Isogenies of elliptic curves and the Morava stabilizer group Mark Behrens 1 Department of Mathematics, MIT, Cambridge, MA 02139, USA Tyler Lawson 1 Department of Mathematics, MIT, Cambridge, MA 02139, USA _____________________________________________________________________________ Abstract Let S2 be the_p-primary second Morava stabilizer group, C a supersingular ellip* *tic curve over Fp, O the ring of endomorphisms of C, and ` a topological generator * *of Zxp(respectively Zx2={ 1} if p = 2). We show that for p > 2 the group O[1=`* *]x of quasi-endomorphisms of degree a power of ` is dense in S2. For p = 2, we show that is dense in an index 2 subgroup of S2. AMS classification: Primary 11R52. Secondary 14H52, 55Q51. Key words: Morava stabilizer group, supersingular elliptic curves, quaternion algebras. _____________________________________________________________________________ Introduction Fix a prime p. Let Fn be the Honda height n formal group over Fpn. The endomorphism ring Op = End (Fn) is the unique maximal order of the Qp- division algebra Dp of Hasse invariant 1=n [18], and the Morava stabilizer group Sn is the automorphism group Aut (Fn) = Oxp. This group is a p-adic analytic group of dimension n2, and is of interest to topologists because it is intimately related to the nth layer of the chromatic filtration on the stable homotopy groups of spheres. We wish to understand the group Sn for n = 2 from the point of view of elliptic curves. _______ 1 The authors were supported by the NSF. Preprint submitted to Elsevier Preprint 2 August 2005 Throughout this paper we let C be a fixed supersingular elliptic curve defined over Fp2._Let O = End (C) be the ring of endomorphisms of the curve C defined over Fp, and let D = O Q be the ring of quasi-endomorphisms. Because C is supersingular, it is known that D is the quaternion algebra over Q ramified at p and 1 [12], and that O is a maximal order of D [23, 3.1], [4]. The reduced norm N : D ! Qxp gives the degree of the quasi-endomorphism. Let bCbe the formal completion of C at the identity. Because C is supersingular,_the formal group bCis isomorphic to the Honda formal group F2 over Fp. In fact, Tate proved that the natural map ae : End (C) Zp ! End (Cb) is an isomorphism [25]. The isomorphism ae extends to an isomorphism ~= ae0: D Q Qp -! Dp, making explicit the fact that D is ramified at p. The map ae0 is compatible with the reduced norm map on the division algebras D and Dp. Fix ` 2 to be coprime to p. As the notation suggests, we intend for ` to be another prime, but this is unnecessary for the results of this paper. Define a monoid = {x 2 O[1=`] : N(x) 2 `Z} O[1=`]x . Then is actually a group: given an endomorphism OE of degree `k, the quasi- endomorphism `-k bOEis its inverse, where bOEis the dual isogeny. Note that if ` is prime, then = O[1=`]x . The group may be regarded as being contained in the group S2 using the map ae. The purpose of this note is to prove the following theorem. Theorem 0.1 Suppose that ` is a topological generator of the group Zxp(re- spectively the group Zx2={ 1} for p = 2). For p > 2, the group is dense in S2. For p = 2, the group is dense in the index 2 subgroup eS2, which is the kernel of the composite S2 N-!Zx2! (Z=8)x ={1, `}. Let Sl2 be the kernel of the reduced norm, so that there is an exact sequence 1 ! Sl2! S2 N-!Zxp! 1. Similarly, let 1 be the corresponding subgroup of , so that there is an exact sequence 1 ! 1 ! N-!`Z ! 1. 2 (We will see in the proof of Theorem 0.1 that N is indeed surjective.) We denote by S02the p-Sylow subgroup of S2, so that there is a short exact sequence 1 ! S02! S2 ! Fxp2-1! 