(t). (see e.g. [Qui71]), and so the result follows from the isomorphism (10.22) O(level_(Z=p, GE)) ~=ß0E[[t]]=

(t). For the general case, we may suppose that A0*( A* is maximal, and so we have * *a pull-back diagram 0 BA0* --i--!B0 ? ? j0?y ?yj BA* --i--!BC* where C ( A is cyclic of order p. The commutativity of the diagram 0*oo_i0*___ B0 ß0EBA + ß0E +______________ ____________________________* *___ ø|| ø|| ___________________________* *________________________ fflffl| * fflffl|________________________* *____________________ ß0EBA*+oo_i_____ß0EBC*+_0_________________________* *_________________________ ___________________________* *____ | | ___________________________* *______ ß| | _________________________ fflffl| ffl___________________________f* *fl| O(level_(A,oG))oO(level_(C,_G)) implies that ß(ø(1)) = 0. 0* BA* 0* * * BA* The result follows, since ß0EBA +is a cyclic ß0E +-module via j , and øis a ma* *p of ß0E +-modules. 9.Cohomology of Thom spectra Suppose that X is a space, and that V is a complex vector bundle over X. Th* *e ß0EX+-module ß0SU(XV , E) is free of rank one (since E is complex orientable) and so can be * *interpreted as the mod- ule of sections of a line bundle L(V ) over XE. The fact that the Thom complex * *of an external Whitney sum is the smash product of the Thom complexes gives rise to a canonical isomorphism L(V W) ~=L(V ) L(W) (9* *.1) This property can then be used to extend the definition of L(V ) to virtual bun* *dles; we define L(V - W) = L(V ) L(W)-1. (9* *.2) If f : X ! Y is a map, and V is a virtual bundle over Y , then there is an isom* *orphism *V X V ß0SU(Xf , E) ~=ß0E + ß0SU(Y , E) ß0EY+ In terms of algebraic geometry, this means that there is a natural isomorphism L(f*V ) ~=(fE)*L(V ). (9* *.3) Here is a series of examples which lead to a fairly complete understanding of* * the functor L(V ). (1)If L denotes the canonical line1bundle over CP1 , then the zero section ide* *ntifies ß0SU(CP1 L, E) with the augmentation ideal in ß0ECP+, and so we have an isomorphism L(L) ~=I(0). (9* *.4) THE oe-ORIENTATION IS AN H1 MAP * * 23 (2)Suppose that V is a line bundle over X, classified by a map b : X ! CP1 . * * In terms of algebraic geometry, the map b defines an XE-value point b = bE of G. It follows from (9.* *3)that L(V ) ~=b*I(0) ~=0*I(-b). (* *9.5) (3)Taking X to be a point and V to be the trivial complex line bundle in (2),* * we have L(V ) ~=0*I(0). (* *9.6) Now L(V ) is the sheaf associated to ß2E, while 0*I(0) is the sheaf of cotange* *nt vectors at the origin of G, isomorphic to the sheaf !G of invariant differentials on G. If f : E ! F is an* * E-algebra (e.g. F = EX+), this gives an interpretation of the homotopy group ß2kF as the sections of f*!* *G. (4)If V is the trivial bundle of dimension k, then by (9.6)and (9.1), L(V ) i* *s just !k. (5)If V = (1-L) is the reduced canonical line bundle over CP1 , then using (9* *.2), (9.4), and (9.6)we have L(V ) ~=ß*0*I(0) I(0)-1 = 1(I(0)), where ß : GE ! SE is the structural map and 1 is defined in (15.2). (6)With the notation of example (2) consider bundle V L over X xCP1 . Then t* *he line bundle L(V L) is pulled back from IG(0) along the map XE x G bx1--!G x G ~-!G. It follows that L(V L) ~=T*bI(0) = IXExG(-b). (* *9.7) P (7)More generally, suppose that V = niLiis a virtual sum of line bundlesPov* *er X. The line bundles Li define points biof G over XE, and the bundle V determines the divisor D = ni* *{bi}. It follows using (9.1) that L(V L) = IXExG(D-1), (* *9.8) P -1 where D-1 = ni{bi }. (8)In fact, by the splitting principle, the line bundle L(V L) can be compu* *ted in this manner even when V is not a virtual sum of line bundles. Indeed, by the splitting principle, th* *ere is a map f : F ! X with the properties that fE is finite and faithfully flat, and f*V is a virtual sum* * of line bundles. The line bundle L(f*(V ) L) can then be computed as O(D-1) as above. But then it is easy to * *check that the divisor D descends to XE x G, even though none of its points do. (9)Let A be a finite abelian group. An element a 2 A can be regarded as a cha* *racter of A*. Let Va be the associated line bundle over BA*. Recall (8.1)that this construction defines a * *group homomorphism Ø : A ! G(BA*E). The line bundle L(Va V L) over BA*Ex XE x G is L(Va V L) ~=T*aI(D-1); taking V to be the trivial line bundle over a point gives L(Va L) ~=T*aI(0) = I(a-1) (10) Now let M Vreg= Va a2A be the regular representation of A*. Over the scheme (BA*)E x G, the line bund* *le associated to the Thom complex of Vreg V L is _ ! O X L(Vreg V L) ~= T*aI(D-1) ~=I T*aD-1 . (* *9.9) a2A a In particular, O L(Vreg L) ~=I(`) = T*aI(0). (9* *.10) a2A 24 ANDO, HOPKINS, AND STRICKLAND (11) Suppose that the map ~Ø: (BA*)E ! hom_(A, G) of (8.2)is an isomorphism. If AT `-!i*G q-!G0 is a level structure with cokernel q over T, then changing base in (9.10)along T x G Ø`-!hom_(A, G) x G (where Ø`is the map classifying the homomorphism `; see (8.4)) and using (12.6* *)gives Ø*`L(Vreg L) ~=q*NqIG(0) ~=q*IG0(0). (9* *.11) (12) Restricting the above example to BA* we find that Ø*`L(Vreg) = 0*Gq*IG0(0)= 0*G0IG0(0) = !G0. This series of examples establishes the following results: Proposition 9.12. For a pointed topological space X, let F be the spectrum EX+* *, and let G = GF = CP1 F be the associated formal group. Attached to each (virtual) complex vector bund* *le V over X is a divisor DV on G, and an isomorphism tV : L(V L) ~=IG(D-1). (9* *.13) The map tV restricts to an isomorphism tV : L(V ) ~=0*I(D-1). Proposition 9.14. The correspondence V 7! DV and the isomorphism (9.13)are det* *ermined by the follow- ing properties i)If V = V1 V2, then DV=DV1+ DV2, and with the identifications L(V1) L(V2)~=L(V ) I(D-1V1) I(D-1V2)~=I(D-1V), there is an equality tV = tV1 tV2. ii)If f : Y ! X is a map of pointed spaces, and if W = f*V , then DW = f*DV, * *and tW = f*tV. iii)If X is a point, and V has dimension 1, then D = {0}, and the isomorphism tL : L(L) ~=I(0) (9* *.15) is given by applying ß0E(-) to the inclusion of the zero section CP1+! CP1 L. Part 3.Level structures and isogenies of formal groups 10.Level structures 10.1. Homomorphisms. Suppose that A is a finite abelian group and G is a forma* *l group over a formal scheme S. Definition 10.1. We write hom_(A, G) for the functor from formal schemes to gr* *oups defined by the formula hom_(A, G)(T) = {pairs(u, `) | u : T ! S , ` 2 hom(A, u*G(T))}. Remark 10.2. We shall use the notation AT `-!u*G to indicate that T is a formal scheme and (u, `) 2 hom_(A, G)(T). THE oe-ORIENTATION IS AN H1 MAP * * 25 Example 10.3. Let G be a formal group over R, and suppose that x is a coordinat* *e on G. Let F be the resulting group law. The "n-seriesö f F is the power series [n](t) 2 R[[t]] de* *fined by the formula [n](x) = n*x, where the right-hand-side refers to the pull-back of functions along the homomo* *rphism n : G -!G. To give a homomorphism ` : Z=n ! G(T) is to give a topologically nilpotent element x(`(1)) of O(T), with the property* * that [n](x(`(1))) = 0; the homomorphism ` is then given by x(`(j)) = [j](x(`(1))). It follows that i j hom_(Z=n, G) = spfR[[x(`(1))]]= [n](x(`(1))) . It is clear from the definition that if B A there is a restriction map hom_(A, G) ! hom_(B, G), and if A = B x C then the resulting map hom_(A, G) ! hom_(B, G) xS hom_(C, G) (10* *.4) is an isomorphism. Also from the definition we see that if j : S0! S is a map o* *f formal schemes, then the natural map hom_(A, j*G) ! j*hom_(A, G) is an isomorphism. Combining these observations with Example 10.3 and the stru* *cture of finite abelian groups gives the