 29 and moreover I.l(1) by vlO. Then the left ideal I. is generated by an unique. d-1 (dllll, each ajEle). Let be ad ==1 and element of the form a(T)==Td+2ajTj J---O. be r=min {7'1aj40, O;Sj<=d}. For r:Sv;$d, tEi le, we put afuv'(t)=,t:. a9-(""j'Ntp("-j)". (if N>O);= S)ag("-""tp(""-d-"" (if N<o), and put aN(t, T)==`S'liiicrfuv'(t)T". j==r v=rr. (if r<d);==O (if r=d). Since afuv'i'(t) = v2) (a9･(""''N)PN(tp("-"'N)p"+a..,t (if N>O);. j=r ;Si)(af("-i')N)pN(tp("'r-clHJ')N)pN+a..,tp('-d)N (if N<O), we have afu"'i)(t)-. =. j==r. (crfuv)(t))p"=tpe("'"'a..,, where e(N, v)=O (if N>O); =N(r-d) (if N<O). There-. ,d-1 fore, in the case of r<d, we have (1-T)･aN(t, T)+afud)(t)･Td==2crfuv)(t)T" v= r. `. d-1 -2(afu")(t))P"･Tv+i+crfud)(t)Td==afu')(t)Tr-[crpu.)(t))p"･Tr+i-aW.+i)(t)･Tr+i]-･･･v=r. [(aWdri)(t))PN.Td-crfud)(t).Td]..a.tPe(N'").Tr+tPe(N'").a..,Tr+i+...+tPe(N'").adTd.. Thus, for the generator cr(T) of I,,, we obtain the equality (1-T)･crN(4 T) +aWd)(t)･Td==tpe("'")･cr(T) in either r<d case or r==d case. Since we have cifXd)(t)==t+{￡iag･(d""'N･tp(d-")" (if N>o); ==age(d-')"･t+.]il ay.(d-j')"･tp(r-")N (if. j=r J--r+1. N<O), ddt crlg) is a non-zero constant and the totality of elements 2 in k satisfying aWd)(2)= O make a (d-r)-dimensional vector space over FpiNi. Let elements zi, ''', zd-r in k be a basis of this vector space. Then, from the above equality, we obtain the equality (1-T)･aN(zi, T)==z,P･e("'V)･a(T) in k[T]. for each i.. We put v9"=aN(2i, T)･v for each i. Then we have (1-T)･vS:)=:O for each i.. We can write cr(T)=TrP(T) for an unique element P(T) in k[T].. We put v(')=T'P(T)･v for 7'=O, 1, ･･･,r-L Then we have Tr･v(J)=O for each ]'. We can verify that the set {v9'), ･･･,v¥el-r)}(d>r) is linearly independent over F,iNE.. In fact, Iet be d-r >i)liv9:'=O (each 2iciF,iNi). Then, since 22icrN(2i,T) -. i=1 i. ==crN(:i.]Ri2i, T) and aN is of degree at most d-1 in T, an element crN( glS;2i2i, T). in ip[T]. belongs to I. and we have aN(1liRi2i, T)==O in k[T].. Therefore .. crX)(glS)iRizi)==a.(ZS.,rR,2,)=o (if N>o);=a.(l2.,r2,2,)P`r-d'"==o (if N<o). si.ce. a.IO, we have d池2,2,=O. Hence each 2, is zero. Consequently, by f(vgz)) i=a. =v9') for each i, the set {vg'), ･･･,vXd-r)} is linearly independent over fe. More-. ver we can verify that the set {v(O), ･･･,v(r-i)} (r>O) is linearly independent. r-1 ver fe. In fact, let be ￡Rjv`j'==O (each 2jEk). By degP(T)<d, we have. j'--o (O' =P(T)･v40. And the orbit k[T].･v(O' is the subspace of V isomorphic to [T]./(Tr). Since the above relation implies the relation (IEi,RjTj)･v(O)=o, we. ave rE)i2jTj=o. Hence each 2i is zero. Since each vSi)G[V]L each v(j) j'=o. [V]f-nii and [V]fA[Y]f-.ii={O}, the set {vX),･･･,v9d'r);v(O),･･･.v(r-i)} is.

(4) K. ToKi. 30. linearly independent over k. Since each element of this set is in le[T].･v and the subspace k[T].･v of V is d-dimensional, this set is a basis of the subspace. k[T].･v in either r>O case or r=O case. Therefore, by vEk[T].･v, we have. VC[V]fO[V]fH.ii. Hence we have V=[V]fO[V]f-.ii. Furthermore it is obvious that each component of this decomposition is stable under the action of f. Particularly let v be in [V]f-.ii. Then we have r>O and we can write v= r-1 Z Zjv(j) (each 2jEle). Since each v(j) satisfies fr(v(j))==O and rEld;:$n, we j'=o. have fn(v(j')==O for each 1'. Therefore v is in Ker f". Q.E.D. REMARK. The idea of above proof for Prop. 1 is due to Hasse-Witt [2], g3 (case of N==1 in Prop. 1).. i. Nextly we shall construct the normal form matrix of f. Let Afv be the. /x,N /.,pNvx lplt,rilf of f" with respect to a basis of V i･e･ f"L li. ?==AfvL x.i.. ?,where. k//.? iS a column vector with respect to a given basis. we put p.(f) ==rankAfv(v==O, 1, 2, ･･･), V(f")=Ker(f":V-V) and A}Pe' the matrix making with Pe-th power of each element of Af. The we have p.(f)=dim V-dim V(f") =rankAfA}P")･･･A}P(V-i'N), p,(f)=dimV=n and p.(f)l.lrp.+i(f) (vllO)･ Since it is obvious that if V(f")= V(f"'i) then V(f"'i) = V(f""2), we have V(f")= v(fn+i). = ･･･ by Prop. 1. Hence a non-negative integer t+1 is uniquely so determined for f that the decrease n= po(f)> pi(f)> ''' pt(f)> pt+i(f)=:Pt+2(f)=: ''' Pn(f) = '''. holds. Therefore,by Prop, 1, we In the case of t).O, let VY' be ={O} and VS`'OV(ft)==V(f`'i). Let t be positive. Then we can. have pt+i(f)==dim[V]f and V(ft'i)=[V]fm.ii. a subspace of V(ft"i) satisfying VSt)AV(f`) If t==O, then we have [V]f-.il=V(f)==V}O'. verify that if a finite set {vi, ･･･,vi} of. elements in V}t) is linearly independent over le then the set {f(vi), ･･･,f<vi)} of. elements in V(f`) is linearly independent over le. In fact, since the relation. tY.,2i.f<vi)=O (each 2sfe) implies the relation.f(tV.,.RPi-"vi)=O, we have ]S] 2i-"viEV}`'AV(f`). Hence each 2i is zero. We have also f(V}t')AV(ftri) i=1. = {O}. Because, if f`-i(f(v)) =O for ]vEV}`' then vEVSt'All(f`). Let VSt-" be a subspace of V(f") satisfying flVSt')cVSH), VSt-i)AV(ft-'i)=={O} and V}t'i). OV(ft-i)=V(f`). Similarly, let YY-2' be a subspace of V(ft-i) satisfying f(Yf(t-i')CVf(tn2', Vf(`-2)AV(f`-2)=={O} and Vf(`-2)OY(f`-2)=V(f`-i),･･･,V}" be a subspace of V(f2) satisfying flVS2')cV}", V}i'AV(f)=={O} and VY'OV(f). =:V(f2). Then, using these V}""s we have [V]fH.ii=VS"'OV}t-i'O･･･OV}" OV}O), f(V}"')cV}""'i) for each v (if t2-}rO); =O (if t=-1). The group-homo-. morphism f:V(f""i).V(f") induces an injective group-homomorphism from V}"' to VS"-i' (v>=1) and f<VS"') is a subspace of VS""i' satisfying dim.IC<VS") =dim V}V). Because, if a finite set {vi, ･･･,vt} in VS"' is linearly independent over fe then the set {f<vi), ･･･, .f<vt)} of elements in VSV-i' is linearly independent. -.