1. Similarly we let Sl02be the subgroup of S02of elements of norm 1. Define to be the subgroup 1 \ Sl02. Theorem 0.1 will follow from the following norm 1 versions. Note that in the following theorem and corollary, ` is only assumed to be relatively prime to p. Theorem 0.2 The group is dense in Sl02. Corollary 0.3 The group 1 is dense in Sl2. We pause to explain the reason why these theorems are interesting from the point of view of homotopy theory. The p-component of the stable homotopy groups of spheres admits an especially rich filtration known as the chromatic filtration [19]. Work of Morava, Hopkins, Miller, Goerss, and Devinatz [14], [20], [8], [5] shows that the group Sn acts on the Morava E-theory spectrum En, and the nth layer of the chromatic filtration is described by the homotopy fixed points EhSnnof this action. The first chromatic layer is completely understood. The second chromatic layer is currently the subject of intense study. Goerss, Henn, Mahowald, and Rezk [7] produced a decomposition of EhS22at the prime 3 in terms of finite homotopy fixed point spectra. The first author gave an interpretation of their work in terms of the moduli space of elliptic curves in [2]. In that paper, a spectrum Q(`) was introduced which was a shown to be a good approximation to EhS22for p = 3 and ` = 2. In future work [3], we will show that the spectrum Q(`) is the homotopy fixed point spectrum Eh . In particular, Theorem 0.1 shows that, in some sense, the spectrum Q(`) is a good approximation for EhS22for all p and suitable `. Gorbounov, Mahowald, and Symonds [9] studied dense subgroups of Sl0n(Sl in their notation), and we prove Theorem 0.2 using their methods. In particular, it is shown in [9] that if p = 3, then there is a dense subgroup Z=3 * Z=3 contained in Sl02. In [3], it will be shown that for ` = 2, the group is Z=3** *Z=3. More generally, the groups and 1 admit explicit presentations as finite amalgamations for any p and `. The authors were alerted by the referee to the related work of Baker [1]. Baker studies, for_primes_p 5, the category whose objects are supersingular elliptic curves over F p, and whose morphisms are the morphisms of the associated formal groups. He then proves an analog of Morava's change of rings theorem: roughly speaking, he shows that the continuous cohomology of this category of supersingular curves computes the E2-term of the K(2)-local Adams-Novikov spectral sequence converging to ss*(SK(2)). The chief difference between this paper and the work of Baker is that we insist on working only with the actual 3 rings of isogenies, and not their p-completions. In Section 1, we recall the relationship between maximal orders of D and the endomorphism rings of supersingular curves at p. In Section 2, we recall the homological criterion that is employed in [9] to detect dense subgroups of Sl02for p > 2. We then extend these methods to give an explicit criterion for density at the prime 2. We use these criteria in Section 3 to prove Theorem 0.1, Theorem 0.2, and Corollary 0.3. The first author would like to thank Hans-Werner Henn for pointing out to him that Theorem 0.1 was true in the case of p = 3 and ` = 2. Thanks also go to Johan de Jong and Catherine O'Neil for helpful discussions related to this paper. 1 Supersingular curves and endomorphism rings In this section we recall the correspondence between endomorphism rings of supersingular curves and_maximal orders of D. Any two supersingular elliptic curves C1 and C2 over F pare isogenous. In fact, Kohel proves the following proposition. Proposition 1.1 (Kohel_[11, Cor. 77]) Let C1 and C2 be supersingular el- liptic curves over Fp. Then for all k 0, there exists an isogeny OE : C1 ! C2 of degree `k. Let Xss_be the collection of isomorphism classes of supersingular curves C0 over Fp. Given an isogeny OE : C ! C0 of degree N, we define a map 'OE: End (C0) ! End (C) Z[1=N] D by 1 'OE(ff) = ___. bOEO ff O OE. N The map 'OEis a ring homomorphism and its image is a maximal order in D. If OE0: C ! C0 is another choice of isogeny then it is easily seen that the maximal order 