p-Divisible Groups 159 Jl:A-+A®A, e:A-+R, and an automorphism i:A~A, describing multi­ plication, neutral element, and invers~, respectively, These are subject to a number of rather obvious conditions (see for example [4J or [6J). Examples. (a) Let r be a finite abstract group oforder m. Let A be the 1 p-Divisible Groups 1 " ring of R-valued functions on T, let (Jlf) (8, t)=f(8t), Jet (if) (8)=f(8- ), and let 8f=f(I). Then r=Spec(A) is a finite group of order mover R, (b) Let A=R [X]/(X" -1), with /l(x)=x@x, where x is the image J. T. TATE of X in A, Then Spec(A) is a finite group of order mover R which is denoted by flnr; it is t.he kernel of the homomorphism m: Gm-rGm, Introduction where Gm denotes the "multiplicative group", viewed over R. (c) Let a, bER with ab=2. Put A=R+Rx, with x 2 +ax=0, and put After a brief review offacts about finite locally free commutative group !lx=x@l+l@x+bx@x. Then G•. ,=Spec (A) is the most general group schemes in § 1, we define p-divisible groups in § 2, and discuss their re­ of order 2 over R whose affine algebra A is free over R. Moreover, Ga,I> lation to formal Lie groups. The § 3 contains some theorems about the and Go'.b' are isomorphic if and only if there is a unit u in R such that action of Gal(KjK) on the completion C of the algebraic closure K of a a'=ua, h' =u-1b. local field K ofcharactelistic 0, In § 4 these theorems are applied to obtain information about the Galois module of points of finite order on a p~ (1.2). Duality divisible group G defined over the ring ofintegers R in such a field K, and From now on we assume all groups to be commutative. Then there is to prove that G is determined by that Galois module, or, what is the same, a duality theory, due to CARTIER (see [3], p. 106). Let G=SpecA be a by its generic fiber G x R K. finite group over R and m:A@A->A define the mUltiplication in A and The ootion ofp-divisible group and the basic theorems of § 2 are the Jl:A->A@A the group law. Put A' =Hom'_m"",[~(A,R). We then get result ofjoint work with SERRE, and § 3 and § 4 owe much to discussions dual homomorphisms with him. Although p-divisible groups are interesting enoughin their own right, our main motivation for studying them has been their applications ,l:A' ®A' -+A', m':A' -rA' ® A'. to abelian varieties. For some ofthese, and for further results' onp-divisi­ fJ.' defines an algebra s1!ucture on A' andit is easy to check that m' defines ble groups, as well as additional bibliography, see SERRE, [10J and [Ill a product on G'=Spec(A'), which makes it into a group scheme. The This text owes much - probably its very existence - to the efforts of order of G' is equal to that of G. G' is called the Cartier dual of G. There T. SPRINGER, who wrote a first draft shortly after the conference. In is a canonical isomorphism G..:},(G'Y. several places, and particularly in § 3, he has made improvements on the If T is a prescheme over R, then the group G' (T) is canonically iso­ original oral exposition. I thank him heartily for his help, morphic to HomT (G, Gm) (homomorphisms of group schemes over T of G x R T into GItI x T, GItI denoting the multiplicative group). § 1. Finite group schemes Examples. (a) Let G=ZjmZ be the cyclic group of order n (as in Example (a) of (1.1». Then G'=Jl,.. (1.1). Let R be a commutative ring (or a prescheme), By afinite group (b) If G=G." (Example (c) of (1.1», then G' =G,.•. oforder mover R we shall mean a group scheme G which is locally free of rank mover R. Such a G is defined by a locally free (sheaf of)aIgebra(s) A ofrank mover R. The group strncture is described by homomorphisms (1.3). Short exact sequences A sequence 1 The research described here bas been partially supported by NSF. O-+G'~G~G"-+O (1) 160 J. TATE p-Divisible Groups 161 offinite R-groups is called exact if j is a closed immersion which identifies § 2. p-divisible groups G' with the keme1 ofj (in the sense of categories), whereas j is faithfully fiat. Ifj is given, it is easy to get G' (it is the inverse image of the unit sec­ (2.1). Definition tion of G'). Given i, one can construct G'. This is more delicate (see [4], Expose V and VI. or [7]). Let p be a prime number, and h an integer ~O. A p-divisible group G If (1) is exact, then we have for the orders In, m', m" of G, G', G" the over R ofheight h is an inductive system relation m=m'm'. This follows from the fact (proved loco ci!.) that the G =(G" i,), v ~ 0, graph of the equivalence relation in G x R G defined by G' is isomorphic where . to GxG.G. '~-----"(i)- -G;-is-a-finite-group-scheme-over-R-or-ordeT p,.IJ;-- ------------.