following. Lemma 10.5. The functor hom_(A, G) is represented by an affine formal scheme ov* *er S. If j : S0! S is a map of formal schemes, then the natural map hom_(A, j*G) ! j*hom_(A, G) is an isomorphism of formal schemes over S0. For formal groups over p-local rings, only the p-groups give anything interes* *ting. Example 10.6. Returning to Example 10.3, the n-series is easily seen to be of t* *he form [n](t) = nt + o(2). If n is a unit in R, then R[[x]]=([n](x)) ~=R so hom_(Z=n, G) is the trivial group scheme over R. Example 10.7. If R is a complete local ring of residue characteristic p, then t* *here is an h with 1 h 1 such that mh h [pm ](t) ffltp + o(tp + 1) mod mR. This h is called the height of G. If h is finite, then the Weierstrass Preparat* *ion Theorem [Lan78, pp. 129-131] implies that there are monic polynomials gm (t) of degree pmh such that [pm ](t) = gm (t) . ffl, where ffl is a unit of R[[t]]. It follows that O(hom_(Z=pm , G)) is finite and * *free of rank phm over R. These examples generalize to give the following. Proposition 10.8. Let G be a formal group of finite height over a local formal * *scheme S. Then hom_(A, G) is a local formal scheme over S. For B A, the forgetful map hom_(A, G) ! hom_(B, G) is a map of formal schemes, finite and free of rank dh, where d is the order of* * the p-torsion subgroup of A=B. 26 ANDO, HOPKINS, AND STRICKLAND Proof.By the product formula (10.4)and Example 10.6 we are reduced to the case * *that A and B are p-groups. By induction it suffices to treat the case that 0 ! B ! A ! C ! 0 is a short exact sequence, where C is cyclic of order p. Let c be a generator o* *f C, a and element of A mapping to c, and let b = pa. Suppose that x is a coordinate on G. If ` : B ! G(T) is a homomorphism, then to give a homomorphism `0: A ! G(T) with `0|B = ` is to give a topologically nilpotent element x(`0(a)) of O(T) wit* *h the property that [p](x(`0(a))) = x(`(b)). It is clear that the universal example of such a situation occurs over the ring O(T) = R[[x(`0(a))]]=([p](x(`0(a))) - x(`(b))), where R = O(hom_(B, G)) and ` : B ! G(R) is the tautological map. The Weierstra* *ss Preparation Theorem implies that O(T) is finite and free over R, of rank ph. 10.2. Level structures. The scheme hom_(A, G) has an important closed subscheme* * level_(A, G), which was introduced by Drinfel'd [Dri74]. Suppose that G is a formal group over a formal* * scheme S. Definition 10.9. Let T be a formal scheme. A T-valued point AT `-!i*G of hom_(A, G) is a level A structure (or level structure or A-structure for sho* *rt) if for each prime q dividing |A|, the subgroup i*G[q] = ker(q : i*G ! i*G) is a divisor on G=T, and there is* * an inequality of divisors X {`(a)} i*G[q] a2Aqa=0 in i*G. The subfunctor of hom_(A, G) consisting of level structures will be den* *oted level_(A, G). Remark 10.10. If we say that AT `-!i*G "is a level structure," we mean that T is a formal scheme, and (i, `) is a T-va* *lued point of level_(A, G). We may omit one of T and i if it is clear from the context. Here are some examples to give a feel for level structures. First of all, onl* *y p-groups of small rank can produce level structures. Lemma 10.11. If |A| is not a power of p, then level_(A, G) = ;. If the height of G is h and the p-rank of A is greater than h, then again level* *_(A, G) = ;. Proof.If |A| is not a power of p, then there is a prime q 6= p such that the di* *visor X {`(a)} qa=0 has degree greater than 1. However, q : G ! G is an isomorphism, so G[q] = {0} * *has degree 1. Similarly, if the height of G is h then the degree of G[p] is ph. THE oe-ORIENTATION IS AN H1 MAP * * 27 A level structure is trying to be a monomorphism; for example if R is a domai* *n in which |A| 6= 0, then a homomorphism ` : A ! G(R) is a level structure if and only if it is a monomorphism (Corollary 10.20). How* *ever, naive monomorphisms from A to G can't in general be a representable functor. Example 10.12. Let bGmbe the formal multiplicative group with coordinate x so t* *hat the group law is x +y = x + y - xy. F The p-series is [p](x) = 1 - (1 - x)p. The monomorphism Z=p -!bGm(Z[[y]]=[p](y)) given by j 7! [j](y) becomes the zero map under the base change Z[[y]]=([p](y))! Z=p y 7! 0. On the other hand, the functor level_(A, G) is representable. Lemma 10.13. Let G be a formal group of finite height over a local formal schem* *e S, and let A be a finite abelian group. The functor level_(A, G) is a closed formal subscheme of hom_(A,* * G). Proof.See Katz and Mazur [KM85 , 1.3.4] or [Str97] (the general assumption in [* *Str97] that S is Noetherian is not used for this result). It is clear from the definition that if j : S0! S is a map of formal schemes,* * then the natural map level_(A, j*G) ! j*level_(A, G) is an isomorphism of formal schemes over S0. 10.3. The Noetherian case. If G is an Noetherian (2.3)formal group of finite he* *ight h over a local formal scheme S of residue characteristic p > 0, then we have the following. Proposition 10.14. Suppose that A is a p-group and |A[p]| ph. i)The functor level_(A, G) is represented by a local formal scheme which is f* *inite and flat over S: indeed O(level_(A, G)) is a finite free O(S)-module. ii)If G is the universal deformation of a formal group over a field (see x14) * *then level_(A, G) is the formal spectrum of a Noetherian complete local domain which is regular of dimension h. Proof.With our hypotheses, we may suppose that G=S is the universal deformation* * of a formal group of height h over a perfect field k of characteristic p; the general case follows b* *y change of base. If A = A[p], then the result is precisely the Lemma of [Dri74, p. 572, in pro* *of of Prop. 4.3]. The proof in the general case follows similar lines and is given in [Str97]. The general case for part i) can be given easily: by definition of level_(A, * *G), the diagram level_(A, G)j----!hom_(A, G) ? ? i?y ?yk level_(A[p],-G)l---!hom_(A[p], G). is a pull-back. Proposition 10.8 implies that k is finite and flat, and so i is* * too. It follows that level_(A, G) is finite and flat over S; for Noetherian complete local rings, finite and flat is* * equivalent to finite and free. 28 ANDO, HOPKINS, AND STRICKLAND 10.4. Level structures over p-regular schemes. In this section, we suppose that* * G is a formal group of finite height over a complete local ring E of residue characteristic p > 0. The* * following description of the subscheme level_(A, G) was found by Hopkins in the course of his work on [HKR00* * ]. Proposition 10.15. Suppose that G is Noetherian, and that p is not a zero divis* *or in E. Let x be a coordinate on G. The scheme level_(A, G) is the closed subscheme of hom_(A, G)* * defined by the ideal of annihilators of x(`(a)), where a ranges over the non-zero elements of A[p]. The proof will be given at the end of this section. Note that the ideal in th* *e Proposition is independent of the coordinate used to describe it. For n 1 let A[n] denote the n-torsion in A. Let R be a complete local E-alg* *ebra, and consider the following conditions on a homomorphism ` : A ! G(R). Again, they are phrased in terms of a choice of a coordinate x on G, but they a* *re easily seen to be independent of that choice. (A)If 0 6= a 2 A[p] then x(`(a)) is regular (i.e. not a divisor of zero). (B)IfQ0 6= a 2 A[p] then x(`(a)) divides p. (C) a2A[p](x - x(`(a))) divides [p](x). (D)The natural map , 0 1 Y Y R[[x]] @ (x - x(`(a)))A! (R[[x]]=(x - x(`(a)))) (10.