(5) On an isogenous invariant of a P-divisble formal group. 31. over le(vll). Therefore, by dimVS"'==p.(f)-p.+i(f), we have p.(f)-p.+i(f) Sp.-i(f)-p.(f). So we obtain the sequence of differences for p,-values: O== ''' == Pn(f)-Pn+i(f) == ''' = Pt+i(f)-Pt+2(f) < Pt(f)- Pt+i(f) == ''' = Pt'+i(f)ptr+2(f)<pt･(f)-ptt+i(f) == ''' = pttt+i(f)-ptrt+2(f) < ･･･ < ptt･･(f)-ptttf+i(f)== ･･･ ==. pi(f)inp2(f)==po(f)-pi(f)･ The decreasing sequence of non-negative integers: t>t'>t"> ･･･t"' is uniquely determined for f: The we have [V]f-.ii=(VSt'O ･･･ OV}`''i))O･･･O(VSt''')O･･･OV}i))OV}O'. We put m(f)=pt(f)-pt-,(f), m'(f) = [Pt'(f) - Pt'+i(f)] - [Pt(f) - Pt+i(f)], M"(f) = [Pt"(f) - Pt"+i(f)] M [Pt'(f) ptt+i(f)], ･･･ i. e. m(f)==pt(f)-pt.i(f), m(f)+m'(f)==pt･(f)-ptt+i(f), m(f)+m'(f). +m"(f)==pt"(f)-pt"+i(f), ･･･, m(f)+m'(f)+ ･･･ +m"'(f)=pt･t･(f)-pt･tt+i(f). Let ,. each .os. be a basis of each V}") satisfying f･vet= ret-i, f･.EBt-i==.cet-2, ''',f'-Mtt+2 =vet'+i;f'-S"t,+iC-S:Bt,;f'-S!)tt=.S{E)tr-b''',f'.!{Bt"+2==.S2E}t"+i;f'.!Bt"+iC.fBt･;''';f'St,tr == ve,,tt-,, ･･･, f･ .gB,t,t., =.!iE),,,t.,;f･ .slEl,t,t.,C .!{Bt,,,; f･ Stt,, = .sZ),t,t.,, ･･･, f･ .gE),= .EB,; f･ .SB,. cSo, and e be a basis of [V]f consisting of vectors fixed by the action of f. Then, by the above decomposition of [V]f-.ii and Prop. 1, we obtain that {e; .EBt, St-i, ･･･, !Bi, .sZ)o} is a basis of V satisfying f･ve.C.s{l).-i, Ol:Sv;:!lt(if tlllO). and E is a basis of V(if t==-1).. PRoposlTIoN 2. Let f: V-V be a's in ProP. 1 and t, e, ve's be as above. Then we have (f･e;f･EBt, ''' , f'vei, f'veo)=(E; .S{Bt, ':', EIBb IEBo)'Af*:. A;=. pn(f)==pt+i(f)<pt(f). lo. m I m' Ir .. .. mltt Il1t. 0 .-. po(f) - pi(f)==dim Ker (f: V---> V),. where I,, I, I', ･･･,I"' are unit--matrices of si2e o.(f), m(f), m(f)+m'(f), ･･･,. m(f)+m'(f)+ ･･･ +m"'(f) resPectively, m==. o. with (t-tX)-l'inzes. IO. .. IO 'I 0 0, m'=. o. with (t'-t")-times O, ･･･, m"'==. Ir o I' O･. o Il11. o If,t. I' O. witht"{-timesOand. the otherPartinAf*consistsofzeroalone,. o Ii/i O mor2over 0 is. the 2ero-matrix. PRooF. Since f･e=e･Io;f･.EBt==.EBt･O+vet-i･I, f･.catri==St-i･O+.flE)t-2･I, ''', f･ .SB,,.,= .EB,t.,･ O + .S,,.,･I; f･ .S,,.,=:.EB,,.,･ O +f･ .!IB,,.,･I; ft.SB,,=:.EB,,･0 + ve,,-,･l',.

(6) 32 K. ToKi '''. ;f'-CRt"+i=-cet"+i'O+f'.EBt"+i'I';''';f･.EBt･tt.iz=.EIZi)tr･･'.,･O-l-f･.EB,tt,.,･I"';f･.gB,,,,=. above. Q･ E･ D･. .cZ)t･,･･O + 9)tt･･.i･J"', ･･･, f･ Si== -g?i･ O +f･.gBi･I"'; f･So== ve,･ O, the marix Af* is as. We call the nxn-matrix Af* the normal form matrix of a pN-th power. semilinear endomorphism f: V--->V over k. REMARK. (i) The matrix AJ*･ is constructed by following the construction. of the normal form of Hasse-Witt matrix A in [2],g3. (ii) If V-V isf'as'm Prop. 2, then we have dim V=p,.,(f)+(t-t')･m(f)+(t'-t")･(m(f)+m'(f))+ ･･･ +t"t･(m(f)+m'(f)+ ･･- +m"'(f))+dim V(f).. f' be two PN-th power semilinear endomorphism over le. Let V->V,fV'->V' In the case of dim V=dim V', we define the relation ft-vf' by the existence of a le-linear isomorphism ￠: V';' ,V satisfying fo￠ =ipof'. Obviously the relation rvis an equivalence relation.. PRoposiTioN 3. Let V, V' be tzvo k-vector sPaces of the same dtrmension and. f f'. V-->V, V'->V' be two p"-th power semilinear endomorPhisms over k. Then the relation fNf' holds ij and only if the equality Af*=A;i holds.. PRooF. Let be frNvf'. Then wehave f'==ip-iofo￠ fora le-Iinear isomorphism ￠:V'[) V Since two square matrices Adi and Aip-i are non-singular, we have. o,(f')=rankAs;-iof"o￠==rankAf,s-iAfvA;?"")=rankAfv==p,(f)forv>=O. Therefore, by Prop. 2, we have Af"=Af*･. Inversely, let be Af*=Aj*". Let T be the t-value for f' and p; be the p.-value for f'. Then, by Prop. 2, if t==-1 then T==-1, if t==O then T=O, and if t>O then T>O. And, in the case of t>=O, we have to;+i=pt+i; T-T/= t-t' and to;- to;+i= tot- tot+i;T'-T"=t'-t" and to;,- to;r+i. pt,-pt,+i;･･･;T"'::=t"' and p;,tr-p;,,t+i==pttf,-pt,t,+i. Therefore we have t=T>t'==T'> ･･･ >t"!==T"' and to;-to;+i= ･･･ ==to;,+i'iO;,+2=:=tOt-tOt+i== ''' ==IOt'÷i - tot'+2, ''', to;･rtH to;m+i= ''' = iol- toS= totm - totrt,+i= ''' = toi- to2, to6p to{= tOo- IOi･. Let e', .gBL's be the basis system e, .EB.'s for f'. From above equalities, we have. dimVY"':=dimV'fi(v' (OS.v$t). By viewing the decomposition V :[V]fOVS`'O ･･･. OVY" (resp. V'==[V']f'OV3`tT'O-･･OVSs9'), we define a k-linear isomorphism. Afy. ip: V!-->V by E';;e, s;'wwVve., .sC-,:;g),-, (foip=￠ef') for OSvgt (convention:. ..--T￠ ge vf'. .os-,== {O}). Then we obtain frvf' via ￠. Q.E.D. g2. An application to P-divisible k-formal groups. lt/. Now let a P-divisible k-formal group G be given, and the product-type up M2-tlmes. - We consider the F-lattice Mo(G)== to isogeny for G be ll(G2×･･･xG2). 2 '. m2-times m2-times. e])[M(CZ/5E5-:':･･･SO[t]il(E5i!(GM] in the F-space o[E(Ztiii5a;:)-d]ii(ti2)o･･･￠E(G￡)]. we know that. both systems of 2's and m2's are uniquely determined for the isogenous class of G (cf. Demazure [1], Chap. IV). If we write F2 for the Frobenius map of.