'OE0(End (C0)) is conjugate to 'OE(End (C0)). Let MD be the collection of conjugacy classes of maximal orders of D. Consider the map , : Xss ! MD given by ,([C0]) = ['OE(End (C0))]. Theorem 1.2 (Deuring [4], Kohel [12]) The map , is a surjection and the preimage of a conjugacy class [O0] of maximal orders consists of either 4 a single class represented by a curve with j-invariant in Fp, or two classes represented by curves with distinct Galois-conjugate j-invariants in Fp2. In the former case, the elliptic curve can be defined over Fp, and in the latter case it can only be defined over Fp2. We choose a preferred set of representatives of conjugacy classes in MD for the remainder of this note. Fix a choice of representative C0 of each isomorphism class [C0] 2 Xss. Using Proposition 1.1, choose for each C0 an isogeny OEC0 : C ! C0 0) 0 of degree `e(C . Define OC0 to be the maximal order 'OEC0(End (C )). By letting C be the representative of its isomorphism class, and fixing OEC = IdC, we can arrange that OC = End (C). The following is immediate from Theorem 1.2. Corollary 1.3 Every maximal order O0 of D is conjugate to one of the form OC0 for some C0 2 Xss. Let O0 be a maximal order of D. Then by Corollary 1.3, cy(O0) = y-1 O0y is equal to OC0 for some y 2 D and some supersingular elliptic curve C0. The map 'bOEO cy : O0 ! End (C) Z[1=`] extends to a norm-preserving ring isomorphism 'bOEO cy : O0 Z[1=`] ! O Z[1=`] with inverse cy-1 O 'OE. In particular, we have the following. Corollary 1.4 Suppose that x0is contained in a maximal Z[1=`]-order O0[1=`] of D. Then there exists an element x 2 O[1=`] with the same minimal poly- nomial as x0. 2 A cohomological criterion for density In this section we recall some material from [9], but we give this material a slightly different treatment. Our reason is that the authors of [9] use results of Riehm [21] on the structure of the commutator subgroups of Sln. Riehm's analysis, however, excludes the case of n = 2 and p = 2, and it turns out that this case has different behavior. The maximal order Op of Dp admits a presentation [18, Appendix 2] Op = W=(S2 = p, Sa = __aS). (2.1) Here W = W(Fp2) is the Witt ring with residue field Fp2, and __adenotes the Galois conjugate (lift of the Frobenius on Fp2) of an element a 2 W. Every 5 element of Op can then be written uniquely in the form a + bS for a, b 2 W. The group S2 = Oxp consists of all such elements where a 6 0 (mod p). Let S02be the p-Sylow subgroup of S2. The group S02consists of all elements a + bS where a 1 (mod p). The subgroup Sl02of elements of S02of norm 1 is the p-Sylow subgroup of Sl2. Suppose that G is a pro-p-group. Let G* be the Frattini subgroup of G, which is the minimal closed normal subgroup that contains Gp and [G, G]. Then we have the following theorem. Theorem 2.1 (Koch [10], Serre [22]) Suppose that H is a subgroup of a pro-p-group G. Then H is dense in G if and only if the composite H ,! G ! G=G* = Hc1(G; Fp) is surjective. Corollary 2.2 Let {ffi} form an Fp-basis of the continuous group homomor- phisms Hom c(G, Fp) = H1c(G; Fp). Then H is a dense subgroup of G if and only if the composite L ffiM H ! G ---i! Fp i is surjective. Every element x 2 S02may be written uniquely in the form x = (1 + pt2 + p2t4 + . .).+ (t1 + pt3 + p2t5 + . .).S, (2.2) where ti = ti(x) are Teichm"uller lifts of elements of Fp2 in W. This is equiva* *lent 2 to saying that the elements ti satisfy tpi = ti. The coefficients ti give rise * *to continuous functions ti : S02! Fp2. Ravenel [16] uses this presentation to express S02as the Fp2-points of a pro- affine group scheme Spec S(2), where S(2) is the Morava stabilizer algebra 2 S(2) = Fp[t1, t2, t3, . .].=(tpi = ti). The algebra S(2) is a Hopf algebra. Equation (2.2) gives an isomorphism of groups S02~=Spec S(2)(Fp2). 