* *16) a2A[p] a2A[p] is a monomorphism. Condition (C) says precisely that there is an inequality of Cartier divisors X {`(a)} G[p]. pa=0 Thus condition (C) is that ` is a level structure. Proposition 10.17. If R is p-torsion free, then these conditions are equivalent. First we prove the following result. It will be convenient to use the symbol * *ffl to denote a generic unit. Its value may change from line to line. Lemma 10.18. Let n = |A[pm ]|. The discriminant of the set {x(`(a)) | a 2 A[pm ]} is Y = ffl x(`(a))n. 06=a2A[pm] Proof.Let F be the group law associated to a coordinate on G. The formula x -y = (x - y)ffl(x, y), F THE oe-ORIENTATION IS AN H1 MAP * * 29 where ffl(x, y) 2 E[[x, y]]x, gives Y = (x(`(a)) - x(`(b))) a6=b2A[pm] Y = ffl (x(`(a)) -x(`(b))) Y F = ffl x(`(a) - `(b)) Y Y = ffl x(`(c)) c6=0a-b=c Y = ffl x(`(c))n. c6=0 Proof of Proposition 10.17.Under the hypothesis that p is regular in R, it is c* *lear that (B) implies (A). Let's check that (A) implies (B). Note that [p](x) = x(p + xe(x)) for some e(x) 2 E[[x]]. For a 2 A[p] we have 0 = [p](x(`(a))) = x(`(a))(p + x(`(a))e(x(`(a)))). If x(`(a)) is not a zero-divisor in R, then we must have p = -x(`(a))e(x(`(a))). Next, let check that (C) implies (B). If (C) holds, then there is a power ser* *ies e(x) 2 E[[x]] such that Y e(x) (x - x(`(a))) = [p](x) = px + o(x2) a2A[p] The coefficient of x on the left is (up to a sign) Y e(0) x(`(a)) 06=a2A[p] so (B) holds. Next let's check that (A) implies (D). With respect to the basis of powers of* * x in the domain and the obvious basis in the range, the matrix of (10.16)is the Vandermonde matrix on t* *he set x(`(A[p])). Condition (A) and Lemma 10.18 together imply that (10.16)is a monomorphism. Finally, let's check that (D) implies (C). Each x(`(a)) is a root of [p](x), * *so the image of [p](x) in the range of (10.16)is zero. If (D) holds then [p](x) is zero in the domain, which * *implies (C). Lemma 10.19. Condition (A) holds if and only if, for all non-zero a 2 A, x(`(a)* *) is a regular element of R. Condition (B) holds if and only if, for all non-zero a 2 A, x(`(a)) divides * *a power of p. Proof.Recall that the p-series [p](x) is divisible by x: let

(x) be the powe* *r series such that [p](x) = x

(x). Thus x(`(pa)) = x(`(a))

(x(`(a))). so if x(`(pa)) divides zero (resp. a power of p), then so does x(`(a)). Corollary 10.20. If R is a domain of characteristic 0, then the conditions (A)_* *(C) hold if and only if ` : A ! G(R) is a monomorphism. Proof of Proposition 10.15.By Proposition 10.14, R = O(level_(A, G)) is a finit* *e free E-module. It follows that p is not a zero divisor in R. Proposition 10.17 implies that R is initial * *among complete local E-algebras satisfying (A). 30 ANDO, HOPKINS, AND STRICKLAND Example 10.21. Let G be a Noetherian formal group of finite height over a p-reg* *ular complete local ring R of residue characteristic p. Suppose that x is a coordinate on G. Let

(t) * *2 R[[t]] be the power series such that t

(t) = [p](t). Proposition 10.15 implies that level_(Z=p, G) ~=spfR[[x(`(1))]]=

x(`(1)). (10.* *22) This calculation occurs as part of the proof of the Lemma in the proof of Propo* *sition 4.3 of [Dri74]. 10.5. Calculations in hom_(Z=p, G) via level structures. Let G be a Noetherian * *(2.3)formal group of finite height over a p-regular complete local ring R of residue characteristic * *p > 0. Let A be a finite abelian group. By construction there is a natural map O(hom_(A, G)) ! O(level_(A, G)). There is also a ring homomorphism O(hom_(A, G)) ! R classifying the zero homomorphism. The proof of Proposition 7.1 uses the follow* *ing result. Proposition 10.23. The natural map O(hom_(Z=p, G)) ! R x O(level_(Z=p, G)) (10.* *24) is injective. Remark 10.25. This result is equivalent to the injectivity for the group Z=p of* * the character map of [HKR00 ]. Proof.Let h be the height of G. Let = (Z=p)h. Let g(x) be the monic polynomia* *l of degree ph such that [p](x) = g(x)ffl where ffl 2 R[[x]]x. Then hom_(Z=p, G) = spfR[[x]]=[p](x) ~=spfR[[x]]=g(x), and Proposition 10.15 (see Example 10.21) implies that O(level_(Z=p, G)) = R[[x]]=

(x). Let D = O(level_( , G)) and let ` : ! G(hom_( , G)) be the tautological homomorphism. By definition, , 0 Ideal obtained by1 D = O(hom_( , G)) @ Q equating coefficientsAin. a2(x - x(`(a))) = g(x) By Proposition 10.14, D is finite and free over R. Therefore, letting Ø denote * *the map (10.24), it suffices to show that Db Ø is injective. Each non-zero a 2 gives a monomorphism Z=p ,! and so a homomorphism O(level_(Z=p,aG))-!D x 7! x(`(a)). We may view these all together as a ring homomorphism Y Db O(level_(Z=p, G)) M-! D[[x]]=(x - x(`(a))). 06=a2 Note that the identity map of D may be written as D F-!D[[x]]=(x - x(`(0))) = D. THE oe-ORIENTATION IS AN H1 MAP * * 31 With this notation, the diagram bØ Db O(hom_(Z=p, G)) -D---!Db (R x O(level_(Z=p, G))) ? ? ~=?y ?yFxM Q Q D[[x]]= a2 (x - x(`(a)))----!a2D[[x]]=(x - x(`(a))) commutes, where the map across the bottom is the evident map (10.16). It is a m* *onomorphism by Proposition 10.17. 11.Isogenies In this section we suppose that G and G0are formal groups over a local formal* * scheme S of residue characteristic p. Definition 11.1. An isogeny is a finite and free homomorphism f : G ! G0of form* *al groups. In particular, kerf is a finite group scheme over S. Isogenies are epimorphisms. Lemma 11.2. If f : G ! G0and g : G ! G00are two isogenies, such that kerf kerg, then there is a unique isogeny h : G0! G00 such that g = hf. A level structure has a cokernel which is an isogeny. Let A be a finite group* *, and let AS `-!G be a level structure on a Noetherian (2.3)formal group of finite height over a * *local formal scheme S of residue characteristic p. Proposition 11.3. Ti)he induced map of formal group schemes AS `-!G (11* *.4) is the inclusion of a sub-groupscheme. ii)The map (11.4)has a cokernel G q`-!G=`(A) which is an isogeny of formal groups. The map q` gives an isomorphism O(G=`(A)) ~=O(G)A. (11* *.5) If x is a coordinate on G, then the isomorphism (11.5)identifies Y T*ax (11* *.6) a2A with a coordinate on G=`(A). Proof.It suffices to prove the Proposition in the case that G is the universal * *deformation a formal group of height h over a field of characteristic p, and R = O(level_(A, G)). In that * *case the result is essentially Proposition 4.4 of [Dri74]: Drinfel'd actually considers a level structure of t* *he form = (Z=pn)h `-!G(R) and a subgroup A , but his argument uses only the A-structure and the fact t* *hat R (G) is a Noetherian p-regular complete local domain. (The existence of the quotient and the coordin* *ate (11.6)in that case is due to Lubin [Lub67].) Strickland [Str97] gives a complete proof in the general* *ity considered here. 32 ANDO, HOPKINS, AND STRICKLAND 12.The norm map Let A be a finite abelian group. Let G and G0be formal groups of finite heigh* *t over a local formal scheme S of residue characteristic p. Let AS `-!G be a level structure with cokernel G q-!G0. An OG0-module W0 gives rise to an A-equivariant OG-module W = q*W0, and W0= (q*W)A. Proposition 11.3 implies the following. Proposition 12.1. The functor W0 7! q*W0 is an equivalence of categories from f* *inite (resp. finite and free) OG0-modules to A-equivariant finite (resp. finite and free) OG-modules. T* *he OG0-module corresponding to an equivariant OG-module W is (q*W)A. In particular q*W0 is a line bundle if* * and only W0 is. Now let L be a line bundle over G. The line bundle O T*aL a2A is equivariant, and so Proposition 12.1 justifies the following. Definition 12.2. The norm of L is the line bundle NL = N`L over G0determined by* * the equation O q*NL = T*aL. a2A Explicitly, we have O (NL) = ( ( T*aL))A. If s is a section of L, then the norm of s is the