(7) On an isogenous invariant of a P-divisble formal group 33 M(G2), the F-map Fo of Mo(G) is expressed as Fo==O(F20･･･OF2), where we. 2 write Fofi for F2 at 2=O. For sake of our aim, we choose' a pre-assigned W(le)-basis {xi, ･･･, x.-s, yi, ･･･,ys} of M(G2) as follows. (2#O case): We put t::=r-s(llilO). Let xi be the class of 1 mod. the left ideal (F`-VS), x2 be F2xi, x3 be Fftxb ･･･, xt be Ftz-ixi, yi be Vfixi, y2 be VSA-ixi, ･･･,y, be V2xi, where V2. is the Verschiebung map of M(G￡). Then we shall check that the set {xi, ･･･, y,} is linearly independent over VV(le). The relation woxi+wix2+'･'+wt-ixt +zaiNi+ '･･ +usys==O in M(G2)(wi, ujG VV(le)) implies the relation wo+wiF+ ･･･. +wt-iFt-i+zaiVS+･･･+u,V==r(F, V)(Ft-VS) in W(le).[F, V]. By FV:VF =P, we can write r(F, V)=(ao+aiF+a2F2+ ･･･)+(biV+b2V2+ ･･･)(a,, b,E W(le)). Let be l==O. Then the set {xi, ･･･,y,} becomes to {yi} and the above relation. becomes to uiV=[(ao+aiF+･･･)+(b,V+b,V2+･･･)] (1-V)in VV(k).[F, V]. From this equality, we have ui=:bi-ao, ao==:Pai, ai=Pa2, a2=Pa3, ･･･ and bi=b2=:b3= '''･. Thus we have aoEAPil?V(k)={O}. Hence ao==ai=='L･･･=Oand uiV=ui(V+V2. i>o +･･･+VM)･(1-V)==zaiV-uiVM"i(mli;1). Hence u,==O. Let be t>O. From the above relation, we have wo=Ptbt-pSas,zvi=Pt"ibt-i-PSas+b･･･,wt-i=Pbi' PSas+thi; ui=Ptbt+s-ao, u2=Ptbt+sHi-Pab ･･･,us==Ptbt+i-PS'ias.i; ao==PSas+t, ai =PSas+tti, ･･･,as-i =PSa,+t+(s-i),･･･ and bi == btbt+s+i, b2==Ptbt+s+2, ･･･, bt-i:==. Ptbt+s+(tmi), bt=Ptbt+s+t, ･･･. Hence each zvb each uj, each ai and each bm EPW(fe). Since M7(k).[F, V]ZM7(k).[F, V] is injective, we have also the relation w6+wlF+ ･･･ wl.iF`-i+ulVS+ ･･･ +ugV=r'(F, V)(F`-VS) in VV(k).[F, V], where wo=Pw6, zoi==Pzvl, ･･･,ui=-pml, ･･･,za,==pug. By repeating this procedure,. we have each w, and each u,E,A>,PZW(k)= {O}. Nextly we shall check that M(GC) is generated over W(fe) by elements xi, ･･･,y,. Let the class of r(F, V). mod. the left ideal (F`-VS) be an arbitrary element of M(G2). Then we obtain that this class is of the form (aoxi+aiF2xi+a2F3xi+ ･･･)+(biV2xi+b2V3xi+ ''')･ Hence, using relations FS=zl/i, F2VR==V2F2::=P, FSxi=Vfixb FS'ixi==PV9-ixi, FS+S-ix,==pS-i]V2x,, FS+Sx,==pSx,, FS+S+ix, = pSFRxl,･ :･･, FS+S+(t-i)xl=pSFft-ix,, ･･･,. and VS,･+ixi=PFS-ixi, -･･, VS+txi=Ptxi, VX+`'ixi=PtV2xi, -･･, VS+t'Sxi=:PtVfixi, ･･',. we see that this class is of the form a6xi+a(FRxi+aSF3xi+･･･+aiHiFir'xi +blV2xi+bSV3xi+･･･+bgVfixi (a:･,bS･EW(k)). Thus the set {xi,･･･,y,} is a W(k)-basis of M(G2). (R =O case):In this case r==1>s=O. Let e be the class of 1 mod. the left ideal (F-1). If w･e==O in M(G2!i) (wEVV(k)) then w=[ao. +aiF+a2F2+･･･)+(biV+b2V2+･･･)] (F-1) in IV(fe).[F, V]. Then we have W==Pbi-ao, ao :ai==a2, ･･･, bi=-Pb2, b2==Pb3, b3==Pb4, ･･･. Hence each bjE A p`l7V(le). i>o == {O}. Therefore we have w=-w(1+F+F2+ ･･･ +FM)(F-1)=w-wFM"i(mlilO). Hence, w =O. Moreover let the class. of r(F, V) mod. the left ideal (F-1) be an arbitrary element of M(G2ii). Then we obtain that this class is of the form. ao+aiFoiie+a2F&ie+･･･+biVo!ie+b2V&ie+･･-. By Fe!i=id. and Fo!iVoii=: Vo/iFo!i=P, we obtain that this class is of the form [(ao+ai+a2+･･･)+P(bi +Pb2+･･･)]･e. Thus e is a W(le)-base of M(G2/i). Let d be the number of.

(8) 34 '･, -･ '-･･･'K.'ToKi distinct factors G2 except the factor Gek/i in the product-type up to isogeny for. G. Then the number d is uniquely determined for the isgenous class of G. Let {Zi, ･･･,2d} be the set of distinct R's except 2=O and ti>um.t2>= ･･･ )-td be. the sequence of t's (ti=ri-si, 2i==si/ri, ri&si are integers satisfying ri>=si>O,. ･(ri, si)=1) in the product-type up to isogeny for G. And we put mi==mA, <1:Si<,=d), mo=mR for 1= O/1. According to Demazure [1], we know that G is decomposed as the product of the 6tale-part Ge (this is isomorphic to a product. of (Qp/Z,)k) and the connected-part GC, where Q, is the p-adic number field. and Zp is the P-adic integer ring. Then, we know that mo=htGe, .E)d miri. z=1. dd. =-=ht GC, mo+ Z miri==･ht G and ]E] misi==dim GC, where "ht" is the abridgement. i=1 i=1 Mi"tlmes. '. N-. of "height". We put M(i)=M(G2i)O･･･OM(GAi) (O;lli;;ld, Ro=O/1)･ Let {2i, ''', em,}, {xfZi2, ･･･, xE,?i, y[el, ･･･,.yEl?i;･･･ ;xSj'.,, ･･･,.tE,1).,, vfl'in,, -･･,yg:I].,} be the VV(le)-. basis of M(o), M(i) (ls-gi5d) constructed by arranging the above W(le)-base {e} of M(G2i'), the above M(le)-basis {x's, y's} of M(G￡) (2=O)respectively. And let F(o),F(i) be the restriction Fo to M(o), to M(i) respectively. Nextly, let V(o), V(i) be a mo-dimensional fe-vector space, a miri-dimensional k-vector space c<1;Si;:iSd) respectively. And sets {e-i, ･･･, e'-.}, {x'i(el, ･･･, XiiZ,'･)i, y-ièl, ･･･,5]gi?i;''' ; x-. i(. eini,･･',x-t(g?.i,:-yi(thi,･･･,.)-,,`l?.,} (1;$i.<..d) denote a le--basis of Vco), of V(i). ･(ISi-<-d) respectively. Moreover we define reductions po:M(o)tV(o), pi:M(i) -->V(i). (ISi<=d) as follows. Let p: VV(k)-fe be the reduction of M7(le) mod. the. maximal ideal pW(k). Then we define po(.MZ..O,waea)==Z.O,p(wa)ed. and pt(t/!.ll,wpixfee'i + t/il.I.l}, uriy"e + ''･ + t/S.,zvpm,xfo?m,+ lli.I,urm,yg'th,) =t,sii.li, p(wpi)xfeei+t/S.l, p(uri). 'bl-. r(. ti si. 31+''`+pE.).,p(Wpmt)x-p(i,)mt+r2=,p(urm,)y-r,m,･ Then po and each pt are well-. defined. Furthemore we define maps F(o):V(o).V(o), F(i):V(t)-->V(t) (15iAfE{d). .. ,by .l7(,)(tM-ipgw.)e-.) =po(F(o)(tM=O,w.'ecr)), i'(z)( t,#.,p(Wpi)X-'p(Z'i+ tpa., P(Urz)Y-r(el+ ''' +. '. tV., p(zve,m. ,)x-pè'm,+ tll.li, p(urm,)YSe'm,)=pt(F(t)( tllS.,wpixfea+ t.lil, uriyY`'i+ ･･･ + tliS., wBm,･ '. XX)mi+rZ=,UrmiYg)mi)) reSPectively･ Then it is obvious that maps i7(,),i7"(i) ･(1-<.im<.d) are well-defined and are P-th power semilinear endomorphism over k of V(o), Va) (1;;li;;ld) respectively. And we define the k-vector space Vo(G) by the direct-sum V(o)(IDVa)O ･･･ OV(d) of fe-modules, the P-th power semilinear endomorphisms fo over le of Vo(G) by fo==F(o)OF(i)O･･･ ([E)F(d), moreover the reduction pG:Mo(G)->Vo(G) by pG=:po(DpiO ･･･ Opd. Then dim V,(G) =ht G and. moreover we have commutative diagrams:.