6 Ravenel shows that this isomorphism gives an isomorphism in cohomology: H*c(S02; Fp2)= H*(S(2)) Fp Fp2 = ExtS(2)(Fp, Fp) Fp Fp2. The Ext group is taken in the category of S(2)-comodules. For an arbitrary element x 2 S02expressed as in equation (2.2), express the norm N(x) 2 Zxpby N(x) = 1 + ps1 + p2s2 + . .,. where the elements si = si(t1, t2, . .).are polynomial functions of the ti, and spi= si are Teichm"uller lifts of elements of Fp (compare with the discussion preceding Theorem 6.3.12 of [18]). Then we may define a quotient Hopf algebra Sl(2) = S(2)=(si(t1, t2, . .).), whose Fp2 points give the subgroups Sl02and for which H*c(Sl02; Fp2) = H*(Sl(2)) Fp Fp2. The following computation is obtained from combining Theorems 6.2.7 and 6.3.12 of [18]. Lemma 2.3 For p > 2, we have H1(Sl(2)) = Fp{h1,0, h1,1}, i where h1,iis represented by the element [tp1] in the cobar complex for Sl(2). Corollary 2.4 For p > 2, the group H1c(Sl02; Fp2) ~=Hom c(Sl02, Fp2) has an Fp2-basis consisting of the continuous homomorphisms t1, tp1: S02! Fp2. Corollary 2.5 (Gorbounov-Mahowald-Symonds [9]) For p > 2, a sub- group H of Sl02is dense if and only if the composite H ,! S02t1-!Fp2 is surjective. 7 Proof. Let ! 2 Fp2 be a primitive p2 - 1 root of unity. We may compute the cohomology with Fp coefficients by taking Gal = Gal(Fp2=Fp) fixed points, and obtain Hom c(S02, Fp) = Hom c(S02, Fp2)Gal. (The Galois group only acts on the coefficient group and not on S02.) Here the i pi+1 Frobenius oe 2 Gal acts by oe(tp1) = t1 for i 2 Z=2. An Fp-basis for this fixed-point module is given by the pair of homomorphisms t1 + tp1, !t1 + !ptp1: S02! Fp2. The result now follows from Corollary 2.2. We now address the case where p = 2. Lemma 2.6 Let p = 2. Then we have H1(Sl(2)) = F2{h1,0, h1,1, h3,0, h3,1}, where the generators are represented in the cobar complex for Sl(2) by i h1,i= [t21], i h3,i= [(t3 + t1t2)2 ]. Proof. We follow the same approach of [18, 6.3] using the May spectral se- quence. (It is important to refer to the second edition of [18]; the previous version, as well as [17], had an error in the restriction formula in the restri* *cted Lie algebras eL(n).) The May spectral sequence for Sl(2) takes the form Es,*2= Hs(E0Sl(2)) ) Hs(Sl(2)). The E1-term may be regarded as the Koszul complex for (E0Sl(2))* E*,*1= F2[hi,j: i 1, j 2 Z=2]=(h2k,j+ h2k,j+1), with differential 8 < P hi ,jhi ,j+i i 4, d1(hi,j) = : i1+i2=i 1 2 1 h2i-2,j+1 i > 4. We see that the only elements of E1,*1that persist to E1,*2are h1,0, h1,1, h3,0, and h3,1. We will show that these elements are permanent cycles in the May spectral sequence by explicitly producing cocycles in the cobar complex that they de- tect. By taking the images of the formulas for the coproduct on BP*BP in 8 [6], we arrive at the following formulas for the coproduct in Sl(2). (t1) = t1 1 + 1 t1, (t2) = t2 1 + t1 t21+ 1 t2, (t3) = t3 1 + t1 t22+ t2 t1 + t21 t21+ 1 t3. Using the relation s1 = t2 + t22+ t31= 0, these formulas may be used to verify that the cobar expressions in the state- ment of the lemma are permanent cycles. Corollary 2.7 For p = 2, the group H1c(Sl02; F4) ~=Hom c(Sl02, F4) has an F4-basis given by the continuous homomorphisms t1, t21, t3 + t1t2, (t3 + t1t2)2 : Sl02! F4. Corollary 2.8 For p = 2, a subgroup H of Sl02is dense if and only if the composite t1 (t3+t1t2) H ,! S02--- - - - -F!4 F4 is surjective. Proof. Let ! 2 F4 be a primitive 3rd root of unity. Just as in Corollary 2.5, we compute the cohomology with F2 coefficients by taking Gal = Gal(F4=F2) fixed points, and obtain Hom c(S02, F2) = Hom c(S02, F4)Gal. (As before, the Galois group only acts on the coefficient group.) An F2-basis for this fixed-point module is given by the homomorphisms t1 + t21, !t1 + !2t21, t3 + t1t2 + (t3 + t1t2)2, and !(t3 + t1t2) + !2(t3 + t1t2)2. 