section O Ns = T*as 2 (NL). a The norm map is not additive, but it is multiplicative, in the sense that if f * *is a function on G, then N(fs) = Nf . Ns. Let ~ be the map of line bundles O ~ : T*aOG ! OG a2A given by the formula ~(f1 . . .fm ) = f1. .f.m. Lemma 12.3. The map ~ is an isomorphism of A-equivariant line bundles over G. Lemma 12.3 and Proposition 11.3 give the following. Proposition 12.4. The map f 7! ~-1(f O q) induces an isomorphism of line bundles OG0~=NOG which restricts to an isomorphism IG0(0) ~=NIG(0). (12* *.5) If s is a coordinate on G, then Ns is a coordinate on G0. THE oe-ORIENTATION IS AN H1 MAP * * 33 Equivalently, q induces an isomorphism q*IG0(0) ~=q*NIG(0) ~=IG(`) (12* *.6) of line bundles over G. 13.Descent for level structures In Definition 3.1 we described "descent data for level structuresä s they ap* *pear on the formal group of an H1 ring spectrum. In this section, we give an equivalent description (Propos* *ition 13.14) which displays the relationship to the usual notion of descent data. In addition to justifyin* *g the terminology, the new formulation simplifies the task of showing that the Lubin-Tate formal groups ha* *ve canonical descent data for level structures (Proposition 14.8). 13.1. Composition of isogenies: the simplicial functor level_*. Let FGpsbe the * *functor from admissible local rings R to sets whose value on R is the set of formal groups G= spfR. If * *f : R ! R0is a map of admissible local rings, then FGps(f) sends G= spfR to f*G= spfR0. Let level_(A) -!FGps be the functor over formal FGpswhose value on R is the set of formal groups G= * *spfR equipped with a level structure A -!G. We define a level_1def=level_(A0); A0 the coproduct is over all finite abelian groups. We have adorned the level_and * *the A with subscripts so that we can make the more general definition a level_n= level_(A0). 0=An An-1... A0 The coproduct is over all sequences of inclusions of finite abelian groups with* * An = 0. With this convention we also have FGps= level_0. We write d0: level_1! FGps (13* *.1) for the structural map. Over level_1(A) we have a level structure A `-!d*0G and an isogeny d*0G qA--!G=`(A) with kernel A. These assemble to give a group G=` and an isogeny d*0G q-!G=` over level_1. We write d1: level_1-!FGps (13* *.2) for the map classifying G=`. 34 ANDO, HOPKINS, AND STRICKLAND Lemma 13.3. Let A `-!G be a level structure. If B A, then the induced map B `|B--!G is a level structure. If q : G -!G0is an isogeny with kernel `|B, then the indu* *ced map `0: A=B -!G0 is a level structure. Proof.The first part is clear from the definition of a level structure (10.9). * *For the second part, consider the diagram A ---`-! G ?? ? y ?yq 0 A=B --`--!G0. Let D be the divisor X D = {`(a)} a2Apa=0 on G; by hypothesis we have an inequality of Cartier divisors D G[p]. It follows that X X T*bD T*bG[p]. b2B b2B The formula (11.6)for the coordinate on the quotient G0shows that the left side* * descends to the divisor X {`(c)}, c2(A=B) pc=0 while the right side descends to the divisor G0[p]. The Lemma gives maps dj: level_n! level_n-1 for 0 j n as follows. For 0 j n - 1, the map dj sends a point 0 = An . . .Aj . . .A0-! G (13* *.4) of level_nto the point 0 = An . .c.Aj. .A.0-!G of level_n-1obtained by omitting Aj. The map dn sends (13.4)to 0 = An-1=An-1 . . .A0=An-1-! G=`(An-1). In the case n = 1 these are just the maps (13.1)and (13.2). We also have for 0 * * j n a map sj: level_n! level_n+1 which sends the sequence (13.4)to the sequence An . . .Aj Aj . . .A0-! G obtained by repeating Aj. It is easy to check that Lemma 13.5. (level_*, d*, s*) is a simplicial functor. THE oe-ORIENTATION IS AN H1 MAP * * 35 13.2. Descent data for functors over formal groups. Now suppose that P -!FGps is a functor over FGps, and if x 2 P(R) is an R-valued point, let's write Gx fo* *r the resulting formal group over spfR. As in the previous section, we define level_(A,=P)level_(A) xFGpsP level_n(P)= level_nxFGpsP and so on. A point (`, x) 2 level_(A, P)(R) is a point x of P(R) and a level st* *ructure A `-!Gx. We write d0: level_1(P) ! level_0(P) = P. (13* *.6) for the forgetful map d0(`, x) = x. We also always have degeneracies sj: level_n(P) ! level_n+1(P) for 0 j n. If (`, x) is an R-valued point of level_(A, P), then we get an isogeny Gx -!Gx=`. Suppose that we have a natural transformation d1: level_1(P) -!P (13* *.7) such that Gd1(`,x)= Gx=`, (13* *.8) or equivalently that the diagram level_1(P)----!level_1 ? ? d1?y ?yd1 (13* *.9) P ----! FGps commutes. Lemma 13.3 then gives maps dj: level_n(P) ! level_n-1(P) for 0 j n. Definition 13.10. Descent data for level structures on the functor P consist of* * a natural transforma- tion (13.7)such that (1)the diagram (13.9)commutes, and (2)(level_*(P), d*, s*) is a simplicial functor. Remark 13.11. It is equivalent to ask for natural transformations dj: level_n(P) ! level_n-1(P) for n 1 and 0 j n, such that (level_*(P), d*, s*) is a simplicial functor* *, and the levelwise natural transformation level_*(P) -!level_* is a map of simplicial functors. 36 ANDO, HOPKINS, AND STRICKLAND For example, let G be a formal group of finite height over a p-local formal s* *cheme S. The formal scheme S has the structure of a functor over FGps: if x : spfR ! S is a point of S, th* *en Gx = x*G; We briefly write G=S for S, considered as a functor over FGpsin this way. The f* *unctor level_(A, G=S) is just the functor called level_(A, G) in x 10; in particular it is represented by the* * S-scheme level_(A, G) of Lemma 10.13. To give maps _`and f`which satisfy condition (1)of Definition 3.1 amount* *s to giving a map d1: level_1(G=S) ! S and an isogeny d*0G q-!d*1G whose kernel on level_(A, G=S) is A. Lemma 13.3 gives maps dj: level_n(G=S) -!level_n-1(G=S) for 0 j n as explained above. With these definitions, parts (2)and (3)of De* *finition 3.1 are equivalent to asserting that (level_*(G=S), d*, s*) is a simplicial functor, and over level_2(G) the diagram d*0d*0G (13.* *12) u III * uuuuuuu Id0qI uuuuuu III uuuu $$I d*1d*0G d*0d*1G d*1q|| |||| fflffl| || d*1d*1GI d*2d*0G IIIII uuu IIIIII uuu IIIIIIzzud*2quu d*2d*1G commutes. A more convenient formulation of Definition 3.1 is the following. Let G=S_to * *be the functor over FGps whose value on R is the set of pull-back diagrams G0 --f--!G ?? ? y ?y spfR --i--!S. such that the map G0! i*G induced by f is a homomorphism (hence isomorphism) of formal groups over spfR. * *For a finite abelian group A, level_(A, G=S_)(R) is the set of diagrams AspfR__`__//_HG0f_//_G HHH | | HHH | | H##Hfflffl|fflffl|i spfR____//_S, where the square part is a point of G=S_(R) and ` is a level structure. To give* * a map of functors level_1(G=S_) d1-!G=S_ (13.* *13) THE oe-ORIENTATION IS AN H1 MAP * * 37 making the diagram level_1(G=S_)d1----!G=S_ ?? ? y ?y level_1--d1--!FGps commute is to give a pull-back diagram G=` ----! G ?? ? y ?y level_1(G=S)----!S; it is equivalent to give a map of formal schemes d1: level_1(G=S) ! S and an isogeny d*0G q-!d*1G whose kernel on level_(A, G=S) is A. Proposition 13.14. Let G be a formal group over an admissible local ring R, and* * let S = spfR. Descent data for level structures on the group G=S is equivalent to descent data for le* *vel structures on the functor G=S_. Proof.One checks that the commutativity of the diagram (13.12)has been incorpor* *ated in the structure of the functor G=S_. 