(9) On an isogenous invariant of a P-divisble formal group. M(o) Fo - M(o),. M(i) Fo - M(i) toiS CIi.,o,l. too-L CI.òoJt. V(o) - V(o). 35. (1 Sig d). V(i) -----------> V(i). fo. ･fo. Mo(G) Fo - Mo(G) ,o.I CLto.I Vo(G) - Vo(G) fo. -. By Foe.=:e., FgxB=PSxB, F5yr==PSyr for Va, "P, Vr, we have･V(o)c[Vo(G)]fo, and. V(i)O'''OV(d)C[Vo(G)]f,-.ii･ Applying Prop. 1 to the P-th power semilinear endomorphism fo over k of Vo(G), we have Vo(G)=[Vo(G)]fOO[Vo(G)]f,-.iiTherefore we have V(o)==[Vo(G)]fo,and V(i)O･･･OV(d)==[Vo(G)]f,-.ii. In each. M(G2) for REO, if h is an integer satisfying h>ti then Fkxp=Fho-`F8xB r=Fft"`Vixp=PF3-`-iVi-ixp, Ftsy, =Fe-`F8y,=Fft-`VzSyrz'=PFah-`:IVfi-iy, for P, r(Z. ==s/r,t==r-s). Therefore we have V(i)O･-'OV(d)CV(f3)a[Vo(G)]f,-.iiJ Thus we have pt,+i(fo)=pt,+2(fo)==･･･==phtG(fo)=phtG+i(fo)=････ And by F8i･xSl). =yil}, F8i'ixS};EPM(i), we have pt,+i(fo)<Pti(fo)<Pti-i(fo)< ''' <Po(fo>. ==htG. Let the sequence ti>= ･･･ lliltd of t's for G be ti= ''' =ta>ta+i=''' == tp>tp+i= ''' tr> ''' ==t6>t6+i= ''' te>tE+i=''' =td(cu<iB<r< ''' <6<e<d). Then we have [Vo(G)]f,..ii=Vo(G)(f8a'i). As to the basis {x-'s, y""'s} of [Vo(G)]f,-.ii, it holds that fo･ov-=O and fo･xHi=nf2, fo･x-2==nf3, ･･',fo'x-t-i =fct, fo'X-t =Yi; fg'X-i. ==nf3,･･･,f3･x-t-i==Yi;･･･;f8･nfi==yi. So let be .St(G)={x-i(ll,･･･,Iflin,;･･･; X-'. i(. fi', ''', X-i(Y&.}, -!iE)t.･-i(G) = {X-2`ll, ''', X-2(lini;''' ;XS%', '''･ X-2(gh.}, ''', -EZ)tp+i(G) =. {1-Ct(icr)-t&i, ''', X-t(Ta)"tp･mi; ''' ; :-Uicta)-tp･i, ''' , X-t(ata'-tp･ma}; Stp(G) == {X-t(ia'-ta+i, ''', X'. a)'tp+i,mi;''' ; X- .)-tp+i,i, ''', X- .Ltp+i,m.IX-i(,ai"), ''', X-i(rvm'.i)+i;''' ; x-Slli', ''', x-i(fihp},. t(i. .. t(ct. t(ct. , 'flBtr+i(G) == {X-t(ia)-tr･i, ''', X-t(ia)-tr･mi;''' ; X-(tcta)-tr･b ''', X-t(cta)-tr･mcr l X-t(ap'-'t)ri, ''',. '''. IiEctp'-1)rnta+i;''' ; X-t(S'--tr･b ''', J)-Ct(Pp)-tr･mp};''' ; E5te(G)={X-t(ia)-te+i･i, ''', X-t(ia)'tE+i･mi; t`6 t(6 ; X- a,'"t,+i,i, ''', X- cr)-te+i･ m6IfiiSFii), ''', X-7(n66'+il;''' ; X-i(91, ''`, X-i(fj;iE}, ''', 'EZ)td+i(G)=. '''. {-Xt(i.)-td,i, ''', X-t(i.)-td,7ni;''' ; :-UE6.)-td,i, '''7 X-t(6.)-td,m6[X-(t2-"`t)d,i, ''', J-Ut(e±ti},ma+i;''' ; X' t(e,)-td,i,. ''' , X- 2)-td,m,} ; ･E8td(G)= {X-t(i.)-tcl+i,i, ''' , X-t(ia)-ta+i,mi; ''' ; X-(te.)-td+i,i, '''r t(. .)-td+i,7n,lx-i(fi'i),''',x-i(fJllY+i;''';x-i(ei),''',:-ui(,dm)cl},-EBtd-i(G)={XEi.)-td+2,i,''',x-t(i.),-td+2,m,;. A-":' t(e. ･･･. ==. ; A-:t(e.)"t,+2,b. ･･･, x- .)-t,+2, .,I )-c2(fi'i', ･･･, x-2`?','.'2.,;r･･ ; J-uSg?, ･･･, x-2èh,}, ･･･, ve2(G) t`S. {:-ut('.)-i,i, ''', x-t('.)-i,m,;''' ; :-ut(e.)-i,b ''', X-t(e.)--i,m,lx-t(ed'-ii),i, ''', x""t(eal+-ii),m,.,;''' ; x-t(dd)-i,i,. ･･･. , x- 7(d,).i,.,}, .!Bi(G)=={x-t(i.',i, ･･･, x-t(i.),m,;''' ; x-t(e.',i, -'', x-t(e.),m,1x-tèd',l', ･･･, x-t(2th),.,;. '''. ;x- t(del',b ''', nf[dd',7nd} ; -eso(G)={:-Yill, ''', .)"s(ii',i; .)-'i(ll, ''', .)-'s('i',2;''' ; Y-i(,im' v ''', )-'s`l)miI. .)"i(?2, ''', :-Ys(Z)i;''' ; .l-Yi(??n2, ''', .l-Ys(:)m2i ''' [ .1-Yi`1i', ''', .)-'t(dd',i;''' ;il-Yi(,dm' v ''`', 5-is(d,',md}.. Then we have fo'St.(G)=vet.-i(G),''',fo'-Stp+2(G)=vetp+i(G);fo'-EZ)tp+i(G)C -S)tp(G); fo'tp(G)=='Stp-i(G), ''', fo''ostr+2(G) =Str.+i(G);''' ; fo''!Z)te(G)=='f{Elte-i(G), '''. , fo'-E!)td+2(G)==-Std+i(G);fo'vetd+i(G)C･EBtd(G);fo'vetd(G) ==-EBtd-i(G), ''', fo'.g)2(G). vei(G); fo･vei(G)C`{Bo(G), and Vo(G)(f8at'i)= Vo(G)(f8ct)O"g?t.(G)k, V,(G)(f8a)==.