3 Proof of Theorem 0.1, Theorem 0.2, and Corollary 0.3 We will make use of the following proposition, which is a special case of Propo- sition 9.19 of [24]. Proposition 3.1 Suppose that f(x) = x2 + a1x + a2 is a monic polynomial over Q that is irreducible over Qp and R. Then there exists an ff in D with f(ff) = 0. If the elements ai are integral over R Q, then ff lies in a maximal R-order of D. 9 Proof of Theorem 0.2 for p > 2. We will use Proposition 3.1 to produce elements x1, x2 in so that t1(x1), t1(x2) form an Fp basis of Fp2. Corollary * *2.5 then yields the result. Choose integers r1 and r2 such that ri 6 0 (mod p), and so that r1 is a square and r2 is not a square in Fp. Let -pri- 2 ffi = _________, `mip(p-1) where the integers mi are chosen sufficiently large so that ff2i< 4. (3.1) We claim that the polynomials fi(x) = x2 + ffix + 1 are irreducible over R and Qp. It suffices to check that the discriminants i = ff2i- 4 are not squares in each of these fields. Condition (3.1) guarantees that i is not a square in R. Over Qp we note that i lies in Zp, so it suffices to check that i is not a square in Z=p2. Because `p(p-1)is congruent to 1 in Z=p2, we have i 4pri (mod p2). As ri is not congruent to 0 (mod p), i is not a square in Z=p2. Applying Proposition 3.1, we see that there exist exiin D so that fi(exi) = 0. The elements exisatisfy monic quadratics over Z[1=`], and so these elements are contained in maximal Z[1=`] orders Oi[1=`] of D. Applying Corollary 1.4, there exist elements xi 2 O[1=`] such that fi(xi) = x2i+ ffixi+ 1 = 0. (3.2) The xi satisfy N(xi) = 1. Therefore, we conclude that the elements xi are contained in the group 1. The images of the ffi in Qp lie in Zp, so the images of the xi in Dp lie in Op. Write xi in the form xi = ai+ biS for ai, bi 2 W. Reducing equation (3.2) modulo the ideal (S), we see that x2i- 2xi+ 1 a2i- 2ai+ 1 0 (mod S). We conclude that ai 1 (mod p). This implies that the elements xi actually lie in , and their images in Op are of the form xi = (1 + pa0i+ biS) 10 for a0i2 W. Equation (3.2) implies that the reduced trace of xi is given by T r(xi) = 2 + p T r(a0i) = -ffi, (3.3) whereas the reduced norm is given by N(xi) = 1 + p T r(a0i) + p2N(a0i) - pN(bi) = 1. (3.4) Substituting the expression for T r(a0i) given by equation (3.3) gives ffi+ 2 N(bi) = pN(a0i) - _______. (3.5) p Note that ffi -2 (mod p). Reducing equation (3.5) modulo p yields N(t1(xi)) N(bi) ri 2 Fp. If t1(x1) and t1(x2) were Fp linearly dependent in Fp2, their norms would lie in the same quadratic residue class. The ri were chosen so that this does not happen, so we conclude that {t1(x1), t1(x2)} forms a basis of Fp2. Therefore, by Corollary 2.5, the subgroup is dense in Sl02. Proof of Theorem 0.2 for p = 2. The proof is similar to the proof for p > 2, but more involved. There is precisely one isomorphism class of supersingular elliptic curve C at p = 2. It follows that D has one conjugacy class of maximal order. By checking the invariants of the division algebra D, it can be shown [15] that D is of the form of the rational quaternions Q=(i2 = j2 = -1, ij = -ji). We may therefore assume that End (C) D is the maximal order O generated by {!, i, j, k}, where k = ij and ! = 1+i+j+k_2. Note that !3 = 1. The automorphism group Aut C = End (C)x is the binary tetrahedral group A"4of order 24 given by the semidirect product Q8 o C3. The cyclic group C3 is generated by ! and the quaternion group Q8 is generated by i and j. We have !i!2 = j and !j!2 = k. Let T be the element i - j 2 O. Then we have T 2= -2 and T ! = !2T . The Witt ring W = W(F4) will be identified with the subring Z2[!] O Z2 = O2. Let z 2 W be an element of norm -1. Then the element S = zT in O2 has the property S2 = 2 and Sa = __aS for a 2 W. This makes explicit the presentation of O2 in terms of S and W given in equation (2.1). 