an abstract characterization of the igusa curves

An Abstract Characterization of the Igusa CurvesAuthor(s): Kevin KeatingSource: American Journal of Mathematics, Vol. 117, No. 2 (Apr., 1995), pp. 419-440Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2374922 .

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AN ABSTRACT CHARACTERIZATION OF THE IGUSA CURVES

By KEVIN KEATING

Let K be a field of characteristic p > 0, let Kalg be an algebraic closure of K, and let KSeP be the separable closure of K in Kaig. Let E be an ordinary elliptic curve over K. Then the group of pn-torsion points of E over Kalg is isomorphic to Z/pnZ. The action of AutK (Kal9) Gal (KSeP/K) on these torsion points defines a character

(0. 1) XE: Gal (KseP/K) zx

In the case where K = Fq is a finite field the character XE is well understood: It maps the Frobenius element of Gal (KSeP/K) to the p-adic unit root of the equation

(0.2) x2 -aqx + q = 0,

where aq = I + q - #E(Fq). When E is generic the situation is more complicated. Let k be a field of

characteristic p, let K = k(j) be the field of rational functions in one variable over k, and let E be an elliptic curve over K with invariant j. Then XE depends on our particular choice of E, but the quotient

(0.3) Xj: Gal (KseP/K) - Zx/(+ 1)

depends only on j. Let K.. C KSeP be the fixed field of the kernel of Xj. In this paper we prove an abstract characterization of K,, which depends on the modular properties of Xj* In particular, we use the fact that K,, is the union of the function fields of the Igusa curves over k. As a consequence of our characterization of K,, we get a characterization of the Igusa curves among all the branched covers of the affine j-line. Our characterization of the Igusa extension KOO/K bears a striking resemblance to Ihara's characterization [5] of the function fields of modular curves of level prime to p over Fp2. However, Ihara's proof is completely different from ours.

Manuscript received December 3, 1992. American Journal of Mathematics 117 (1995), 419-440.

419



420 KEVIN KEATING

We now give an outline of the contents of the paper. In ? 1 we state the main result in two different forms. In ?2 we prove a local version of the theorems in ?1. In ?3 we prove some facts about cyclic isogenies whose degree is a power of some fixed integer N. In ?4 we use the results from ?2 and ?3 to prove the theorems in ?1.

A version of Theorem 1.11 appeared in the author's 1987 Harvard Ph.D. thesis and was stated, but not proved, in [7]. The author thanks Prof. Dick Gross for his indispensable guidance while the thesis was being written.

1. Statement of the theorem. In order to state the main theorem we need to define some modular curves in characteristic p, especially the Igusa curves. More detailed discussions of the Igusa curves, including proofs of the facts stated here, may be found in [4] and [6, Chap. 12].

We work over an arbitrary field k of characteristic p > 0. Given an elliptic curve E over a k-algebra R, define E(Pn)/R to be the base change of E by the iterated Frobenius endomorphism x _* xPn of R. For n > 1 let Xn0 be the k-scheme which represents the moduli problem of (isomorphism classes of) pairs (E, ir), where E is an elliptic curve over a k-algebra R and 7r is an element of order pn in the group E(Pn)(R). Then Xn? is a smooth affine curve over k. The Igusa curve Xn of level pn is defined to be the unique smooth projective curve over k which contains Xn? as a dense open subvariety. We say that Xn is the compactification of Xn. For each n > 1 there are natural maps from Xn+l to Xn and from Xn to the projective j-line Xo. These branched coverings are Galois, with

(1.) Gal(Xn/Xo) c(Z/pfZ)x/( i 1).

Let P be any geometric point on the j-line X0, and let Q be a point on Xn lying over P. If P corresponds to a supersingular elliptic curve then Q is totally ramified over P. If P corresponds to an ordinary elliptic curve then Q is unramified over P, except that if P = 0 then the ramification degree of Q over P is 3, and if P = 1728 then the ramification degree of Q over P is 2. The point P = oo splits completely in Xn, so Q is unramified over P and k-rational in this case.

Our characterization of the Igusa curves will depend on the isogeny invariance of the p-adic Galois character associated to an elliptic curve in characteristic p. In order to exploit this invariance we need to work with_generic elliptic curves which have nontrivial isogenies. Therefore choose N > 1 which is prime to p and consider the moduli problem (El, ?2, E5), where El, E2 are elliptic curves over a k-algebra R and 5: El -- E2 is a cyclic N-isogeny defined over R. This moduli problem is not representable, but there does exist a coarse moduli scheme Yo? over k which corresponds to it (cf. [6, Chap. 8]). Since Yo? is a smooth affine curve over k we may define Yo to be the compactification of Yo?. The smooth projective curve Yo is an analog over k of the modular curve Xo(N) over C. There is a natural map from Yo to the projective j-line Xo given by the function jl = j(Ei).



CHARACTERIZATION OF THE IGUSA CURVES 421

For n > 1 let Yn? be the k-scheme which represents the moduli problem (El, E2, ?, i), where 7r is a point of order pn on E(PL ) and El --? E2 is a cyclic N-isogeny. As above Yn, is a smooth affine curve over k so we may define Yn to be the compactification of Yn*. For each n > 0 there is a natural map from Yn to Xn which makes Yn a branched cover of Xn of degree

(1.2) d=N J7J(1+1-1). 1IN

In addition there are maps from Yn+l to Yn for each n > 0. The curve Yn is Galois over Yo, with

(1.3) Gal(Yn/YO) Gal(Xn/XO)

-(Z/pnZ)x/(? 1).

The action of the Galois group (1.3) on the moduli problem of Yn is given by

(1.4) (r) -(El, E2, 0, 7r) = (El, E2, 0, r * 7r) (r E (Z/pnZ)x).

The Fricke involution Wv of Yo may be described in terms of its action on the moduli problem associated to Yo. Specifically, wv maps the triple (El, E2, q$) onto (E2, EL, tq5), where 0q: E2 -* El is the isogeny which is dual to 0. Let Z be the smooth curve over k which is the quotient of Yo by the action of Wv.

The curves we have constructed and their associated covering maps are shown in the following diagram.