13.3. Noetherian rings and artin rings. Suppose that D is a subcategory of the * *category of admissible local rings. If P is a functor from complete local rings to sets, let PD denote* * its restriction to D. Definition 13.15. Descent data for level structures on PD consists of a natural* * transformation d1: level_1(P)D d1-!PD , such that the restriction to D of the diagram (13.9)commutes, and such that the* * (level_*(P))D, d*, s*) is a simplicial functor. For example, let N be the category of Noetherian complete local rings, and le* *t A be the category of Artin local rings. If S and T are Noetherian local formal schemes, then the natural m* *aps (formal schemes)(S, T) ! (functors)(SN , TN ) ! (functors)(SA(,1* *TA3).16) are isomorphisms. Proposition 13.17. If G is a formal group over a Noetherian local formal scheme* * S, then the forgetful maps, from the set of descent data for level structures on G=S_to the set of de* *scent data for level structures on G=S_Nand on G=S_A, are isomorphisms. Proof.If G is a formal group over a Noetherian local formal scheme, then by Pro* *position 10.14, level_(A, G) is also a Noetherian local formal scheme. The result follows easily from the is* *omorphism (13.16). 14.Lubin-Tate groups Let k be a perfect field of characteristic p > 0, and let be a formal group* * of finite height over k. In this section we shall prove that the universal deformation of has descent for leve* *l structures. 38 ANDO, HOPKINS, AND STRICKLAND 14.1. Frobenius. Let k be a perfect field of characteristic p > 0, and let be* * a formal group of finite height over k. The Frobenius map OE gives rise to a relative Frobenius F map as in the* * diagram OE_____________ ____________________________________________* *_________________________________________________________________________@ ______________________________________________* *_________________________________________________________________________ _______________""______________________________* *__________________________* __F_//_DDOEk___// DD | | DDD | | D!!Dfflffl|Offlffl|Ek speck____//_speck. The Frobenius map F is an isogeny of degree p. 14.2. Deformations. If T is a local formal scheme, then we write T0 for its clo* *sed point. Definition 14.1. Let T be a local formal scheme. A deformation of to T is a t* *riple (H=T, f, j) consisting of a formal group H over T and a pull-back diagram HT0 --f--! ?? ? y ?y T0 --j--!speck, such that the induced map HT0! j* is a homomorphism (and so isomorphism) of fo* *rmal groups over T0. The functor from complete local rings to sets which assigns to R the set of def* *ormations of to spfR will be denoted Def( ). From the definition it is clear that if (H=T, f, j) is a deformation of , th* *en there is a natural transformation H=T_! Def( ). Lubin and Tate [LT66] construct a deformation (G=S, funiv, juniv) with an isomo* *rphism S ~=spfWk[[u1, . .,.uh-1]] (14* *.2) inducing juniv: S0~=speck such that the natural transformation G=S_N ! Def( )N (14* *.3) is an isomorphism of functors over FGps. 14.3. Descent for level structures on deformations. We continue to fix a formal* * group of finite height over a perfect field k of characteristic p > 0. Let A be a finite abelian group. If R is a complete local ring, then a point* * of level_(A, Def( )) is a commutative diagram A --`--!H ---- HT0 --f--! ?? ? ? y ?y ?y (14* *.4) T ---- T0 --j--!speck, consisting of a deformation (H, f, j) of to R and a level structure ` on H. T* *he level structure in (14.4)of Def(A, )(R) has a cokernel H q-!H0 If x is a coordinate on H, then x(`(a)) is topologically nilpotent in O(T). It * *follows that x(`(a)) = 0 on T0, and so there is a canonical isomorphism _qmaking the diagram r can f HT0_F__//_(OEr)*HT0//_HT0____//:: uu qT0|| u_uuu || || || fflffl|quufflffl|uOEfflffl|rjfflffl| H0T0______//T0______//T0___//speck THE oe-ORIENTATION IS AN H1 MAP * * 39 _ commute. In other words (H0, fcanq, jOEr) is a point of Def( )(T), and we have* * constructed a natural transformation level_1(Def( )) d1-!Def( ). (14* *.5) satisfying (13.8). The fact that OErOEs= OEr+sthen implies Lemma 14.6. The map d1 is descent data for level structures on the functor Def(* * ). Now let (G=S, funiv, juniv) be Lubin and Tate's universal deformation of = s* *peck. Using the isomor- phism (14.3)and Proposition 13.17 we may trade G=S_for Def( ) in (14.5)to get a* * map d1: level_1(G=S_) -!G=S_ (14* *.7) such that Proposition 14.8. The natural transformation d1 is descent data for level struc* *tures on the functor G=S_, and so gives descent data for level structures on the formal group G=S. 14.4. Comparison to the descent data coming from the E1 structure of Goerss and* * Hopkins. The construction of the descent data in Proposition 14.8 uses the equality pr{0} = kerFr : ! (OEr)* (14* *.9) of divisors on and the equation Fr+s= ((OEr)*Fs)Fr : ! (OEr+s)* (14.* *10) More generally, to give descent data for level structures on G=S_is equivalent * *to giving a collection of isogenies Fr: ! (OEr)* for r 1 satisfying the analogues of (14.9)and (14.10). The descent data in th* *e Proposition are uniquely determined by the choice Fr= Fr. Now let E be the homogeneous ring spectrum such that G = GE is Lubin and Tate* *'s universal deformation of , so SE = S = spfWk[[u1, . .,.uh-1]]. In work in preparation, Goerss and Hopkins [GH02 ] have shown that E is an E1 r* *ing spectrum; by Theorem 3.26 it follows that there is a map dhg1 level_1(G=S_) --!G=S_ giving descent data for level structures on G=S_. Let A be a finite group of order pr, and let AspfR`-!i*G be a level structure on G. Reducing modulo the maximal ideal in the constructio* *n of _E`(3.9), one sees that _E`= OEr: SE ! SE. Examination of the construction (3.14)of _G=E`shows that (_G=E`)S0= Fr : GS0! (OEr)*GS0. Thus we have the following result. Proposition 14.11. If E is the spectrum associated to the universal deformation* * of a formal group of finite height over a perfect field k, then the descent data for level structure* *s on GE provided by the E1 structure of Goerss and Hopkins coincide with the descent data in Proposition 1* *4.8. Remark 14.12. At the time of the writing of this paper, the result of Goerss an* *d Hopkins is not published. The arguments of this section do not depend on their result beyond the existenc* *e of the H1 structure, so a cautious statement of the Proposition is that "the descent data for level struc* *tures on GE provided by any H1 structure on E coincide with the descent data in Proposition 14.8." 40 ANDO, HOPKINS, AND STRICKLAND Part 4.The sigma orientation 15. k-structures 15.1. The functors k. Suppose that G is a formal group over a formal scheme S,* * and suppose that L is a line bundle over G. Definition 15.1. A rigid line bundle over G is a line bundle L equipped with a * *specified trivialization of 0*L. A rigid section of such a line bundle is a section s which extends the specifie* *d section at the identity. A rigid isomorphism between two rigid line bundles is an isomorphism which preserves th* *e specified trivializations. Definition 15.2. Suppose that k 1. We define the line bundle k(L) over Gk by* * the formula O |I| k(L) def= (~*IL)(-1) . (15* *.3) I {1,...,k} If s is a section of L, then we write ks for the section O |I| ks = (~*Is)(-1) . I {1,...,k} of k(L). We define 0(L) = L and 0s = s. For example we have *0*L 1(L)= ß____L 1(L)a= L0_L a 2(L)a,b= L0__La+b_L a Lb 3(L)a,b,c= L0__La+b__La+c__Lb+c_L. a Lb Lc La+b+c We observe three facts about these bundles. (1) k(L) has a natural rigid structure for k > 0. (2)For each permutation oe 2 k, there is a canonical isomorphism ,oe: ß*oe k(L) ~= k(L). Moreover, these isomorphisms compose in the obvious way. (3)There is a canonical identification (of rigid line bundles over Xk+1) k(L)a1,a2,... k(L)-1a0+a1,a2,... k(L)a0,a1+a2,... k(L)-1a0* *,a1,...