-.-....,- Finally, the dual of an exact sequence (1) is also exact. (ii) for each v;;'O, i" p" 0-1' G., -+ G'I'+l -+ G"+1 (1.4). Connected and etale groups is exact (i.e., G" can be identified via I" with the kernel of multiplication In this section we suppose R is a complete noetherian local ring. If G by pI' in G'I'+1)' is a group of finite order over R, there is a canonical exact sequence These ax.ioms for ordinary abelian groups would imply o-+ GO ~ G ~ Get -+ 0 , G,c;;;(Zlp"zl" and G=limG,=(Q,/Zp)". -+ ;'here GO is connected and Gel is elale over R. lithe corresponding affine A honwmorphismf: G-+ H ofp-divisible groups is defined in the obvious O O rings are A , A, and Aft, then A is a product of local R-algebras, and A way: if G=(G" i,), H=(H" i,) thenfis a system of homomorphisms. is the local quotient of A through which the map e:A-+R factors, while f":G'I'-+H,, .of R-groups, which is such that l.,.f~=f.'+l·il' for all v~ 1. A et is the maximal etale subalgebra of A. The map i is an open and.c1osed By iteration, one gets from the I., closed immersions immersion, and GO is the maximal connected subgroup of G. For varying iV ,J1: G" -+ G + I, G, the functorsGI---+ GO, and GI---+Get are exact. St G is connected if GO = G. In that case, the order of G is a power of the for all!" v;;,O, which ideutify G, with the kernel of multiplication by p' characteristic exponent of the residue field k of R, i.e., is I ifchar(k)=O, in all G.+>. It follows that the homomorphism and is a power ofp if char Ie = p > 0. This follows from the theory offinite pSt: G/ + -+ G/ +" group schemes over k (or even over k). 1 V l Gis etale if G=G". The functor Gf-> G(k:) is an equivalence of the can be factored through G, and then a consideration of orders shows that category of etale R-groups of finite order aud finite n-modules ou which we have an exact sequence n operates continuously, where n=Ga1(k:lk) is the fundamental group of J•• , °~ G'St---+•. , G{I+V-)- G,,~°, (2) R. Given such an-module r, the etale finite R-group r corresponding to it is given by r=SpecA, where A is the ring ofall functions r-+Ret which wherejSt,,, is the unique homomorphism such that i",St·jSt,,,=p"". commute with n, and where Ret is a "maximal local eWe ~ntegral exten~ Examples. (a) Let X be an abelian scheme over R of dimension d. Let sion" of R (I.e., a maximal unramified extension of R in the old terminol­ X, be the kernel of multiplication by n. Then (Xp '" i,) (i, denoting the ogy, if R is a complete discrete valuation ring), and where 11; operates on obvious iuclusion) is a p-divisible group X(p), with height h = 2d. R er in the unique way compatible with its operation on the residue field (b) The same construction can be performed for other groups over R. extension k,,11e (the residue field Ie", of R", being a separable algebraic If one takes X=Gm, the resulting p-divisible group is Gm(p)=(I'p" i,.). closure of k). For general (not necessarily etale) G, we have G"=G(k:). Its height is I. 162 J. TATE p-Divisible Groups 163 Question: Are there any p-divisible groups over Z other than products ofR so that, with the previous notations,mJi'+1= M, say, is the maximal of powers of Gm(p) and of QplZp1 ideal of d. Lemma O. The ideals ml'd+Jv constitute a fundamental system of (2.2). Relations with formal Lie groups neighborhootk of0 in the M-adic topology ofd. Indeed, dl(m'A+J,)=A,/m'A, is an Arlin riug, so the ideals in ques­ In this section we assume R complete, noetherian, local, with residue tion are M-adically open.