11 Claim 1: Let a, b, 2 W be such that x = (1 + a) + bS 2 O2 has minimal polynomial x2 + 1. Then we must have a 0 (mod 2) and 2(b) = 0. Proof of Claim 1: In order for x to have this minimal polynomial, we must have the following: T r(x)= 2 + T r(a) = 0, (3.6) N(x) = 1 + T r(a) + N(a) - 2N(b) = 1. (3.7) Reducing equation (3.6) mod 4, we see that T r(a) 2 (mod 4). Substituting this into equation (3.7) and reducing mod 2, we see that N(a) 0 (mod 2). Write a = 2a0. Then we have N(a) = 4N(a0). Therefore, when we reduce equation (3.7) modulo 4, we get N(b) 1 (mod 2), and we conclude that 2(b) = 0. Claim 2: Suppose a, b 2 W and ff 2 Z2 are such that x = (1 + a) + bS 2 O2 has minimal polynomial x2 + ffx + 1. Then if ff satisfies ff 6 (mod 16), we must have a 0 (mod 2) and 2(b) = 1. Proof of Claim 2: Because ff 6 (mod 16), we can write 2 + ff = 4fl, where fl 2 (mod 4). In order for x to have this minimal polynomial, we must have T r(x) = -ff and N(x) = 1. As a result, we have the following: T r(a)= -4fl, (3.8) N(a) = 2N(b) + 4fl. (3.9) Reducing equation (3.9), we find that N(a) 2N(b) (mod 4). (3.10) This shows that N(a) 0 (mod 2). Writing a = 2a0 and substituting back into equation (3.10), we find that N(b) 0 (mod 2), so b = 2b0. Re-expanding equations (3.8) and (3.9) gives the following: 12 T r(a0)=-2fl, (3.11) N(a0) = 2N(b0) + fl. (3.12) From equation (3.11), we find T r(a0) 0 (mod 2). Write a0= a1 + a2! 2 W. Because T r(a0) = 2a1 - a2 0 (mod 2), we findpa2_ 0 (mod 2). As a result we can write a0= u + vd for u, v 2 Z2 and d = -3 = 2! + 1. We then have N(a0) = u2 + 3v2. Therefore, equation (3.12) can be reduced as follows: u2 - v2 2N(b0) + 2 (mod 4). (3.13) However, the equation u2 - v2 2 (mod 4) has no integer solutions. In order for equation (3.13) to hold, we must have N(b0) 1 (mod 2), or equivalently 2(b) = 1 + 2(b0) = 1. Claim 3: Given x 2 Sl02, let x0= !2x!. Then x0 is in Sl02and we have 8 < ti(x) i even, ti(x0) = : !ti(x) i odd. This claim is immediate from the definition of the functions ti given in Sec- tion 2. We now complete the proof of Theorem 0.2 for the case p = 2. Consider the polynomial 6 f(x) = x2 + __x + 1 `4 with discriminant = 4(9=`8 - 1). We have that < 0, so the polynomial f is irreducible over R, and f is irreducible over Z2 because 2( ) is odd. By Proposition 3.1 and Corollary 1.4 there exists an element y of O[1=`] so that f(y) = 0. Because N(y) = 1, the element y lies in 1. In order to show that satisfies the hypotheses of Corollary 2.8, we claim that the elements i, k, y, y0 = !2y! lie in , and their images under the homomorphism t1 (t3 + t1t2) : Sl02! F4 F4 13 form an F2-basis of F4 F4. Claims 1, 2, and 3 imply that these elements do lie in , and the functions ti evaluated on them satisfy t1(i)6= 0, t1(k) = !t1(i), t1(y) = 0, t1(y0)= 0, t3(y) 6= 0, t3(y0)= !t3(y). These conditions are sufficient to conclude that their images give a basis. Corollary 2.8 now implies that is dense in Sl02. Proof of Corollary 0.3. There is a short exact sequence 1 ! Sl02! Sl2! Cp+1 ! 1, where the cyclic group Cp+1 is the group of elements of Fxp2= (Op=(S))x of Fp-norm 1. It therefore suffices to show that we can lift the generator of Cp+1 to an element of 1. Let __y2 Fxp2be a generator of the norm 1 subgroup, with minimal polynomial __ f(x) = x2 + ax + 1 over Fp. Let "a be an integer that reduces to a modulo p, and define ff = "a=`m(p-1) where m is chosen sufficiently large so that ff2 < 4. Then the poly- nomial f(x) = x2 + ffx + 1 is irreducible over Qp and R. Just as in the proof of Theorem 0.2, Proposi- tion 3.1 and Corollary 1.4 may be used to show that there exists an element y 2 O[1=`] so that y reduces to a generator of Cp+1. Because y has norm 1, it lies in 1. Proof of Theorem 0.1. Assume that p > 2. In light of Corollary 0.3 and the short exact sequence 1 ! Sl2! S2 N-!Zxp! 1, we must show that there exists an element x of so that N(x) is a topological generator of Zxp. Because ` was assumed to be a topological generator, it suffices to show there exists an x so that N(x) = `. By Proposition 1.1, for m sufficiently large there exists an endomorphism ff 2 End (C) of degree `2m+1 . Then the element x = `-m ff 2 has norm `. The argument for p = 2 is identical, except that we use the short exact sequence 1 ! Sl2! eS2N-!Zx2={ 1} ! 