~=1.(15.4) Definition 15.5. A k-structure on a line bundle L over G is a trivialization s* * of the line bundle k(L) such that (1)for k > 0, s is a rigid section; (2)s is symmetric in the sense that for each oe 2 k, we have ,oeß*oes = s; (3)we have s(a1, a2, . .). s(a0+ a1, a2, . .).-1 s(a0, a1+ a2, . .). s(a0,(* *a1,1.5.).-1=.16) under the isomorphism (15.4). A 3-structure is known as a cubical structure [Bre83]. We write Ck(G; L) for * *the set of k-structures on L over G. Note that C0(G; L) is just the set of trivializations of L, and C* *1(G; L) is the set of rigid trivializations of 1(L). We also define a functor from rings to sets by C_k(G; L)(R) = {(u, f) | u : spec(R) -!S , f 2 Ckspec(R)(u*G; u** *L)}, and we recall the following. THE oe-ORIENTATION IS AN H1 MAP * * 41 Proposition 15.7 ([AHS01]). Let G be a formal group over a scheme S, and let L * *be a line bundle over G. The functor C_k(G; L) is represented by an affine scheme over S, and for j : S0* *! S, the natural map C_k(j*G; j*L) ! j*C_k(G; L) is an isomorphism. 15.2. Relations among the k: the functor . Definition 15.8. If M is a line bundle over Gn, then we define M to be the rig* *id line bundle over Gn+1 given fiberwise by the formula Ma1,a3,...,an+1 Ma2,...,an+1 Ma1,a2,...,an+1= _________________________M. a1+a2,a3,...,an* *+1 M0,a3,...,an+1 If s is a section of M then we write s for the rigid section s(a1, . .,.an+1) = ___s(a1,_._.).__s(a2,...)_s(a 1+ a2, .* * .). s(0, a3, . .). of M. The following can be checked directly from the definitions. Lemma 15.9. i) is multiplicative: if M is a line bundle over Gn then there i* *s a canonical isomorphism of rigid line bundles (M1 M2) ~= (M1) (M2). (15.* *10) ii)Under the identification (15.10), one has (s1 s2) = (s1) (s2). iii)If L is a line bundle over G then for k 2 there is a canonical isomorphi* *sm of rigid line bundles kL ~= k-1L. (15.* *11) iv)If s is a section of L then under the isomorphism (15.11), one has ks = k-1s 16.The norm map for k-structures Given a line bundle M over Gn, n > 0, and a section b of M, let ~TbM be the l* *ine bundle whose fiber over (a1, . .,.an) is T~bMa1,...,andef=M(b+a1,a2,...,an)_. (16* *.1) M(b,a2,...) If s is a section of M, define ~T*bs by ~Tbs(a1, . .,.an) = s(b_+_a1,_._.,.an)_. s(b, a2, . .,.an) There is a canonical identifications ~Tb(L1 L2)~=~TbL1 ~TbL2, (16* *.2) ~Tb~TcL~=~Tb+cL, (16* *.3) and with respect to these identifications one has T~b(s1s2)= ~Tb(s1)T~b(s2) ~Tb~Tc(s)= ~Tb+c(s). The operation ~Tbcommutes with k. 42 ANDO, HOPKINS, AND STRICKLAND Proposition 16.4. For a line bundle L over G and for k 1 there are natural is* *omorphisms T~b kL ~= k~TbL ~= kT*bL (16* *.5) of rigid line bundles over Gk. Proof.One checks directly that if L is a line bundle over G and M is a line bun* *dle over Gn, then there are natural isomorphisms of rigid line bundles T~b 1L~= 1~TbL ~= 1T*bL T~bM ~=~Tb M. The proof follows by induction, using the isomorphism (15.11) kL ~= k-1L. Now suppose that A is a finite abelian group, and AS `-!G q-!G0 is a level structure with cokernel q. Suppose that k 1, and that L is a line * *bundle over G. N Lemma 16.6. The rigid line bundle b2A~Tb kL over Gk is equivariant with respe* *ct to the action of Ak. Proof.Suppose that a 2 A and 1 i k. For a 2 A let Ti,a: Gk ! Gk denote translation by a in the i coordinate, and, for a line bundle M over Gk l* *et ~Ti,aMa1,...,ak= Ma1,...,ai+a,...,ak_. Ma1,...,a,...,ak The symmetries of the line bundles kL together with the isomorphism (16.5)impl* *y that there is a canonical isomorphism ~Ti,b kL ~= k(T*bL) and so, again using (16.5), O O ~Tb k(L) ~= T~i,b k(L). b2A b2A Of course ~Ti,a+b= ~Ti,a~Ti,band so for any line bundle M over Gk there is a ca* *nonical isomorphism O O ~Ti,a ~Ti,bM ~= T~i,bM. b2A b2A Finally _ ! _ ! O O O T*i,a ~Ti,bM ~= T~*i,aT~i,bM (T~i,bM)a1,...,a,...,ak b2A a1,...,ak b2A a1,...,akb2A _ ! ~= O T~i,bM O Ma1,...,a+b,...,ak_. b2A a1,...,akb2AMa1,...,b,...,ak The last factor has a canonical trivialization. The Lemma together with Proposition 12.1 justifies the following. Definition 16.7. If M is a line bundle over the form kL, then we define ~NM to* * be the line bundle over G0ksuch that O q*N~M = ~TbM. b2A THE oe-ORIENTATION IS AN H1 MAP * * 43 There is a canonical isomorphism N~(M1 M2) ~=~NM1 ~NM2. (16* *.8) Proposition 16.4 implies Proposition 16.9. There is are natural isomorphisms ~N kL ~= kN~L ~= kNL of rigid line bundles over G0k. As with the ordinary norm, the map ~Ncan be extended to sections. If s is a s* *ection of M, we define a section ~Ns of ~NM by O ~Ns = ~T*as. a2A As with the ordinary norm, this is not linear, but satisfies ~N(s1 s2) = ~Ns1 ~Ns2. under the isomorphism (16.8). The norm map ~Ncommutes with in the sense that N~ s = N~s. With all this it is straightforward if tedious to verify Proposition 16.10. If s is a k-structure on L, then ~Ns is a k-structure on N* *L. 17.Elliptic curves Definition 17.1. An elliptic curve is a pointed proper smooth curve _0______________________________________ ""_______________________________________* *_________ C ____//_S whose geometric fibers are connected and of genus 1. Much of the theory of level structures, isogenies, k-structures, which we ha* *ve described in detail in this paper for formal groups, carries over to elliptic curves. In this section we br* *iefly recall some results which we will need. 17.1. Abel's Theorem. Note that the discussion of the line bundles kL in x15 a* *pplies to abelian groups in any category where the notion of line bundle makes sense. The first result a* *bout elliptic curves is that they are group schemes. Theorem 17.2 (Abel). An elliptic curve C=S has a unique structure of abelian gr* *oup scheme such that the rigid line bundle 3(IC(0)) is trivial. The (necessarily unique) rigid triviali* *zation s(C=S) of 3(I(0)) is a cubical structure. Proof.See for example [KM85 , p. 63]. Remark 17.3. The theorem of the cube says that any line bundle over an abelian * *variety has a unique cubical structure. A general enough statement of the theorem of the cube, toget* *her with the group structure on elliptic curves, implies Theorem 17.2. We have stated Theorem 17.2 to emphas* *ize that the group structure on an elliptic curve is constructed to trivialize 3(I(0)), so that by the time* * you get around to applying the theorem of the cube, you already know the conclusion for I(0). 44 ANDO, HOPKINS, AND STRICKLAND 17.2. Level structures on elliptic curves. Abel's Theorem 17.2 means that it ma* *kes sense to study level structure on elliptic curves. Katz and Mazur do this in [KM85 ]. Let C be an elliptic curve over a scheme S, and let A be an abelian group. Definition 17.4. A group homomorphism A `-!C(S) is a level A-structure (or level structure or A-structure) if the Cartier divis* *or {`} is a sub-groupscheme. Lemma 17.5. Let C be an elliptic curve over an admissible local ring R. If ` : A ! bC(R) is a level structure on the formal group of C, then A ! bC(R) ! C(R) is a level structure on C. Proof.This follows from the definition and Proposition 11.3. 17.3. Isogenies. The definition of isogeny (11.1)is the same if G and G0are tak* *en to be elliptic curves over a scheme S. Proposition 17.6. Let A `-!C(S) be a level structure on an elliptic curve C ove* *r a scheme S. The induced map AS `S-!C of groups schemes has a cokernel C q-!C=`(A) which is an isogeny of elliptic curves. The structure sheaf of C=`(A) is (q*OC)* *A; that is, OC=`(A)(U) = OC(q-1U)A for an open set U C=`(A). Proof.See for example [Mum70 , p. 111]. Corollary 17.7. Let C be an elliptic curve over an admissible local ring R of r* *esidue characteristic p > 0, and let A `-!bC be an A-structure on its formal group. The natural map Cb=`(A) -!C"=`(A) is an isomorphism of formal groups. 