(10) K. ToKi. 36. Vo(G)(f8at-')OEBt.-i(G)le, ''', Vo(G)(f8P'2)== Vo(G)(f8P'i)e3Z)tp+i(G)k; Vo(G)(f8P'i) = Vo(G)(f8P)(DEBtp(G)le, ''', Vo(G)(f8r'2)==Vo(G)(fòr"i)(Dvet,+i(G)k;'''; Vo(G)(f6E'i)==. Vo(G)(f6e)Ovet,(G)k, -･･, Vo(G)(f8d"2)==Vo(G)(f8d'i)O,.!Bt,+i(G)k; Vo(G)(f8d"i)= I/o(G)(f8d)OE{Bt,(G)le,･･･, Vo(G)(fg) =Vo(G)(fo)O!Bi(G)k, Vo(G)(fo)==So(G)le, where. each ve,(G)k is the subspace of Vo(G) generated over k by g),(G). Therefore we may consider sZ),(G)k as Vo(G)%' for each v and, by viewing the dimension of both hands sides, we have pt.(fo)-pt.+i(fo)== ''' ==ptp+2(fo)-Ptp+i(fo)=Mi+ ''' +Ma< Ptp(fO)- PtP+i(fO)= ''' == Ptr+2(fO)- Ptr+i(fO)== Mi+ ''' + MP < PtE(fO)- Pts+i(fO)== ''' == ". Pt.+i(fo)-Ptd+2(fo)=Mi+'''+ME>Ptd(fo)pPtd+i(fo)='''=Pi(fo)-P2(fo)==Mi+'''. +nzd;Spo(fo)-pi(fo)==misi+･''+mdsd=dimGC. And we have EVo(G)]f,-.ii= (vo(G)Yt,a) O ･･･ O Vo(G)}t,P "))O(V,(G)St,F)O ･･･ O V,(G)St,r"i))O ･･･ O(V,(G)}t,s' O ･･･ O V,(G)}t,d'i))e(V,(G)St,d'(D ･･･ O Vo(G)S',')O Vo(G)}O,'. We put E(G) == {e-'i, ･･･, e-.,}. Then. fo(e'.) e-. for each a and E(G)le==[Vo(G)]fo. Thus, for a p-divisible le-formal group G, we can construct the matrix Af*, by means of the basis {e(G), .vet.,(G), ･･･. , vei(G), reo(G)} of 'Vo(G). So we write rrtG for Af",.. PRoposiTIoN 4. Let G be a'Pdivisible fe-formal grouP for which Product-. .d. type up to isogeny is (GOkii)Mox Z(G2i)Mi, where Ri=si/ri, ri&si are integers. ' ' ･ i=:1 .. satisfying ri>=si>O, (ri, si)= 1(1;Si-:E{d) and Ri ik2j (i 7Ej). And let the sequence. of t's for G be of the form ti = ''' =ta>ta+i= ''' =tp>tp+i== ''' ==tr>tr+i= '''. >t6+i== ''' =tE>ts+i=`'' ==td (ti=ri-si)･ Them mG is a sequare matrix of size. htG of the form Io. Mo 11Tcv. I. mp.. JB '' Me. I, md. I, .'. o. d. ]Z misi, i==1. where Io, I., Ip, ･･･, I,, ld are unit-matrices of si2e mo, mi+ ''' +m,r, mi+ ''' +. mp, ''',mi+ ''' +me, mi-E ''' +md resPectively, m.= O with. ..----" ,. Ia O Ia O 'Icv O with (tp-tr) -times O, ''',. (t,,-tp)-times O, mp = O. Ip O / IpO I -/. Ip O.

(11) On an isogenous invariant of a P-divisble formal group 37. "m,= O zvith (t,-td)-times O, ntd =O with td-times. f,0 I,OI,0 IdO. leO ldO. O and the othe Part in me consists of 2ero alone, moreover 0 is the square 2ero-. matrix. And rttG is an isogenous invariant of G. i .. '. PRooF. Since {s(G), vet.(G), ･･･,.ceo(G)} is such a basis {e, s).'s} as in g1. of Vo(G) for a P-th power semilinear endomorphisma fo over k, the matrix rttG is a nxn-matrix of the above form (n==dim Vo(G)). By dim Vo(G)==ht G, its size is of htG. And since both systems of 2's and m2's are uniquely determined up. toisogenyforG,thematrixmGisanisogenousinvariantofG. Q.E.D.. REMARK. (i) If G is as in Prop. 4, then we have htG =mo+(t.-tp)(mi+ '･･ +Ma)+(tP-tr)(Mi+ ''' +MP)+(te-td)(Mi+ ''' +ME)+td(Mi+ ''' +Md)+(MiSi+ ''' +mdsd). (ii) Though, for two P-divisible k-formal group G, G', if G is isogenous to G' then rttG==trtG･, the inverse is false. For example, let be G=Gli4xG2le!3,. G'=Gki2xGZi5.ThenGisnotisogenoustoG'andmG=O =mG,(be10 1 OO oo 10 o Ol cause, for G&G', mo==O, ti==3>t2==1, mi=m2==1, misi+m2s2==3)･ (iii) MG=. .. ifandonlyifGisisogenousto(G2ii)Mx(Gkii)",whereIisthe. (i o]Mn unit-matrix and O is the zero-matrix. t. Nextly let a P-th power sdmilinear endomorphism f of a positive finite dimensional le-vector space V be given. If the map fsatisfies the relation. V.Vrv. f fo.. Yo(G) ' Vo(G)in the sense of g1 for a P-divisible k-formal group G then we say that f:V->V comes from G (in this case, dim V=htG). Let t(f). ff. be the t-value for V-V. In the case of t(f)==-1 (resp. O) V->V comes from <Go,ii)dimV (resp. (GO,ii)pi(f)×(Gkii)dimV(f)) up to isogeny by the above remark (iii). and Prop. 3. So we are concerned with the case of t(f)>O. In this case, it does not necessarily holds that G is determined when V-->Y comes from G,. . fo >Vo(G) up to isogeny. For example, the above remark (ii) shows that Vo(G) comes from two non-isogenous P-divisible le-formal groups G, G'. We put ti =t(f). Now, as to differences of p,(f)'s let be pt,(f)-pt,+i(f)== ''' =Pt2+2(f)'. Pt2+i(f)<Pt2(f)-Pt2+i(f)=='''<0td(f)-Ptd+i(f)=='''=Pi(f)-P2(f)･ We PUt Mi(f) =Pt,(f)MPt,+i(f), Mi(f)==[Pti(f)-Pt,+i(f)]-[Pt,-,(f)-P,,.,.i(f)] (2$iSd).. PRoposlTIoN 5. Letf be a P-th Power semilinear endomorPhism of a fe-.