Yn

1 \

(1.5) YO Xn

z xo

In [7, ?6] the function fields of these curves are computed explicitly in the case p = 3, N = 2, n = 2.

PROPOSITION 1.6. Yn is abelian over Z.

Proof. We begin by showing that Yn is Galois over Z. Since Yn is Galois over Yo it suffices to show that W E Gal (Yo/Z) lifts to an automorphism of Yn. We describe a lifting v6N E Aut (Yn/Z) of wV in terms of its action on the moduli problem associated to Yn: v1v maps (El, E2, 0, 7r) onto (E2, El, t, (pnP )(X)). Next, by considering the action (1.4) of Gal (Yn/YO) on the moduli problem associated to Yn, we easily see that v6N commutes with the elements of Gal (Yn/Yo). Thus, Gal (Yn/Z) is abelian.



422 KEVIN KEATING

Remark 1.7. Proposition 1.6 expresses the fact that if E is an elliptic curve with invariant j and E' is an elliptic curve with invariant j' which is related to E by a cyclic N-isogeny then Xj = Xj'* This isogeny invariance property of the character Xj is the key to Theorems 1.9 and 1.11.

Remark 1.8. The automorphism T - vWNTvN- 1 of the abelian group Gal (Yn/Yo) induced by v6 depends only on Wv and not on the choice of lifting wvi. By Propo- sition 1.6 this automorphism is the identity.

We now state our characterization of the Igusa curves.

THEOREM 1.9. Let k be afield of characteristic p > 0 and choose N > 1 such that gcd (N, p) = 1. Let Xn, Yo, Z be the modular curves over k defined above: Xn is the Igusa curve of level pn, Yo is the analog over k of the modular curve Xo(N), and Z is the quotient of Yo by the action of the Fricke involution WV. Let C be an irreducible smooth curve over k which is a (possibly branched) cover of XI, and assume that

(1) C is abelian over Xo.

(2) C XXO Yo is abelian over Z.

(3) The point oo of Xo splits completely in C.

Then for some n > 1 the curves C and Xn are isomorphic as covers of XI.

Remark 1.10. Since C is assumed to be abelian over Xo, and Yo has no quotient which is a nontrivial abelian cover of Xo, the fiber product C xxo YO is a geometrically connected curve over k which is abelian over YO. Therefore VWI induces an automorphism of Gal (C xxo Yo/Yo), as in Remark 1.8. By hypothesis 2 this automorphism is the identity.

To facilitate the proof we rephrase Theorem 1.9 in terms of the function fields of the curves involved. For n > 0 let Kn = k(Xn), Ln = k(Yn), and M = k(Z) be the function fields of Xn, Yn, and Z. Then Kn, Ln, M are global fields whose places correspond to the closed points of the curves Xn, Yn, Z. Since Xn+I is a branched cover of Xn we have Kn C Kn+I for n > 0. Let K,, be the union of Kn for all n > 0. Then by Proposition 1.6, LoK,, is an abelian extension of M.

We now reformulate the definition of the character Xi of (0.3) in order to make clear the relation between this character and the function fields defined above. Let E be an elliptic curve over Ko = k(j) with invariant j and let KoeP be a separable closure of Ko which contains K,. Then for each n > 0 the pn-torsion subgroup of E(pn) is a cyclic group of order pn whose elements are rational over KoeP. The action of Galois on this group defines a character Xn Gal (KseP/Ko) >

(Z/pnZ)x; by taking the inverse limit of these characters we get the character XE of (0.1). The quotient of XE by ?I is the character Xj of (0.3), which is independent of the choice of elliptic curve E with invariant j. It is not hard to




see that the fixed field of the kernel of Xj is Ko, the union of the function fields of the Igusa curves.

We now state our characterization of the Igusa extension K,/Ko. Theo- rem 1.11 below is equivalent to Theorem 1.9, thanks to the one-to-one corre- spondence between finite extensions of Ko and branched covers of Xo. The proof of Theorem 1.11 (and hence of Theorem 1.9) will be given in ?4.

THEOREM 1.11. Let k be afield of characteristic p > 0 and choose N > 1 such that gcd (N, p) = 1. Let IC/Ko denote the maximum subextension of KseP/Ko such that:

(1) AC is abelian over Ko.

(2) ICLO is abelian over M.

(3) The place oo of Ko splits completely in IC.

Then IC = K.

Remark 1.12. If k is not algebraically closed then hypothesis 3 excludes from IC those extensions of Ko which come from extensions of k. But hypothesis 3 is still necessary even when k is algebraically closed: For instance, if p -1 (mod 4) and N = 3 then the field k(vj`-1728) satisfies hypotheses 1 and 2, but has degree 2 over Ko and therefore cannot be a subfield of K,. On the other hand, if k is algebraically closed then the maximum pro-p subextension of KoeP/Ko satisfying hypotheses 1 and 2 is equal to the maximum pro-p subextension of KO/Ko.

Remark 1.13. The combination of hypotheses 1 and 2 of Theorem 1.11 is very strong in that the intersection of the function fields Ko and M is the constant field k. From this point of view it is surprising that there should be any extensions of Ko which satisfy hypotheses 1 and 2, other than those which come from extensions of k. In characteristic 0 the curves X0 = X(1), Yo = Xo(N), and Z = XO(N)/IW are modular curves corresponding to the subgroups SL2(Z), Fo(N), and

F'=Fo(N) U Fo(N) [N/2 -N 1/2

of SL2(R). In this setting the statement Ko n M = k is equivalent to the fact that the subgroup of SL2(R) generated by SL2(Z) and V is not commensurable with SL2(Z).

Remark 1.14. The proof of Theorem 1.11 requires some special arguments when p = 2 or p = 3. Some of the details in these cases will be omitted.

We now wish to say a few words about the possible interpretations of Theo- rems 1.9 and 1.11. By Proposition 1.6 the field Ko, has the property that KooLo



424 KEVIN KEATING

is abelian over M; hypothesis 2 of Theorem 1.11 stipulates that IC must have the same property. As pointed out in Remark 1.7, Proposition 1.6 expresses the invariance of the character Xj under cyclic N-isogenies. Thus hypothesis 2 may be interpreted as a requirement that the extension IC/KO have an isogeny invariance property similar to that of the Igusa extension K, /Ko. Theorem 1.11 says that the Igusa extension K,,/Ko is characterized by this isogeny invariance.