1. 14 References [1] A. Baker, Isogenies of supersingular elliptic curves over finite f* *ields and operations in elliptic cohomology. Glasgow University Mathematics Department preprint 98/39. [2] M. Behrens, A modular description of the K(2)-local sphere at the prime 3. * *To appear in Topology. [3] M. Behrens, Isogenies of elliptic curves, Buildings, and the K(2)-local sph* *ere. In preparation. [4] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenk"orp* *er. Abh. Math. Sem. Hansischen Univ. 14, (1941). 197-272. [5] E.S. Devinatz, M.J. Hopkins, Homotopy fixed point spectra for closed subgro* *ups of the Morava stabilizer groups. Topology 43 (2004), no. 1, 1-47. [6] V. Giambalvo, Some tables for formal groups and BP . Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 169-176, Le* *cture Notes in Math., 658, Springer, Berlin, 1978. [7] P. Goerss, H.-W. Henn, M. Mahowald, C. Rezk, A resolution of the K(2)-local sphere. To appear in Ann. of Math. [8] P.G. Goerss, M.J. Hopkins, Moduli spaces of commutative ring spect* *ra. Structured ring spectra, 151-200, London Math. Soc. Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004. [9] V. Gorbounov, M. Mahowald, P. Symonds, Infinite subgroups of the Morava stabilizer groups. Topology 37 (1998), no. 6, 1371-1379. [10]H. Koch, Galoissche Theorie der p-Erweiterungen. Mit einem Geleitwort von I. R. ~Safarevi~c. Springer-Verlag, Berlin-New York; VEB Deutscher Verlag d* *er Wissenschaften, Berlin, 1970. [11]D.R. Kohel, Endomorphism rings of elliptic curves over finite fields. Ph.D. Thesis, University of California at Berkeley. [12]D.R. Kohel, Hecke module structure of quaternions. Class field theory_its centenary and prospect (Tokyo, 1998), 177-195, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001. [13]J.S. Milne, Points on Shimura varieties mod p. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 165-184, Proc. Sympos. Pure Math* *., XXXIII, Amer. Math. Soc., Providence, R.I., 1979. [14]J. Morava, Noetherian localisations of categories of cobordism comodules. A* *nn. of Math. (2) 121 (1985), no. 1, 1-39. [15]A. Pizer, An algorithm for computing modular forms on 0(N). J. Algebra 64 (1980), no. 2, 340-390. 15 [16]D.C. Ravenel, The structure of Morava stabilizer algebras. Invent. Math. 37 (1976), no. 2, 109-120. [17]D.C. Ravenel, The cohomology of the Morava stabilizer algebras. Math. Z. 152 (1977), no. 3, 287-297. [18]D.C. Ravenel, Complex cobordism and stable homotopy groups of spheres. Second edition. AMS Chelsea Publishing, Amer. Math. Soc., Providence, RI, 2004. [19]D.C. Ravenel, Nilpotence and periodicity in stable homotopy theory. Appendix C by Jeff Smith. Annals of Mathematics Studies, 128. Princeton University Press, Princeton, NJ, 1992. [20]C. Rezk, Notes on the Hopkins-Miller theorem. Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), 313-366, Contemp. Math., 220, Amer. Math. Soc., Providence, RI, 1998. [21]C. Riehm, The norm 1 group of a p-adic division algebra. Amer. J. Math. 92 1970 499-523. [22]J.P. Serre, Cohomologie galoisienne. Cours au Coll`ege de France, 1962-1963. Seconde 'edition. With a contribution by Jean-Louis Verdier. Lecture Notes * *in Mathematics, Vol. 5. Springer-Verlag, Berlin-Heidelberg-New York 1964. [23]J.H. Silverman, The arithmetic of elliptic curves. Graduate Texts * *in Mathematics, 106. Springer-Verlag, New York, 1986. [24]R.G. Swan, K-theory of finite groups and orders. Lecture Notes in Mathemati* *cs, Vol. 149. Springer-Verlag, Berlin-New York, 1970. [25]W.C. Waterhouse, J.S. Milne, Abelian varieties over finite fields. Proc. Sy* *mpos. Pure Math. XX (1971), 53-64. 16