(12) 38 ' K.ToKi. f. vector sPace V(t(f)>O) and be n=dim V>O. Then we obtain that (i) V-÷V comes from a P-divisible k-formal grouP of htn for which the sequence of t's coincides with that sequence t(f)=:ti>t2> ･･･ >td (>O) for f:V-->V, if and only if there exist Positive integers si, s2, ･･･, sd satieflying (ti, si)==(t2, S2) = ･･･ =(td, Sd>. df. ==1 and po(f)-pi(f)== >i] mi(f)･si, (ii) ij' dim V(f2)==2dim V(f) then V->V comes. i =1 from (GO,fi)Pn(D x(GY(i+ti))Mi(f) × ･･･ (Gikl(i'td))Md(f) uP to isogeny.. PRooF. First, we shall verify the "if-part" of (i). By the assumption. ". (ti, si)==1 (1;:lli::Sd), we have (si, si+ti)=,1 (1;!li.Sd). So let be G=(GOk/i)p.(f). ×(Gkgif(Si+ti))m,(f)×･･･×(GZd/(Sd."d))md(f). By the assumption mi(f)si+･･･+ md(f)sd= po(f)- pi(f), we have ht G= pn(f) + Z dmi(f)(si+ti) = p.(f)+(ti- t2)mi(f>. t. i=1. +(t2't3)(Mi(f)+M2(f)) + ''' + (td-i-td)(Mi(f) + ''' + Md-i(f)) + td(Mi(f) + ''' +. md(X"))+po(f)rpi(f)). By Prop. 2, remark (ii), the third expression of this equality is equal to dim Y. Hence htG==n. The sequence to t'sfor Gobvious-. ly coincides with that sequence for f:V"V. And, by mo =pn(f), mt(f)=mi (lg!ii:{.d) for G, we have Af"-'mmG. Therefore, by Prop. 3, it is obtaiined that. f V-->V comes from G. Second, we shall verify "only if-part of (i). Let G' be a p-divisible fe-formal group satisfying given conditions, and be isogenous to (G,o/i)m6x (G2i)m'i × ･･･ × (G2a･)Ma･, where t(l.i) ･･･ lllta･(tt･=:rt-sC･, 2:･==s:･/rC･, r:･ &s:･. are integers satisfying (rC･,sC･)=1, r:･>=st･>O, 2:･ik2S･(i#1')). Then, since the. sequence of t's for G' coincides with ti> ･･･ >td by the assumption, we have. d'=d and t{･=ti (1.<=i.<.d). Moreover we have Af*=mG by the assumption v-{>vtNvv,(G')-l9'>v,(G') and Prop. 3. Therefore we have po(f)-pi(f)== S3 m:･s:･. i=1. and mi･=m,i(f) (1$iSd). So put si =s:･ (1:Sli.Sd). Then (t6 sC･)=1 by (rC･, sC･)=1.. d Hence (ti,si) =1 (IEi.Sd). And moreover po(f)-pi(f)=2mi(f)si. Third i= 1. we shall verify (ii). By the assumption 2 dim V(f) =dirp V(f2), we have p,(f>. -p2(f)=-pe(f)-pi(f)･ So, put si=:･･･=isd==1. Then, since we have p,(f> -P2(f)==Ptd(f)-Ptd+i(f) i= i.lli=,mi(f>,,, we have po(f)-pi(f)==,;., mi(f)st. Therefore. V-LV satisfies assumptions in the "if-part" of (i). Hence there exists a p-divisible. k-formal group G' of htn for which the sequence of t's coincides with ti> ･･･. f comes from G'; Then the product-type up to isogeny for･ G' >td, and V-V is of the form (G2ii)Pn(f'×(G2i)Mi(f'× ･･･ ×(G2a)Md(f', where 1;-==s;･/r:･, rt･&s,C are integers satisfying (r:･, s:･)==1, r:･l.lls:･>O, r:･-s:･=ti, R:･ iERS･ (i 7Ej). Since we. have mi(f)sri+ ･･･ +md(f)sa=po(fo)-pi(fb)==Po(f)-pi(f)==pi(f)mP2(f)== mi(f)+ ･･･. ' each 'S' :･>O, we obtain sl±= ;･･sa==1. Hence each R;･ is ''equal to +md(f) and. 1/(1+t,). '' '-' i･-･ .･ / Q.E.D. /± tt t t tt ttttttt t ttt ttt t tt tlttt tt /t t ttt tl tit/t / , 1･ ,V?' ': ''1･:' ,,'4. '. '. '.

(13) On an isogenous invariant of a P-divisble formal group 39 g3. Case of p-divisible groups of abelian varieties.. Here let le be Fp. From now on, we shall observe the matrix mG in the case of G==X(P)(g)le, where X(p) is the p-divisible group of an abelian variety. X of positive dimension g defined over a finite field I7... Henceforth we write simply X(P) for X(P)opk. Let PFk(T) be the characteristic polynomial of. l. the Frobenius endomorphism Fk relative to I7p. of X, and elements Ti, ･'',T2g in the algebraic closure of Q, be all roots of PFk(T)==O. Then, according to Manin [4], we know .that there exist positive integers b,e satisfyingb=-O <moda) and all Ti, ･･･,T2.GW(I7',b)[piie]. Let v, be the p-adic valuation in VV(F.b)[Pi/e], and A[z(2) be the number of i satisfying v.(Ti)/a==Z for a rational. g number 2 in O;$R;Sl. Then since we know PFk(T)=T2g+ ･･･ +P"g= ,].IE=,(T-Ti) ･(T-paq･ i), we obtain that (a):Nx(Z)=Nx(1-2), (b): Nx(O)+ 2 Alx(2)==2g and. 2 ;o (c):Ni(O)+ :E) Ni(2)+(1/2)Nz(1/2)=g. And, according to Manin [4], by. O<2<112 considering the Newton-polygon of PFk(T), we know that each 2Nx(2) is a nonnegative integer. Furthermore, due to Manin-Demazure (cf. [1], [4],Chap.IV), we know the fact that the set {v.(Ti)/ali-p-1,.･･･,2g} coincides with the set of. 1's counting multiplicities in the product-type up to isogeny for X(P), where. we mean the multiplicity of 2 as the number mR･htG2. Then, through the expression 2==2Nx(2)/Nx(R), we have htX(p)=2g by the equality (b) and dimX(P)C=g by equalities (a)&(c). Let o(X) be the P-rank of X. Then, according to Demazqre [1], we know X(p)==(Q,/Z,)Z(i)×.2e, where X)O is the. connected formal completion of X. For Nx(2);O, we put the g.c.d. (Ni(R), RIVi(R)) as mi(R). Then the number mx(2) is equal to the multiplicity m2 of of the simple factor G￡ at Z for the product-type up to isogeny for X(P). We. note that mx(O) =iVx(O), mx(1)=Nx(1) and mi(1/2)==(1/2)Nx(1/2). From the l. above known-fact, we see that X(p) is isogenous to the product-type (G2fi)Mi(O) ×[(Gkii)MZ(O)×,<2ll<,!,(G￡xGikfi2)Mx(2)×(Gilei2)Mx(if2)]. In this product, the factor. <G2ii)Mi(O) is 6tale and the factor [ ] is connected. Since k is algebraically closed, G2ii is isomorphic to (Q,/Z,)le. Therefore, by considering the F--space. E(X(P)), we have a(X)=mi(O). From these notes, we see that mi(,) is a 2g ×2g-matrix with the sequence of square submatrices on the diagonal:Io of size o(X), ･･･,O of size g. So the number 4 for X(P) is at most g and we have o(X)==pt,+i(A)== ''' =pg+i(fo) =pg+2(A)== '''･. PRoposlTIoN 6. Let X, Y be two abelian varieties of Positive dimensions dojined over a finite field of characteristic P. Let all simPle factors at 2 tO, 1, 1/2 of the Product-tyPe zaP to isogeny for X(P) (resP. Y(P)) be of the sanze height. Then we obtain that X(P) is isogenous to Y(P) if and only if rrtx(p) is eqzaal to my(p).. PRooF. By Prop. 4, it remains to verify that the "if-part" holds. By assumptions, the sequence of distinct 2's in RtO, 1, 1/2 for X(P) (resp. Y(P)) is.