In [5] Ihara proved a characterization of certain modular curves of level prime to p over Fp2 which is in many ways similar to Theorem 1.11. The proof of this result, however, is completely different from the proof of Theorem 1.11. Ihara uses the congruence subgroup property of SL2(Z[1/p]) to prove an analog of his theorem in characteristic 0, and then uses Grothendieck's theorem on liftings of etale covers from characteristic p to characteristic 0 to pass from the characteristic-0 version of his theorem to the characteristic-p version. The proof of Theorem 1.11 given in ?4 is unfortunately not so conceptual as Ihara's proof. Different arguments are used for the pro-p and prime-to-p parts of the theorem, and the pro-p part is proved by reducing to a local problem. One would prefer a more unified approach to proving Theorem 1.11 than that presented here; however, since there is no analog of the Igusa curves in characteristic 0, it seems unlikely that there is an interpretation of Theorems 1.9 and 1.11 which relates these results to geometry in characteristic 0.

2. A local result. Theorem 1.11 may be viewed as a global version of the local characterizations proved in [2], [7], and [9]. These local theorems charac- terize the Galois extension generated by the p'-torsion points of a certain class of formal groups over R = k[[t]]. In this section we prove a variant of these local characterizations in which the constant field k is algebraically closed, rather than finite. The proof, based on Artin-Schreier theory, uses the same techniques as the proof of [10, Th. 2].

Let Go be a one-parameter formal group law of height 2 over the algebraically closed field k of characteristic p. Then Go is given by a power series in two variables over k

(2.1) Go(x,y)=x+y+y ,

and multiplication by p in Go is given by a power series of the form

(2.2) [P]Go(X) = CoXp + . . .,

with co E kX. The endomorphism ring of Go is isomorphic to the maximal order B in the quaternion division ring over Qp.

Let F = k((t)) be the field of formal Laurant series in one variable over k. Then F is a local field, with ring of integers R = k[[t]] and valuation VF

normalized so that vF(t) = 1. For a a continuous k-linear automorphism of F




define the ramification number

(2.3) i(a) = VWU - )t) - 1.

This does not depend on the choice of uniformizer t for F. For a E FX we have

(2.4) VF((c - I)a) > vF(a) + i(o)

with equality if i(a) > 0 and vF(a) - 0 (modp). Let G be a formal group law over R whose reduction (mod (t)) is Go. Then

(2.5) [P]G(X) = d0XP + ...

for some do E tR. Assume that vF(do) = 1; then G is universal in the category of deformations of Go to complete Noetherian local k-algebras (cf. [12]). In particular, for any 0 E Aut (Go) there is a unique continuous k-linear automorphism cr, of R such that b lifts to an isogeny

(2.6) q: G - ?G,

where u?G denotes the base change of G by u?,: R -* R. See [7, ?4] for an example where u? is determined explicitly.

The map R x Aut (Go) -* R given by (f(t),q ) F-* f%(ugt) is a right action of Aut (Go) on R. This action extends uniquely to an action of Aut (Go) on F. The element 0 E Aut (Go) acts trivially on F if and only if q E Aut (G) Zpx . Therefore the action of Aut (Go) on F factors through Aut (Go)/Zpx.

Now let Falg be an algebraic closure of F and let FS"P be the separable closure of F in Falg. The p'-torsion subgroup of G(Falg) is isomorphic to Z/p'Z, so the action of AutF (Falg) Gal (FSPP/F) on the p'-torsion of G defines a character

(2.7) Xn Gal (FSPP/F) (Z/pnZ)x.

Taking the inverse limit of the Xn gives a character

(2.8) XG: Gal (FSP/F) ZX

which is a local analog of the character XE of (0.1). If the formal group laws G, G' are isogenous over R then by the definition

of the characters XG, XG' we have XG = XG/' In particular, by (2.6) we have XG = XJ,,G for all 0 E Aut (Go). Let %o be any automorphism of FSPP whose restriction to F is u?. By the functoriality of the definition of XG we have

(2.9) X,OG(T) = XG(jI Tdo)



426 KEVIN KEATING

for all T E Gal (FSeP/F). Combining this with the identity XG = X,"G and replacing do with its inverse we get

(2.10) XG(T) = XG(joTU&1)

for all b E Aut (Go) and T E Gal (FSeP/F). This formula expresses the isogeny invariance of the character XG-

Let FOG C FS"P be the fixed field of ker XG c Gal (FSeP/F). It is known (see [4, p. 100] or [3, Th. 3.5]) that XG is onto, and hence that Gal (FOG/F) !

Zpx. If p > 2 we have

(2.11) ZP =up-I x (I +pZp),

so there is a unique subextension Fw /F of FOG/F such that

(2.12) Gal(FW /F) (1 +pZp)x

= zp.

By (2.10) we have &OF, = Fw, so T H- *-O,d,1 defines an automorphism of Gal (Fw /F). Since Gal (Ft /F) is abelian this automorphism depends only on u? and not on the choice of %?. From (2.10) we see that the automorphism of Gal (Fw /F) induced by u? is the identity.

PROPOSITION 2.13. Letk be an algebraically closedfieldofcharacteristicp > 3, let d > 3 be a divisor of p + 1, and let H be the (unique) closed subgroup of index (p + 1)/d in Aut (Go). Let F/F be the maximum abelian pro-p subextension of FSeP/F such that, for all q E H, &js.F = F and %o induces the identity automorphism of Gal (TF/F). Then F = Fw . In particular we have Gal (YF/F) Zp.

Remark 2.14. If p = 3 and H is a subgroup of Aut (Go) of index < 2 then

n = Fw and Gal (F/F) Z3. If p = 2 and H contains the elements of Aut (Go) of reduced norm 1 then F = FOG and Gal (Y/F) - Z2 . We omit the unpleasant proofs of these special cases.

Proof. Denote the group of continuous homomorphisms from Gal (Y/F) into Z/pZ by

(2.15) X = Hom( Gal (Y/F), Z/pZ).