17.4. The norm map. Proposition 12.1 carries over verbatim to the case that A `-!G q-!G0 is an isogeny of elliptic curves with an A-structure on its kernel (see for exa* *mple [Mum70 , p. 111]). It follows that the discussion of norms including Proposition 12.4 carries over to this si* *tuation as well. Explicitly, if L is a line bundle on G, then we get a line bundle NL on G0suc* *h that O q*NL = T*aL; a2A if U G0is an open set then (NL, U) = (L, q-1U)A. The norm map applies to sections: if U 2 G0and s 2 (L, q-1U), THE oe-ORIENTATION IS AN H1 MAP * * 45 then O A Ns = T*as 2 (L, q-1U)= (NL, U). The isomorphism NOG ~=OG0 of Proposition 11.3 induces an isomorphism NIG(`) ~=IG0(0). The discussion of the reduced norm ~Nof x 16 applies to elliptic curves as we* *ll. The main point is that, if L is a line bundle on the elliptic curve G, then the isogeny q gives isomorp* *hisms of rigid line bundles ~N kL ~= kN~L ~= kNL over G0as in Proposition 16.9, and if s is a k structure on L, then ~Ns is a * *k-structure on ~NL, as in Proposition 16.10. 17.5. The Serre-Tate theorem. Let C0 be an elliptic curve over a field k of cha* *racteristic p > 0. Definition 17.8. A deformation of C0 is a triple (D=T, f, j) consisting of an e* *lliptic curve D over a local formal scheme T of residue characteristic p > 0 and a pull-back diagram DT0 --f--! C0 ?? ? y ?y T0 --j--!speck of elliptic curves. A map deformations (ff, fi) : (D, f, j) ! (D0, f0, j0) is a pull-back square D ---ff-!D0 ?? ? y ?y T ---fi-!T0 such that the diagram f ___________________________________________* *_________________________________________________________________________@ _____________________________________________* *_________________________________________________________________________@ _ffT0_________""______________________________* *________0f0 DT0_____//DT00___//_C0 | | | | | | fflffl|f|fflffli0fflffl|j0 T0_____//_______<<________T00//_speck ______________________________________________* *_______________________________________________ ____________________________________________* *_________________________________________________________________________@ _________________________________________* *________________________________________________________ j commutes. Let C0 be a supersingular elliptic curve over a field k of characteristic p >* * 0. Theorem 17.9 (Serre-Tate). The natural transformation Def(C0)N ! Def(cC0)N is an isomorphism of functors over FGpsN. Suppose that k is perfect, and let G=* *S be the universal deforma- tion of the formal group cC0. Then there is a deformation (C=S, funiv, juniv) o* *f C0 to S such that the natural maps C=S_N! Def(C0)N ! Def(cC0)N G=S_N are isomorphism of functors over FGpsN. 46 ANDO, HOPKINS, AND STRICKLAND Proof.The Serre-Tate Theorem as stated in [Kat81] proves that the forgetful nat* *ural transformation induces an isomorphism Def(C0)A ! Def(cC0)A (17.* *10) of functors of Artin local rings. On the other hand, the functor Def(C0) is eff* *ectively prorepresentable: there is deformation (C0=S0, f0, j0) with S0~=spfWk[[u]], such that the natural map (C0=S0_)A ! Def(C0)A (17.* *11) is an isomorphism (see for example [DR73 ]). It follows that (C0=S0_)N ~=Def(C0)N . Combining the isomorphisms (17.10)and (17.11)with the isomorphism Def(cC0)N ~=G=S_N gives an isomorphism of formal schemes ~ S =-!S0 and, if C is the elliptic curve over S obtained from C0=S0by pull-back, an isom* *orphism C=S_N! G=S_N. Example 17.12. In characteristic 2 the elliptic curve C0 given by the Weierstra* *ss equation y2+ y = x3 is supersingular (e.g. [Sil99]). The universal deformation of its formal grou* *p is a formal group G over S ~=spfZ2[[u1]]. It is well-known (e.g. by the Exact Functor Theorem [Lan76]) t* *hat there is a spectrum E with GE=SE = G=S : it is a form of E2. The Serre-Tate Theorem endows E with the structure of an el* *liptic spectrum: if C=S is the universal deformation of C0 to S, then there is a canonical isomorphism GE = G ~=bC of formal groups over SE. 17.6. Descent for level structures on a Serre-Tate curve. Since C=S_is a functo* *r over FGps, Definition 13.10 provides a notion of descent for level structures on C=S_. Proposition 13* *.17 and the descent data (14.5) give such descent data d1: level_1(C=S_) ! C=S_ for C=S_. Explicitly, suppose that AT `-!i*bC is a level structure over a Noetherian local formal scheme T. The descent data * *provide an isogeny of formal groups i*bCf`-!_*`bC (17.* *13) over T with kernel `. It is natural to ask for an isogeny of elliptic curves i*C g`-!_*`C extending f`. This corresponds, in the language of x13, to replacing the functo* *r FGpswith the functor Ell whose value on a ring R is the set of elliptic curves C= specR. Thus we shall r* *efer to descent data for level structures on C=S_together with isogenies g` extending f` as descent data for l* *evel structures on C=S_over Ell. Now by Proposition 17.6 we have an isogeny of elliptic curves over T i*C q-!C0 THE oe-ORIENTATION IS AN H1 MAP * * 47 with kernel `. Corollary 17.7 gives a canonical isomorphism cC0~=_*`bC (17.* *14) of formal groups over T. Theorem 17.9 implies that there is a unique isomorphis* *m of elliptic curves C0~=_*`C extending (17.14); put another way, we have the following. Corollary 17.15. The functor C=S_has descent data for level structures over Ell* *, whose restriction to Cb=S_= G=S_are the descent data given by Proposition 14.8. In particular, for e* *ach level structure AT `-!i*bC, there is a canonical isogeny of elliptic curves g` making the diagram i*bC----! i*C ? ? f`?y ?yg` _*`bC----!_*`C commute. 18.The cubical structure of an elliptic curve is compatible with descent Proposition 18.1. Let C be an elliptic curve, and let s(C=S) be the cubical str* *ucture of Theorem 17.2. If i*C ! _*C is an isogeny, then _*s(C=S) = ~Ni*s(C=S). Proof.Let f : C ! C0 be an isogeny of elliptic curves over S. The uniqueness o* *f the cubical structure implies that s(C0=S) = ~Ns(C=S). 19.The sigma orientation Suppose that E is a homogeneous ring spectrum and let G = GE. Let V be the li* *ne bundle Yk V = (1 - Li) j=1 over (CP1 )k. In Lemma 6.1 we observed that Proposition 9.12 gives an isomorphi* *sm tV : L(V ) ~= k(IGE(0)), and if g : MU<2k> ! E is an orientation, then the composition (CP1 k)V ! MU<2k> g-!E represents a rigid section s of k(IG(0)). In fact it is easily seen to be a k* *-structure, that is a ß0E-valued point of C_k(G; IG(0)). Similarly, if g : BU<2k>+ ! E is a homotopy multiplicat* *ive map, then the composite CP1 k! BU<2k> ! E represents a k-structure on the trivial line bundle OG, and so a point of C_k(* *G; OG). In [AHS01] we proved 48 ANDO, HOPKINS, AND STRICKLAND Theorem 19.1. If E is a homogeneous spectrum and k 3, then these corresponden* *ces induce isomorphisms RingSpectra(MU<2k>, E) ! C_k(G; IG(0))(ß0E) (19* *.2) and RingSpectra(BU<2k>+, E) ! C_k(G; OG)(ß0E). Now suppose that (E, C, t) is an elliptic spectrum: that is, E is a homogeneo* *us ring spectrum, C is an elliptic curve over SE, and t is an isomorphism t : GE ~=bC of formal groups over SE. Abel's Theorem 17.2 gives a cubical structure s(C=S) * *on C, which gives a cubical structure t*bs(C=S) on GE. Definition 19.3. [AHS01] The sigma orientation for (E, C, t) is the map of ring* * spectra oe(E, C, t) : MU<6> ! E which corresponds to t*bs(C=S) under the isomorphism (19.2). Now suppose that E is a homogeneous H1 spectrum, with the property that ß0E i* *s an admissible local