(14) 40 K. ToKI. of the form: si/r, (r-si)/r, s2/r, (r-s2)/r, ･･･; r>2, r>si>O, (r, s,) =1 (resp. s{/r', (r'-s()/r', sS/r', (r'-sS)/r', ･･･;r'>2, r'>sl･>O, (r',sC･)=1). And the t-. sequence in R#O, 1, 1/2, for X(P) (resp. Y(P)) is of the form: ti>t2> ''' >td. (>O); dEO(mod2) (resp. tl>tS>･･･tat(>O); d'iO(mod2)). Now let be mx(,) =my(p). Then we have immediately Nx(O)=IVY(O) and dimi¥==dim Y. It is not the case that one of A[x(1/2)&Ny(1/2) is zero and the other of them is zero. In fact, let be IVx(1/2)=O and Ny(1/2)>O. Then the t-sequence for X(P) is of the form a) ti> ･･･ >td(==1) or b): t,> ･･･ >td(>1), if a(X)=O; is of the form cr):ti> ･･･ >tcl(=1)>td+i(==O) or P):ti> ･･･ >td(>1)>tcl.,(=O), if a(X)>O, and the t-sequence for Y(P) is of the form a'):t(> ･･･ >ta･= ta･.,( =1) or b'):tl> ･･･ >ta･>ta･.,( =1), if a(Y)=O; is of the form a'):tl> ･･･ >ta･==ta･.,(==1)>ta･.,( =O). ,. ". or P'):tl>･･･>ta･>ta･+i(==1)>ta･.,(=O), if a(Y)>O. Consistences of a) & b'), b)&b')y cr)&P') and P)&P') are not cases respectively, because the contradiction d==d'+1 happens by the coincidence of both m-sequences respectively. The consistence of a') & b)(resp. a') & P)) is not the case, because the contradic-. tion td=ta,+i happens by the coincidence of both m-sequences.' Suppose that that consistence of a)& a')(resp. a)&a')) is the case. Then, by the coincidence of both m-sequences, we have d==d' and ti==tl･(IEIi;.Sd), mx(2i)=my(Rt･)(1:;li-Sd-1),. mx(2d)==my(2a)+(1/2)Ny(1/2). On the other hand, by htX(p)=ht Y(p), we have. r{mx(2i)+･･･+mx(Rd)}=r'{my(RI)+･･･+my(2a)}+Ny(1/2). Since 2d==(r-1)/r and 2a=(r'-1)/r', we have r-1==ri-s' (for is').1) and r'--1==r-s (for is).1).. Therefore we have r=r'. Hence we have r･my(2'd)+(r/2)Ny(1/2)=r･mx(Rd) =r･my(Ra)+Ny(1/2). Therefore we have r=2. That is not the case. Thus we have Nx(1/2)==AiY(1/2)=O or IVx(1/2)Ny(1/2)7EO. Now let be Aix(1/2)== NY(1/2)=O. Then the t-sequence for X(P)(resp. Y(P)) is of the form ti> ･･･ > td(>O)(resp. tl> ･･･ >ta･(>O) in the case of o=O, and is of the form ti> ･･･ > ･ld>td+i(==O)(res. 'tl> ･･･ >tat>ta･+i(==O)) in the case of a>O. Therefore, in both cases of a, we have d= d' and ti=t:･, mx(2i)==my(2C･)(1;lli:Ild) by the coincidence f. of both m-sequences for X(P)& Y(p). On the other hand, by htX(P)=ht Y(P), we have r･{mx(2i)+ ･･･ +mx(2d)}=r'･{my(Rl)+ ･･･ +my(2a)} in both cases of a. Therefore we have r=r' and ti/r=tC･/r'(ISiSd) in both cases of a. Thus we have 2i=RC･, mx(2i)==my(Rt･)(1;;li:Eld). Therefore, from moreover Nx(O)==ATlr(O),. X(P) is isogenous to Y(P) in both cases of a. Next, let be Nx(1/2)AIY(1/2)#O. Then the t-sequence for X(P)(resp. Y(P)) is of the form a): ti> ･･･ >td>td+i (=1) of b): ti> ･･･ >td-i>td =td+i(= 1)(resp. a'): tl> ･･･ >tat>t6-t+i(=1) or b'): tf> ･･･ >ta･.i>ta･=ta,+i(=1)) ip the case of a=O, and is of the form cr): ti> ･･･. >td>td+i(==1)>td+2(= O) Or P): ti> ･･･ >td==td+i(==1)>td+2(==O) (reSP. cr'): t{> ･･･ >ta･>ta,.,(==1)>tat.,(==O) or Pi): tf> ･･･ >ta,= ta,.,(= 1)>ta,.,(==O)) in the case. of a>O. Then consistences of a)&b'), b)&a'), a)&P') and P)&a') are not cases respectively, because the contradiction d-d'E!1(mod2) happens by the coincidence of both m-sequences respectively. Let the consistence of a)&a') (resp. cr)&a')) be the case. Then we have d==d' and ti==tl･, mx(Ri)==my(Rl･) (1:$i<=,d+1) by the coincidence of both m-sequences. Sirice htX(P)==ht Y(P),. N.

(15) On an isogenous invariant of a P-divisble formal group. 41. mx(Rd+i)=(1/2)Nx(1/2) and my(2a+i)=(1/2)A7Y(1/2), we have r==r' and IVx(1/2) :A7Y(1/2). Hence Ri==ti/r=t;･/r'==R!･, mx(2D==my(2;･)(1;Si=<d+1). Therefore, from moreover Nx(O)=AIY(O), X(P) is isogenous to Y(P). Let the consistence of b)&b')(resp. P)&P')) be the case. Then we have d==d' and ti==t:･, mx(Ri) -my(lt･)(1;ISi,<.,d-1), Mz(Rd)+mi(2d+i)=my(2a)+my(Ra+i) by the coincidence of pm. both m-sequences. On the other hand, we have r･{mx(2i)+･･･+mx(Rd)}+ Ni(1/2) =r'･ {my(1{)+･･･ +my(2a)} +AiY(1/2) by htX(P)=ht Y(P). Since2d=(r-1)/r ,. and 2a==(r'-1)/r', we have r-1=r'-s' (for is')1) and r'-1==r-s (for ]sll). Therefore we have r==r'. Since mx(ld+i) = (1/2)IVx<1/2) and my(R'd+i)==(1/2)AJti(1/2),. we have 7'･mx(Zd)+(r/2)･IVx(1/2)=r･my(2a)+(r/2)･Alir(1/2)==r･my(2d)+Nx(1/2) `. -NY(1/2)+(r/2)･Ny(1/2). Therefore we have Nx(1/2)==Ailr(1/2) and mx(2d) =77zy(Ra). Thus we have Ri==ti/r==t:･/r'==R:･ (1;;ii:.Sd+1), IVx(1/2)=AIY(1/2),. Nx(O)=NV(O) and mz(2i)==my(2:･) (ISi.Sd). Therefore X(p) is isogenous to. Y(P). Thus we completeaproof of Prop. 6. Q.E.D. REMARK. (i) Due to T. Honda (cf. [3]), it is known as an aflirmative answer of Manin's conjecture (cf. [4] g5) that there exists an abelian variety defined over a finite field of characteristic P whose connected formal completion is isogenous to a given P-divisible k-formal group of the form G2xGL-i(2 =s/r,. r & s are positive integers satisfying r-s>s>O, and (r, s)==1), through his construction of a F..-simple abelian variety Xo of dimension r defined over Fp. corresponding to the conjugacy class of Weil-number of order r with respect. to P which is a root of the quadratic equation: x2-PSx+p'=O. By the way, we not that Nx,(O)==Nx,(1/2)==O for such Xo. (ii) From the formula(c), the equality Nx(O) =o(X) and the number RNi(R) being integer, as for the number Nx(1/2), we obtain that if a(X)==g(resp. g-1, resp. g-2, resp. g-3, resp. g-4, resp. g-5) then Nx(1/2)=O(resp. 2, resp. 4, resp. O or 6, resp. O or 2 or 8, resp. O or 2 or 4 or 10). '. ExAMpLE. We shall show the possible forms of X(P) up to isogeny and mx(,) in the case of a(X)l.lg-5(g=dimX). We put pi=pi(fo):. X(p) up to isogeny nt.(p) 1.. Ifa(X)==g, (GOk!ixGL!i)g ×. t, Iml=g ==O XI ×. g= pl. Og. lx N. == g-1, (G2ii×Gkii)g"i×GLi2. (t, =1> t, =O. mi==1, m2=g-1. g-1= p2. N N. N,1. 1=P1-P2==m1. o 1. o. g.