To prove the proposition it suffices to show that the order of X is < p. For every + E X there exists a E K such that +b is equal to the Artin-Schreier character




<)a associated to the equation xP - x - a = 0. Let px = xP - x; then l/a = <ba' if and only if a' - a = pb for some b E F. Let 4'a E X; then for every 0 E H and T E Gal (FSeP/F) we have

(2.16) <)a(T) = fa(&fIT&O)

= f?,oa(T)-

Therefore (u? - 1)a = pbt,, for some bt E F. For +b E X define

(2.17) v(+)= inf{-vF(a): 4a = b}

and for n > 0 let

(2.18) Xn= {b E XX: v(fb) < n}.

The following general properties of the groups Xn depend only on the assumption that k is algebraically closed.

LEMMA 2.19.

(a) Xo is trivial.

(b) Xn = Xn- if n_ 0 (modp).

Proof. (a) Since k is algebraically closed we have pR = R. Thus if vF(a) > 0 then 4'a = lbo is trivial.

(b) Choose 4'a E X such that vF(a) = -n < 0 with n 0_ (modp). Since k is perfect there is b E F such that a' = a + pb satisfies vF(a') > -n. Then

a = fba' E Xn-1

To prove Proposition 2.13 it suffices to prove the following additional properties of the groups Xn. These properties depend on the cr,-invariance (2.16) of the elements of X.

LEMMA 2.20.

(a) Xn = Xn- if n - 1 (modp).

(b) Xn = Xn- 1 if n _ 0 (mod d).

(c) X = X2p.

(d) Xp+l has order < p.

Proof. We begin by choosing two elements which generate a dense subgroup of H and studying the automorphisms of F induced by these elements. Let 0 be the ring of integers in a ramified quadratic extension of Qp. Then 0 embeds as a subring of End (Go), and for each such embedding we have O9x C H. Therefore



428 KEVIN KEATING

there exists 0 E H such that Zp[0] (0. The assumption p > 3 implies that 0 generates the Zp-module 0</Zp' zp, and so by [8, Th. 1.1] we have

(2.21) i(Uo)= 1

i(Cr) = 2p + 1.

Since Aut (Go) B> has elements of order p2 _ 1, there exists ij E H whose image in H/Zp1 has order d. The automorphism a,, E Autk (F) induced by rq also has order d.

Choose V/a E X such that vF(a) = -n < 0 Then there exist bo, b7l E F such that

(2.22) (co - I)a = pbo

(U77 - I)a = pb?l

We are now ready to prove the claims made in the lemma. (a) If n- 0 (modp) use Lemma 2.19(b). If n - 0 (modp) then by (2.4) and

(2.21) we have

(2.23) VF((uo - I)a) = -n + 1.

By (2.22) we get -n + 1 = vF(pbo) = pvF(bo), and hence n -1 (mod p). (b) By Kummer theory there is a uniformizer u for F such that u,,U = (u with

E k a primitive dth root of unity. If n - 0 (modd) then VF((u?l - I)a) = -n. But (2.22) implies -n = vF(pb?) = pVF(b?), and hence n- 0 (modp). Therefore by Lemma 2.19(b) we get 4'a E Xn-I

(c) By Lemma 2.19(b) we may assume n - 0 (modp). Then applying (co - I)P-1 to both sides of the equation (co - I)a = pbo we get

(2.24) (cro - I)a = (o- I)P- pbo

= g(uo - 1)PK'bo.

By (2.4) and (2.21) the valuation of the left side of this equation is -n + 2p + 1. To compute the valuation of the right side observe that by (2.22) and (2.23) we have vF(be) = (-n + 1)/p. Therefore by (2.4), (2.21), and the assumption p > 3 we get

(2.25) VF((uo - I)P-bo) > (p - 1) +-n+ p

2p-n + 1 p




If n > 2p + 1 then VF(p(ao - I)P-bo) > -n + 2p + 1, a contradiction. Therefore n < 2p.

(d) Let ?/'a, 0/a be nontrivial elements of Xp+i. As in (b) we write a, a' in terms of the uniformizer u for F; by replacing a with a + pc and a' with a'+ pc' for appropriately chosen c, c' E F, we may assume that the coefficients of u-P in a,a' are both 0. By (a), (b), and Lemma 2.19(a) we have vF(a) = vF(a') = -p-1. Therefore there exists r E kX such that vF(a' - ra) > -p.

Using (2.23) we see that (co - I)a = pbo and (co - I)a' = pb'b with vF(bo) =

vF(b9) = -1, so we may write bo = b1t-I + and bl = b' t- + with

bi, b' E kx. Thus we get the formula

(2.26) (cor - 1)(a' - ra) = pb' -rpbo

= (blP -rbP )t-P + (rbl -b )t-1 +*-

Since vF(a' - ra) > -p we have bIP - rbP, = 0, and hence

(2.27) VF((ro - 1)(a' - ra)) >-1.

Recall that ur7U = (u with ( a primitive dth root of unity. Since p + 1 0 (mod d) we have VF((u?l - I)a) > -p, and hence by our assumption about a we get VF((u?l - I)a) > -p. Applying this inequality to the formula (a,, - I)a = pb? we get vF(pb?) > -p, and hence vF(b7,) > 0. A similar argument shows that

(ar - )a' = pbr with vF(b1) > 0. Assume that vF(a' - ra) < 0. The right side of the equation

(2.28) (U77 - 1)(a' - ra) = pb -rpbrB

lies in R, so by the formula ur,u = (u we see that vF(a' - ra)- 0 (mod d). Therefore we have vF(a' - ra) < -d, so by (2.23) we get

(2.29) VF((ro - 1)(a' - ra)) < -d + 1.

Combining this inequality with (2.27) we get d < 2, a contradiction. It follows that vF(a' - ra) > 0, and hence a' = ra + pc for some c E R. Using (2.26) we get b'P = rbP,, b' = rbl, rP = r, and r E Fp. Therefore 40a' = O)ra = r/a. This completes the proof.