ring of residue characteristic p > 0. Let S = SE. Suppose that (E, C, t) is an * *elliptic spectrum. In particular, the G = GE is Noetherian and of finite height. By Theorem 3.26, the H1 structur* *e on E gives descent data for level structures on G. Definition 19.4. An H1 elliptic spectrum is an elliptic spectrum (E, C, t) whos* *e underlying spectrum E is a homogeneous H1 spectrum E as above, together with descent data for level stru* *ctures on C=S_, considered as a functor over Ellas in x17.6, such that the diagram of functors over FGps level_1(C=S_)t----!level_1(G=S_) ? ? d1?y ?yd1 C=S_ --t--! G=S_ commutes. Proposition 19.5. Let (E, C, t) be an H1 elliptic spectrum, and suppose in addi* *tion that p is regular in ß0E. Then the oe-orientation oe(E,C,t) MU<6> -----!E is an H1 map. Proof.By Proposition 7.1, it suffices to show that, for each level structure AT `-!i*bC, we have ~Ng`s(C=SE) = _E`*s(C=SE), where g`is the isogeny of elliptic curves making the diagram i*GE ----! i*C ? ? _G=E`?y ?yg` * * _E` GE ----! _E` C commute. Proposition 18.1 gives the result. THE oe-ORIENTATION IS AN H1 MAP * * 49 Now let (E, C, t) be the elliptic spectrum associated to the universal deform* *ation of a supersingular elliptic curve C0 over a perfect field k of characteristic p > 0. For example, we may ta* *ke C0 to be the Weierstrass curve y2+ y = x3 over F2 (Example 17.12). Applying the Proposition, Corollary 17.15, and Proposi* *tion 14.11 gives the Corollary 19.6. The orientation MU<6> oe(E,C,t)-----!E is an H1 map. Appendix A. H1 -ring spectra Given an integer n 0, let Dn : SU ! SU be the functor E 7! L(n) ^ E(n), n where L(n) = L(Un, U) is the space of linear isometric embeddings from Un to U. An E1 -ring spectrum is a spectrum with maps Dn(E) ! E, n 0, making the following diagrams commute: {1U} o E----! D1E DnDm E ----! Dn+mE ?? ? ?? ?? y ?y y y (A.* *1) E _______E DnE ----! E. An H1 -ring spectrum is a spectrum E together with maps DnE ! E such that the d* *iagrams (A.1)commute up to homotopy. The category of E1 -ring spectra is naturally enriched over topological space* *s. The space of E1 -maps from E to F is the subspace of all maps consisting of those which make the diag* *rams DnE ----! DnF ?? ? y ?y E ----! F commute. For a topological space X, the spectrum which underlies the üf nction* * object" is simply the spectrum EX+. The spectrum which underlies E X is more difficult to describe.* * If E is only an H1 -ring spectrum, the spectrum EX+ is still H1 . These remarks actually depend very little on the construction of the functor * *Dn and are mostly matters of pure category theory. Indeed, the map DnE ! E can be regarded as a natural t* *ransformation of functors SU(F, E) ! SU(DnF, E). Given a topological space X, we can use (2.4)to define a transformation SU(F, EX+) ! SU(DnF, EX+). (A.* *2) as the composite SU(F, EX+)~=Spaces(X, SU(F, E)) ! Spaces(X, SU(DnF, E)) ~=SU(DnF, EX+). 50 ANDO, HOPKINS, AND STRICKLAND A more subtle property is that the transformation (A.2)is also given by SU(F, EX+) ~=SU(F ^ X+,-E))-!SU(Dn(F ^ X+), E)) -diag-!S (A.* *3) U(Dn(F) ^ X+), E) ~=SU(DnF, EX+). An important property of the functors is summarized in the following result o* *f [BMMS86 ]. Proposition A.4. There is a natural weak equivalence ` L(2) ^ Di(E) ^ Dj(F) ! Dn(E _ F). i+j=n Furthermore, the ij-component of ` Y Dn(E) Dn(r)----!Dn(E _ E) ! L(2) ^ Di(E) ^ Dj(E) ! L(2) ^ Di(E) * *^ Dj(E) i+j=n i+j=n is the transfer map Trijwith respect to the inclusion ix j n. Note also that if W is a virtual bundle of dimension 0 over a space X, then D* *A(XW ) is the Thom spectrum the virtual bundle Vreg W over DA(X), where Vregis the regular representation * *of A*. References [Ada74] J. Frank Adams. Stable homotopy and generalised homology. Univ. of Chic* *ago Press, 1974. [AHS01] Matthew Ando, Michael J. Hopkins, and Neil P. Strickland. Elliptic spec* *tra, the Witten genus, and the theorem of the cube. Inventiones Mathematicae, 146:595-687, 2001. DOI 10.1007/s002* *220100175. [And95] Matthew Ando. Isogenies of formal group laws and power operations in th* *e cohomology theories En. Duke Math. J., 79(2):423-485, 1995. [BMMS86]R. Bruner, J. P. May, J. E. McClure, and M. Steinberger. H1 ring spectr* *a, volume 1176 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. [Bre83] Lawrence Breen. Fonctions th^eta et th'eor`eme du cube, volume 980 of L* *ecture Notes in Mathematics. Springer-Verlag, Berlin, 1983. [DR73] P Deligne and M Rapoport. Les schemas de modules de courbes elliptiques* *. In Modular functions of one variable II, volume 349 of Springer lecture notes, 1973. [Dri74] V. G. Drinfeld. Elliptic modules. Math. USSR-Sb., 23(4):561-592, 1974. [GH02] Paul Goerss and Michael J. Hopkins. Realizing commutative ring spectra * *as E1 ring spectra, 2002. Preprint. [EGAI] A. Grothendieck. 'El'ements de g'eom'etrie alg'ebrique. I. Le langage d* *es sch'emas. Inst. Hautes 'Etudes Sci. Publ. Math., (4):228, 1960. [HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Generaliz* *ed group characters and complex oriented cohomology theories. J. Amer. Math. Soc., 13(3):553-594 (electronic), 2* *000. [Hop95] Michael J. Hopkins. Topological modular forms, the Witten genus, and th* *e theorem of the cube. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994),* * pages 554-565, Basel, 1995. Birkhäuser. [Kat81] Nicholas M. Katz. Serre-Tate local moduli. In Surfaces alg'ebriques: s'* *eminaire de g'eom'etrie alg'ebrique d'Orsay 1976-1978, volume 868 of Lecture Notes in Mathematics, pages 138-202. S* *pringer, 1981. [KM85] Nicholas M. Katz and Barry Mazur. Arithmetic moduli of elliptic curves.* * Princeton University Press, Princeton, NJ, 1985. [Lan76] Peter S. Landweber. Homological properties of comodules over MU*MU and * *BP*BP. Amer. J. Math., 98:591-610, 1976. [Lan78] Serge Lang. Cyclotomic fields. Springer-Verlag, New York, 1978. Graduat* *e Texts in Mathematics, Vol. 59. [LT66] Jonathan Lubin and John Tate. Formal moduli for one-parameter formal li* *e groups. Bull. Soc. math. France, 94:49-60, 1966. [Lub67] Jonathan Lubin. Finite subgroups and isogenies of one-parameter formal * *Lie groups. Ann. of Math. (2), 85:296-302, 1967. [McC86] James E. McClure. Power operations in Hd1ring theories. In H1 ring spec* *tra and their applications, volume 1176 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. [Mum70] David Mumford. Abelian varieties. Published for the Tata Institute of F* *undamental Research, Bombay, 1970. [Qui69] Daniel Quillen. On the formal group laws of unoriented and complex cobo* *rdism theory. Bull. Amer. Math. Soc., 75:1293-1298, 1969. [Qui71] Daniel Quillen. Elementary proofs of some results of cobordism theory u* *sing Steenrod operations. Advances in Math., 7:29-56 (1971), 1971. THE oe-ORIENTATION IS AN H1 MAP * * 51 [Sil99] Joseph H. Silverman. The arithmetic of elliptic curves. Springer-Verlag* *, New York, 199? Corrected reprint of the 1986 original. [Sin68] William M. Singer. Connective fiberings over BU and U. Topology, 7, 196* *8. [Str97] Neil P. Strickland. Finite subgroups of formal groups. J. Pure and Appl* *ied Algebra, 121:161-208, 1997. [Str98] N. P. Strickland. Morava E-theory of symmetric groups. Topology, 37(4):* *757-779, 1998. Department of Mathematics, The University of Illinois at Urbana-Champaign, Ur* *bana IL 61801, USA E-mail address: mando@math.uiuc.edu Department of Mathematics, Massachusetts Institute of Technology, Cambridge, * *MA 02139-4307, USA E-mail address: mjh@math.mit.edu Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, En* *gland E-mail address: N.P.Strickland@sheffield.ac.uk