(16) 42. K. ToI<I. ,. g-2,(GO,ii×Gi,ii)g-2×(GLi2)2. t,=1>t,==O ( mi= 2, m2=g-2 == g-3, (G%ii×GLii)g-3×GLf3×GZi3. lk. oo 11. (. mi==1, m2=1, m3=r-g-5. 1= P2 - P3 == M1. 2=p1-p2 == m1+ m2. o.. g. g-4=P3. -1×. Nl. 1=P2-P3 == m1. 9o. oo 11. 3= P1-P2 :M1+ M2+ M3. iO. lx. g. g-4==P2. Nl. oo. 4==p1-p2 == ml. oo. 11. iO. g. g-5=ps. 1`s.1. 999oo. mi =1, m2=1, m3==g-5. t, ==3> t, ==2> t, ==O. 1=P3-P4 == m1. o 10 1OO 11. mi=4, m2=g-4. or (G2i'× Gk/i)gm5 × G2i5 × G3k!5. g. g-4=P4. x. t,I=4> t,==1> t, ==O. o. sl. mi==1, m2==m3==1, m4=g-4. g. 3=p1-p2= m1. lx. (. g. g-3=p2. 11. t, ==2> t, -- t, =l> t, =O. =g-5,(G2/i×Gk/i)g-5×Gk/5×Gt/5. o. 10. mi=1, m2==1, m3 =g-4. t, ( =1>t,==O. .. oo. t,=:3> I t, =1> t, ==O. or (G2/i× Gk/i)g-4×(Gki2)4. 1=p2-p3=m1 2= P1 --- P2 = M1+ M2. xl. mi=3, m2==g-3. g. g'3==P3. lx. t, ( ==1>t, =O. or (G2/i× GS/i)gH4× GY3× GZ13× Gk/2. o. 2o 10 1. mi==1, m2==1, m3=g-3. =g-4, (G￡1i× Gk/i)g-4× Gk/4× G3,14. 2=p1-p2=m1. IKxNl. t,=2>t,=1>t,==O (. or (G2fi×Gi,fi)g-3×(Gy2)3. g'2==P2. ×Nl. 11 O 1's,1. 1=p4-Ps==m1 1=p3-P4= m1 1==p2-P3=m1. 2=pl-p2=m1+m2. g. g-5==p4. o i2oo. ii9 o. 1=P3-P4==m1. 2= P2 -P3 = ml+ m2. 2=P1-P2= m1+ m2 g. '. '.

(17) On an. isogenous invariant of a P-divisible formal group. 1/2 or (G2ii×Gk!i)gm5×Gki4×Gle314 ×Gle. {mi=1, m2=m3==1, m4==g-5. 1. s. t,==3>t,==-t,==1>t,==O. or (G2ii×GYi)g-5×GY3×GZf3×(GY2)2. I. o. ls sl. or (G2/i× Glii)gm5×(Gi/2)s. =1>t,=-O ( t, mi==5, m2=gL5. 3=P1-P2=m1+M2+M3 g. gff5=P3. 1=p2-p3=m1 4=P1-P2=M1+M2'+M3. o IOO. oo. mi==1, m2== 1, m3==2, m4==g-5. t. 1=p3-p4=ml 1=P2-p3=m1. 111 O. t, ==2 > t, = t, ==1> t, =O L,. g-5=:P4. Nlo. i9oo. s. 43. 1111 O ls..1. g. g-5= P2. oo. 5=pl-p2=ml. ooo. Illll O By observing this list, for two abelian varieties '' X,. g. Yof the same dimension g. defined over a finite field, we can note that if a(X) ==a(Y) =g (resp. g-1, resp.. g-2) then X(P) is isogenous to Y(P), and if a(X)==o(Y)==g-3 (resp. g-4) & Ni(1/2) =AG(1/2) then X(P) is isogenous to Y(p). Furthermore we shall talk about the matrix mx(.) for X =1(C), where C is a complete non-singuler curve of genus g(ll) defined over a finite fie!d F. pa. with F..-rational points and J(C) is its jacobian variety (this is defined over F.a)･ Due to Serre, we know that a(X) is equal to the Hasse-Witt invariant a(C) for C (cf. Serre [5] or S-S [6]), and due to Hasse-Witt [2] we know that the number a(C) iS the size of the firstunit submatrix of the normal form. A" of matrix A in [2], Satz 11. Therefore the first unit submatrix Jo of t. t. mJ(a)(p) is of size o(C) and coincides with the first unit submatrix of A* for C.. Now let Fa be the p-th power semilinear endomorphism of k-vector space R!(R(O)+k(C)) of dimension g induced by the p-th power Frobenius map of le(C), where R(resp. le(C)) is the ring of repartitions on C(resp. the functionfield of C) and R(O) is the subring of R consisting of all elements whose com-. ponents at any point P of C is in the valuation ring Op. Then each number pi(Fc)-pi+i(Fc)(i>.,.O) means the difference 6i+i in the sense of [2], Satz 11 and 3= ''' the eqqality g-o(C)==6i+S2+53+･'' (Si)--62)-S>. lll5g=`'5g+i =6g+2 = ''' ==O),. holds. And we have pi(fo)-pi+i(fb)>=pi(Fa)-pi+i(Fc)for i=:O & i>.=g, becauSe SiS-g and pg(Fc)==p,+i(Fc)= ･･･ =o(C), where .ICb is the P-th power semilinear endomorphism of Yo(J(C)(P)) in the sense of S2 induced by the F-map Fo of Mo(f(C)(P)). By observing the above list, we can note that these inequlitiesfor ISiS.g-1 are also true in the case of gS.5, because 6252, 6i$1(i=3, 4, 5) in the c.ase of g$5. Thus, in the case of g$5, we can observe A* for C as a.

(18) 44 K. ToKi. minor of iUJ(c)(p) as follows: A*== Io a(C) , --i-. ･- -. O 6i., 1= =o 'i-t''' O 5,. where 6i+i 6i O. S,O. 6i., '･ g. 'O pi(fo)-pi+i(fo), g..p,(Of,)-p,(f,). 1 ll. f. s. r'. o .... 'o. i. 1.. 6i+1 '･. '1 1. --e 1 'i. g. ln MJ(c)(p)･. However we have no comments on these observations for any g.. '"ij. References '. [1] M. DEMAzuRE, Lectures on P-divisible groups, Lecture Notes in Math., 302, Springer-Verlag, 1972.. [2] H. HAssE und E. WiTT, Zyklische unverzweigte Erweiterungs-k6rper vom Primzahlgrade P Uber einem algebraischen Funktionenk6rper der Characteristik P, Monath. Math., 43 (1936), pp. 477-492. [3] T. HoNDA, Isogeny classes of abelian varieties over finite fields, J. Math. $oc. Japan, vol. 20, 1968, pp. 83-95.. [4] Yu. I. MANiN, The theory of commutative formal groups over fields of finite characteristic, Usp. Math. Nauk, vol. 18, 1963, pp. 3-90; English translation: Russian Math. Surv. vol. 18, 1963, pp. 1-83. [5] J.-P. SERRE, Sur la topologie des vari6t6s alg6briques en caract6ristique P, Sympo-. sium of Topology, Mexico, 1956. [6] Sugaku-Sinkokai, Algebraic geometry and its application, Summer Seminar II.. Akakura, July, 1957 (in Japanese). .. ! 1 ;. 1.

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