3. Cyclic N-isogenies. Since Theorem 1.11 characterizes the Igusa extension in terms of its invariance under cyclic N-isogenies, we need to study these isogenies more closely. More generally, we need to consider isogenies 5: El -* En which can be expressed as the composition

(3.1) = n-I 0*..001



430 KEVIN KEATING

of cyclic N-isogenies Xi: Ei - E?i+j. In this section we show that certain pairs of elliptic curves E, E' in characteristic p are related by isogenies which are the composition of cyclic N-isogenies. We also show that if E is a supersingular elliptic curve then certain elements of End (E) can be p-adically approximated by isogenies which are the composition of cyclic N-isogenies. The material in this section may be well-known, but we include it here for the sake of completeness.

As in ?2 we let k be an algebraically closed field of characteristic p > 0. We begin by considering ordinary elliptic curves, which for our purposes are easier to handle than supersingular curves.

PROPOSITION 3.2. Let El be an ordinary elliptic curve over k. Then there are infinitely many elliptic curves E2 over k which are related to El by an isogeny which is the composition of cyclic N-isogenies.

Proof. The endomorphism ring 0 of El is isomorphic to an order (say of conductor c) in an imaginary quadratic field K. Given n > 1 there is an elliptic curve E2 and a cyclic Nn-isogeny 0q: El - E2 such that End (E2) is isomorphic to the order of conductor cNn in K. Then q is the composition of n cyclic N- isogenies, and the elliptic curves E2 corresponding to different values of n are distinct.

To deal with supersingular elliptic curves we need some preliminary lemmas.

LEMMA 3.3. Let 1 be a prime different from p, and let E be a supersingular elliptic curve over k. Then there exists an integerf > 0 with the following property: If the quadraticfield K embeds as a subring of End (E) 0 Q, then the order Z + lf OK embeds as a subring of End (E).

Proof. Let D be the quaternion algebra over Q which is ramified at p and oo. Then End (E) 0 Q D, and End (E) is isomorphic to a maximal Z-order R1 in D. The quadratic field K embeds as a subring of D if and only if K is imaginary and p is ramified or inert in K.

Since D 0 Qi - M2(Q1), the strong approximation theorem [13, III, Th. 4.3] implies that there is only one conjugacy class of maximal Z[1/l]-orders in D. Thus we may choose representatives R1, R2,.. .,Rh for the conjugacy classes of maximal Z-orders in D such that the associated Z[1/l]-orders Rj[1/1] are the same for every i. Therefore there exists f > 0 such that lfRi C RI for all i. If the field K embeds as a subring of D, then the ring of integers OK embeds as a subring of some Ri. Therefore Z + lf OK embeds as a subring of R1 End (E).

Let E denote the formal group of E. Then E has dimension 1 and height 2, and the endomorphism ring

(3.4) End (E) End (E) 0 Zp




of E is isomorphic to the maximal order B in the quaternion division ring D X Qp over Qp. Every 0 E End (E) satisfies a quadratic equation

(3.5) 92 - bq + c =O,

where b = Tr 0 E Zp is the reduced trace of 0, and c = Nr 0 E Zp is the reduced norm of 0. If 0 E End (E) then Tr q E Z, Nr 0 E Z, and Nr q is equal to the degree of q.

LEMMA 3.6. Let E be a supersingular elliptic curve over k with formal group E, let r be a positive integer, and let N be a positive integer such that gcd (N, p) = 1. Then for all sufficiently large integers n we have the following: For every 0 E

End (E) with reduced norm N , there is a cyclic isogeny q' E End (E) such that deg q' = Nn and Tr q' _ Tr 0 (modpr).

Proof. We will only consider the case p > 2; the proof for p = 2 is a slight modification of the argument given here. We may assume that r is odd. Letf be as in Lemma 3.3 and choose a prime 1% 2pN such that N is a square (mod 1). Let n be any integer such that

(3.7) 4Nn > p2r+2 N 2 . 14f

and choose q E End (E) such that Nr q = Nn. Then by the Chinese remainder theorem there exists an integer b such that

b Trq5 (modpr)

vp(b2- 4Nn) < r

(3.8) gcd (b, N) = 1

b2 - 4N _ 0 (mod 12f)

b2- 4Nn < 0.

If vp(b2 - 4N) < r then Qp(vb2 - 4Nn) Qp (O), while if vp(b2 - 4N) = r then Qp(\/b2 - 4Nn) is a ramified quadratic extension of Qp, since r is odd. In either case p does not split in the imaginary quadratic field K = Q(Vb2 -_4Nn), and hence K embeds as a subring of D. Therefore by Lemma 3.3, Z + lf OK embeds as a subring of End (E). The isogeny q' E End (E) corresponding to

(3.9) b + 2 E Z + lf OK

is cyclic and satisfies the equation

(3.10) q12 - bq5' +Nn = 0.



432 KEVIN KEATING

This proves the lemma.

PROPOSITION 3.11. Let E1, E2 be supersingular elliptic curves over k. Then there exists an isogeny q5: El - ?2 which is the composition of cyclic N-isogenies.

Proof. We proceed by cases according to the form that N takes. Let q be a prime dividing N.

Case 1: N = q. The endomorphism ring of El is isomorphic to a maximal order R1 in the quaternion algebra D over Q ramified at p and oo. We may identify Hom(El, E2) with a right ideal I of R1. Define d > 0 by letting dZ be the fractional ideal of Q generated by {Nr a: ae I I}, and choose 6 E D such that Nr 6 = d. By replacing I with 6-1I we may assume that d = 1. In that case the isogeny associated to ao E I has degree Nr ae. Since D 0 Qq M2(Qq), the right ideal I X Z[l/q] of R[1/q] is principal [13, III, Th. 5.7], and is generated by some aoo e I. The assumption d = 1 implies Nr aoo Z[1/q] = Z[1/q], and hence Nr auo is a power of q. It follows that the degree of the isogeny 00: El -4 E2

corresponding to auo is also a power of q, so qo is the composition of cyclic q-isogenies.

Case 2: N = q2. By Case 1 there exists an isogeny qo: El -- E2 whose degree is a power of q. By Lemma 3.6, El has an endomorphism ql whose degree is an odd power of q. Thus, replacing qo by qo o X1 if necessary, we may assume that the degree of q0 is an even power of q. Then 00 factors as qo = a o [qS]1E, with b: El -4 E2 a cyclic isogeny whose degree is an even power of q. Hence q is the composition of cyclic q2-isogenies.

Case 3: N = qS. By Case 2 we are reduced to the situation where there is a cyclic q2-isogeny q$o El -- E2. Let H be the kernel of the isogeny qo o [qS-l ]E

El -- E2; then

(3.12) H - (Z/qS-lZ) x (Z/qs+lZ).

Let Ho be the subgroup generated by the element of H which corresponds to (1,q). Then we have

(3.13) Ho - Z/qsZ

H/Ho - Z/qsZ.

Therefore there exists a (supersingular) elliptic curve E3 over k and cyclic qS-

isogenies Xl: El -* E3 with kernel Ho and q2: E3 -1 E2 with kernel H/Ho such that q2 o = o00 o [qS-l]E. Hence El and E2 are related by an isogeny which is the composition of two cyclic qS-isogenies.




Case 4: N = qSn with gcd (q, n) = 1. Using Case 3 we may assume there is a cyclic q2S-isogeny q05 El -+ E2. By applying the same arguments we used in Case 3 we see that bo o [n]E, = a2 q Xl, where q1: El -+ E3 and 02: E3 -* E2 are cyclic N-isogenies. It follows that Ei and E2 are related by an isogeny which is the composition of two cyclic N-isogenies.

PROPOSITION 3.14. Assume p > 2 and let E be a supersingular elliptic curve overk withformal group E. Let C be the closure of the subgroup ofAut (E) generated by Zpx and the elements of End (E) which are the composition of cyclic N-isogenies. Then C has index 2 in Aut (E) if N is a square (modp), and C = Aut (E) if N is not a square (modp).

Remark 3.15. If p = 2 then C contains the subgroup of index 4 in Aut (E) generated by Z2' and the elements of reduced norm 1.

Proof. End (E) is isomorphic to the maximal order

(3.16) B= {x E D0Qp: Nrx E Zp}

in the quatemion division ring D 0 Qp over Qp. There is a filtration BX D B1 D B2 D on Aut (E) - B> defined by

(3.17) Bn = {x E B: Nr(x-1) E pnZp} (n > 1).

This filtration has quotients

(3.18) Bx/BI 'L Fx2 p

BnlBn1 l F 2

Let n be a large integer such that Nn 1 (modp3) and let u be an element of zpx which is not a square. Then there are 0b1, ?b2, q3 E Aut (E) which satisfy the equations

2-(2 +p)ol +Nn = 0 (3.19) q2- (2 + up)2 + Nn = 0

?}2 - (2 + Up2)3+ =0 3 )03 + Nn= 0.

By Lemma 3.6 there are cyclic isogenies Xl, q2, q$ E End (?) of degree Nn such that

Tr Xl 2+p (modp2) (3.20) Tr q 2 + up (modp2)

Tr q! 2 + up2 (modp3).



434 KEVIN KEATING

Then {b, 02} generates the quotient B1 /B2 and {IO, l+p} generates the quotient B2/B3. Therefore, since p > 2, the set {q$, O/, q$, 1 + P} generates a dense subgroup of B1. Since X1, X2, q$, and I +p are all elements of C we have B1 c C.

Let S be the cyclic subgroup of B x /B1 F><2 of orderp+ 1, let ( be a generator p2

of S, let b be any element of Zp whose reduction (modp) is ( + (P E Fp, and let n be a large integer such that Nn_ 1 (modp). Then there is q4 E Aut (E) which satisfies the equation

(3.21) X2 - bO4 + n = O.

By Lemma 3.6 there is a cyclic isogeny q$ E End (E) of degree Nn such that

(3.22) Tr X4 = b (modp).

Then X4 E C and the image of X4 in BX/BI is ( or 01. The image of zpx in BX/B- F? is F1<. Since S and Fp< generate a subgroup of index 2 in BX/BI, we conclude that C contains the unique closed subgroup of index 2 in Aut (E).

We denote the subgroup of index 2 in Aut (E) by A. Given q E Aut (E) we have q E A if and only if Nr q is a square in Zp. Thus if N is a square (modp) then C c A and hence C = A. If N is not a square (modp) let El be a supersingular elliptic curve over k which is related to E by a cyclic N-isogeny. Then by Proposition 3.11, El and E are related by an isogeny which is the composition of cyclic N2-isogenies. Therefore there is an element b E End (E) which is the composition of an odd number of cyclic N-isogenies. Since q E C and Nr X is not a square in Zp<, we have C t A. Thus C = Aut (E) in this case.

4. Proof of the theorem. In order to prove Theorem 1.11 we divide it into two independent statements, one characterizing the prime-to-p part of IC/Ko, the other characterizing the pro-p part of IC/Ko. Thus let KI/Ko be the largest subextension of IC/Ko whose Galois group is the projective limit of groups of order prime to p, and let ICw/Ko be the largest subextension of IC/Ko whose Galois group is a pro-p group. Then I(t/(W = /C, ct n ICW = Ko, and

(4.1) Gal (KZ/Ko) - Gal (CIt/Ko) x Gal (C'w/Ko)

To prove Theorem 1.11 it suffices to prove the following.

PROPOSITION 4.2. Assume p > 2 and let Kt/Ko and /Cw/Ko be the maximum prime-to-p and pro-p subextensions of IC/Ko. Then

(a) KCt = K




Remark 4.3. If p = 2 then (b) is replaced by the statement Gal (/(w/Ko) ?x Z. The proof in this case is similar to that given below, with Remarks 2.14 and 3.15 replacing Propositions 2.13 and 3.14.

Remark 4.4. By hypothesis 3 of Theorem 1.11 we may assume that k is algebraically closed, as we will throughout this section.

To prove Proposition 4.2 we introduce an auxiliary level structure. Let 1 > 3 be a prime such that gcd (1, pN) = 1 and let ( E k be a primitive lth root of 1. Define (Xo)0 to be the k-scheme which represents the moduli problem (E, AI, A2), where E is an elliptic curve and Al, A2 are i-torsion points of E whose Weil pairing (AI, A2)1 is equal to (. Then (X0)0 is a smooth affine curve over k, so we may define X0 to be the compactification of (X6)0. Then X0 is a smooth projective curve over k whose function field Ko is a Galois extension of Ko.

Let (Y6)0 be the scheme over k which represents the moduli problem (El, E2, , A1, A2), where : El -> E2 is a cyclic N-isogeny and Al, A2 are i-torsion points

of El such that (A1, A2)1 = (. Once again (Y6)0 is a smooth affine curve, and we define Y6 to be its compactification. Then Y6 is a smooth projective curve over k. Let LI = k(Yo) be the function field of Yo. We have then LI = LoKO.

The involution vW of Yo lifts to an involution wjN of Yl, defined by its action on the moduli problem associated to Y0,

(4.5) wN(EI, E2, , AI, A2) = (E2, Ei, t', (A2), r q (A I)).

Here t E2 El is the isogeny which is dual to b, and r is any integer satisfying rN -1 (mod 1). Let Z' be the smooth curve over k which is the quotient of Yl by the action of wjN, and let M' = k(Z') be the function field of Z'. Since the restriction of wk to Lo is vW, by hypothesis 2 of Theorem 1.11 we see that wjN induces the identity automorphism of Gal (ILI/LI), and hence that KCLI is abelian over M'.

Proof of Proposition 4.2(a). Let J/Ko be a cyclic subextension of ACt/Ko of degree n > 1. It suffices to show that n divides [KI: Ko] = (p - 1)/2 and that J is determined by n. By Kummer theory J = Ko(a /n) for some a E Kox. Let

(4.6) (a) = E cpP

be the divisor of a in Ko; then cp = 0 for all but finitely many places P of Ko. Let P' be a place of Ko lying over the place P of Ko. Then the coefficient of P' in the divisor of a in Ko is epcp, where ep is the ramification degree of P' over P. Thus if P :' 0, 1728, oo then ep = 1, while if p > 3 then eo = 3 and e1728 = 2. If P :/ oo and Q' is a place of LI lying over P' then Q' is unramified over P'. Therefore the coefficient of Q' in the divisor of a in LI is epcp.



436 KEVIN KEATING

Let El, E2 be elliptic curves over k which are related by a cyclic N-isogeny, and let PI, P2 be the corresponding places of Ko. Then we may choose places Q'1, Ql of LI lying over PI, P2 such that Ql = wNQ'I. Since ICtLl is abelian over MI we have

(4.7) wNa E (LIx)n. a

It follows that

(4.8) ep, cpl -eP2CP2 (modn).

By hypothesis 3 of Theorem 1.11 we have c,, --0 (modn). If El is an ordinary elliptic curve then by Proposition 3.2 there are infinitely many elliptic curves E2 related to El by an isogeny which is the composition of cyclic N- isogenies; it follows by (4.8) that epcpl C- 0 (modn) if El is ordinary. By Proposition 3.11 and (4.8) there is an integer c such that for all supersingular places P of Ko we have epcp = c (mod n).

Suppose now that c -0 (mod n), so epcp -0 (mod n) for every place P of

Ko. Then cp -0 (modn) for P #/ 0, 1728, and by using the formula Zcp = 0 it is easy to see that co C 1728 0 (mod n) as well. It follows that a is an nth power in Kox, contrary to our assumptions. Therefore we can't have epcp 0 O (mod n) for every P. As a consequence we see that c is relatively prime to n (otherwise replacing n by gcd (c, n) gives a contradiction). In addition, if b is an element of Kox whose divisor E dpP satisfies epdp _ epcp (mod n) for all P then a/b E (Kox)n and J = Ko(bl-n).

It follows easily from the preceding paragraph that the cyclic subextension J/Ko of KV/K0 is determined by its degree n. To complete the proof it suffices to show that n divides the degree (p - 1)/2 of the extension K1/Ko. Combining the congruences above with the formula E Cp = 0 we get

(4.9) E 6- c_=0 (mod n), P s.s. ep

where the sum is taken over the supersingular places of Ko. By Eichler's mass formula E ep-1 = (p-1)/12 we see that c(p-1)/2 _ 0 (modn). Since gcd(c,n) =

1 we conclude that n divides (p - 1)/2.

Proof of Proposition 4.2(b). Since LI/KO and K1o/Ko have no abelian subextensions, the restriction maps

(4.10) Gal (/CWL/LI) , Gal (/CwWKO/KO) - Gal (/CW/Ko)

are isomorphisms. We denote each of the isomorphic groups in (4.10) by g. Let P :/ oo be a place of Ko, let P' be a place of Ko lying over P, and let Q' be




a place of Lo lying over P'. Let gp, gpi, gQ' denote the inertia groups of g at P, P', Q'. Then the restriction maps

(4.11) 9QI gpi gp

are isomorphisms, since Q' is unramified over P', and the inertia group of P' over P has no quotients of order p. (This claim is obvious if p > 3, since in that case the ramification degree of P' over P is < 3, but it is true in general.)

LEMMA 4.12. Let Po be a supersingular place of Ko. Then g = gpo.

Proof. The field Ko has no unramified extensions, since it is the function field of a curve of genus 0 over the algebraically closed field k. Therefore g ? Gal (C'w/Ko) is generated by its inertia subgroups. To prove the lemma it suffices to show that the inertia groups of g at the supersingular places of Ko are all identical, and that the inertia groups of g at the other places of Ko are trivial.

Let Pi 7 oo be a place of Ko and let El be the corresponding elliptic curve over k. Let : El -> E2 be a cyclic N-isogeny, let Ql be the place of Lo lying over PI which corresponds to the triple (El, E2, s), and let Q' be a place of Lo lying over Ql. Then by (4.5) the restriction of Q' = wNQ' to Ko is the place P2 which corresponds to E2. Since ICWL' is abelian over M' we have gQ' = gQ. Since the natural maps gQ' -> gp, and gQ2 -> P2 are isomorphisms, it follows that gp, = gP2

By Proposition 3.11 and the preceding paragraph the inertia groups gp at the supersingular places P are all identical. By hypothesis 3 of Theorem 1.11 the inertia group gOO is trivial. Let P1 be a place of Ko which corresponds to an ordinary elliptic curve El over k. Then by Proposition 3.2 there are infinitely many places P2 whose corresponding elliptic curves are related to El by an isogeny which is the composition of cyclic N-isogenies. By the preceding paragraph we have gpl = gP2 for each such place P2. Therefore gpl is trivial, lest Ko have a finite separable extension ramified at infinitely many places. This completes the proof of the lemma.

To complete the proof of Proposition 4.2(b) we introduce the universal elliptic curves associated to our moduli problems. Let 1 be the coordinate ring of the moduli space (Xo)0 and let S be the coordinate ring of (Y6)0. Defined over 1i there is a universal elliptic curve E with i-torsion points AI, A2 such that (AI, A2)1 = (

Over S there are universal elliptic curves E1 = E Ojz S and E2 = wNEI, where wkNE denotes the base change of El by wNj S -> S. These are related by a cyclic N-isogeny

(4.13) D:El E2.

Let PI be a place of 1 and let Qi be a place of S lying over P1. Set Q2 = wNQi, and let P2 be the restriction of Q2 to RZ. For i = 1, 2 let Rpi - k[[t]]



438 KEVIN KEATING

be the completion of R. at Pi and let Fp, be the completion of Ko at Pi (so Fpi is the fraction field of 1pi). Similarly, let SQi be the completion of S at Qi and let FQi be the completion of Lo at Qi. Since Qi is unramified over Pi we have

Rpi = SQi and Fp, = FQ,. Let Ei be the elliptic curve over k which is the reduction of E (mod Pi). Then

for i = 1, 2 the elliptic curve Epi = E OR 1Zp, is a universal deformation of Ei to complete noetherian local k-algebras. Hence by the Serre-Tate lifting theorem (see [11] or the appendix to [1]) the formal groups EP , EP2 are universal deformations of El, E2 to complete noetherian local k-algebras.

By taking the completion of (4.13) at Qi we get a cyclic N-isogeny

(4.14) 4!Q,: El Os SQI -) (wEl) Os SQI

over SQI whose special fiber is the cyclic N-isogeny El -+ E2 over k. Since SQi is identified with Rpi we may rewrite (4.14) as an isogeny of elliptic curves over lZp,

(4.15) q:Ep, -*/3EP2

Here P3 1Zp2 -* 1Zp is induced by wN. By the universal property of Ep2, the map P: 1Zp2 -* 1Zp is the unique k-algebra homomorphism such that qE I __ E2

lifts to an isogeny

(4.16) ( Ep1 - 1 /3EP2

For i = 1, 2 let Fpi/Fpi be the local extension induced by the global extension

IZKO/KO. Then Gal (Jpj/Fpj) ? gp, and hence by Lemma 4.12 we have canonical isomorphisms

(4.17) Gal (p/Fpi) l g

Since I3L' is abelian over M', wN induces the identity on g - Gal (lwL'/L'). Therefore the isomorphism between Gal ( Ip2/Fp2) g and Gal (pI /pl ) lp g induced by /3 induces the identity on g.

Now let P1,P2 be places of 1R corresponding to isomorphic supersingular elliptic curves EL,E2, and let t: E2 -) El be an isomorphism between these curves. By the universal property of Ep, there is a unique k-algebra map -y: lZp -+ lZp2 such that t lifts to an isomorphism t: EP2 -* 7Epl. On the other hand, for every r E Gal (K6/Ko) there is an isomorphism e: E -+ rE. We may choose r and e so that TP1 = P2 and the reduction of 6 (modP2) is equal to t. Then the localization of 6 at P2 is an isomorphism CP2: EP2 -* Epl, where 6: lZp1 -+ ZP2 is induced by r. Since the special fiber of 6P2 is t, the




universal property of -y implies that 6 = -y. Then since r induces the identity automorphism of g Gal (IW/Ko), we see that 7y also induces the identity on

G Gal(Yp, /Fp) ) Gal (Yp2/Fp2). Fix a place P1 of R whose associated elliptic curve E1 is supersingular. Let

E E End(EI) be an endomorphism of E1 which is the composition of cyclic N-isogenies. Then E may be factored as

(4.18) 0=u? n-I 0 . 0b,

where Xi: Ei -* Ej+I is the cyclic N-isogeny associated to a place Qi of S, and u: Et -- E1 is an isomorphism.

For 1 < i < n - 1 let /i3: RpjI -+ lpi be the unique k-algebra map such that Xi-: Ei Ej+I lifts to an isogeny qj : Epi -+ iEpi+ Also, let -y .Zp, -* 1Zpn be the unique k-algebra map such that u lifts to an isomorphism u: Epn -y Ep,. Then by (4.18),

(4.19) ao = , ? ? on-I ? 7

is the unique k-automorphism of Zp, such that 0 lifts to an isogeny q : Ep, aoEp,. Since /3i and -y induce the identity automorphism of g, it follows that ao induces the identity on g 5 Gal (.Fp /Fp) as well.

Let C denote the closure in Aut (EL) of the subgroup generated by Zp and the set

(4.20) {q E End (El): q is the composition of cyclic N-isogenies}.

For any 0 E C we let ac be the unique k-automorphism of 1Zp, such that q$ lifts to an isogeny q$ tp -+ a EPI. If qb E x then ao is the identity, while if X is an element of the set (4.20) then the arguments above show that cro induces the identity on Gal (Yp1 /Fp, ). Therefore by the continuity of the action of Aut (El) on Fp, we see that ao induces the identity on Gal (YFp, /FpI ) for every q E C. By Proposition 3.14, C is a subgroup of index 1 or 2 in Aut (El). Therefore by Proposition 2.13 (or Remark 2.14 if p = 3) we have 9 Gal (,Fp, /Fp,) Z Zp. This proves Proposition 4.2(b), and thus completes the proof of Theorem 1.11.

DEPARTMENT OF MATHEMATICS, THE JOHNS HOPKINS UNIVERSITY, BALTIMORE, MD 21218

Current address: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF FLORIDA, GAINESVILLE,

FL 32611



440 KEVIN KEATING

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an abstract characterization of the igusa curves

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