A supersingular congruence for modular forms

Transcription

1 ACTA ARITHMETICA LXXXVI.1 (1998) A suersingular congruence for modular forms by Andrew Baker (Glasgow) Introduction. In [6], Gross and Landweber roved the following suersingular congruence in the ring of holomorhic modular forms for SL 2 (Z) with q-coefficients in the ring of -local integers Z () for a rime > 3: (0.1) u 2 ( 1 ) (2 1)/12 mod (, u 1 ). The regular sequence, u 1, u 2 is defined using the canonical formal grou law F associated to the universal Weierstraß cubic whose -series has the form (0.2) [] F (X) = X u 1 X u 2 X 2 + (higher order terms) u 1 X u 2 X 2 + (higher order terms) mod () u 2 X 2 + (higher order terms) mod (, u 1 ). In fact, u 1 is essentially the Eisenstein function E 1, in the sense that u 1 E 1 mod (). The main result of this aer is the following suersingular congruence for E +1 which is closely related to (0.1): (0.3) (E +1 ) 1 ( 1 ) (2 1)/12 mod (, E 1 ). These congruences are equivalent to equations that hold in the field of definition of a suersingular ellitic curve over a finite field of characteristic greater than 3, and our roof is couched in terms of this interretation. It turns out that our result is related to one conjectured by de Shalit [12] and described by Kaneko and Zagier in [4]. In fact, our original attemt at roving Theorem 1.4 involved a reduction to equation (6.1). This unsuccessful strategy was aborted when Don Zagier ointed out the equivalence of the two results! 1991 Mathematics Subject Classification: 14K22, 11G20. Key words and hrases: modular forms, suersingular ellitic curves. [91]

2 92 A. Baker Our original motivation in studying this question, and more generally isogenies of suersingular ellitic curves, lies in ellitic cohomology. Insired by results of Robert [8] and of Gross and Landweber, we have determined the recise relationshi between the category of isogenies and the stable oeration algebra of suersingular ellitic cohomology. The details will aear in [1] which is currently in rearation. Like the resent work, this makes use of Tate s theory, articularly that of the -rimary Tate module (never formally ublished by him but described in [17], see also the Woods Hole Notes [7]). I would like to thank K. Buzzard, F. Clarke, I. Connell, J. Cremona, R. Odoni, R. Rankin, J. Tate and D. Zagier for hel and encouragement on the subject matter of this aer and esecially G. Robert for a never forgotten conversation of many years ago. 1. Recollections on modular forms and ellitic curves over finite fields. Background material for this section can be found in the articles of Serre [10, 11], Katz [5] and Tate [14]; see also the books by Husemoller and Silverman [3, 13]. Throughout, let > 3 be a rime and let S(Z () ) (resectively M(Z () ) ) denote the graded ring of modular forms for SL 2 (Z), holomorhic (resectively meromorhic) at and with q-coefficients in the ring of -local integers Z (). We will make use of the following modular forms which are determined by their q-exansions: P = E 2 = r σ 1 (r)q r, Q = E 4 = r σ 3 (r)q r, R = E 6 = r σ 5 (r)q r, = Q3 R , j = Q3, A = E 1 = 1 B = E +1 = 1 2( 1) B 1 2( + 1) B +1 σ 2 (r)q r, 1 r σ (r)q r. Here, Q and R are modular forms of weights 4 and 6, while P is almost modular of weight 2. 1 r

3 Theorem 1.1. As graded rings, Suersingular congruence 93 S(Z () ) = Z () [Q, R], M(Z () ) = Z () [Q, R, 1 ]. Also, S(Z () ) 0 = Z (), M(Z () ) 0 = Z () [j]. There is a derivation on M(Z () ) which restricts to S(Z () ) and satisfies P = Q P 2, Q = 4R, R = 6Q 2, = 0, j = 12 Q2 R. Theorem 1.2. For the rime > 3, 1. In the ring S(F ) = S(Z () ) /(), is not a factor of A. 2. In the ring S(F ) each irreducible factor of A has multilicity one, hence the same is true in the ring M(F ) = M(Z () ) /(). 3. In each of the rings S(F ) and M(F ), every irreducible factor of A has one of the forms Q, R, Q 3 α, Q 6 + β Q 3 + γ 2, where α, β, γ F with α 0 and X 2 + βx + γ F [X] irreducible. We also note the following calculational result. Proosition 1.3. For a rime > 3, in the ring M(F ) we have the identities modulo : B A, B QA. Now let F q be the finite field of order q = d where we continue to assume that > 3. An ellitic curve E over F q is determined by its Weierstraß form, E : y 2 = 4x 3 ax b. The non-singularity of E is equivalent to the existence of a classifying ring homomorhism θ E : M(F ) F q for which θ E (Q) = 12a and θ E (R) = 216b. The curve is suersingular if θ E (A) = 0, or equivalently θ E factors through a homomorhism M(F ) /(A) F q (which we will also denote by θ E ). For x M(Z () ) or M(F ) /(A) we will often write x(e) = θ E (x). Given u F q, the curve E u : y 2 = 4x 3 au 2 x bu 3 is the u-twist of E. It is isomorhic (as an abelian variety over F q ) to E if and only if u is a square in F q and in that case, an isomorhism is rovided by the comletion of the affine ma ϕ v : (x, y) (v 2 x, v 3 y) where v 2 = u. Associated to the Weierstraß form is the canonical invariant differential ω E = dx y.

4 94 A. Baker Notice that when u = v 2, Also, if F M(Z () ) k, then ϕ vω E u = v 1 ω E. F (E u ) = v k F (E). Using the above notation we restate our main result as the following: Theorem 1.4. For the rime > 3, in each of the rings S(Z () ) and M(Z () ) we have the congruence B 1 (2 1)/12 mod (, A). Equivalently, for a suersingular ellitic curve E over a finite field F d, B(E) 1 = (E) (2 1)/ Some suersingular isogeny invariants. Recall that for two ellitic curves E 1, E 2 defined over a field k, an isogeny ϕ : E 1 E 2 over k is a non-zero morhism of abelian varieties. Using the dual isogeny ϕ : E 2 E 1, it is easily seen that the existence of an isogeny E 1 E 2 is equivalent to the existence of an isogeny E 2 E 1. Hence the notion of isogeny defines an equivalence relation on ellitic curves. The next imortant result due to Tate [15] (see also [3], Chater 3, Theorem 8.4) allows us to determine isogeny classes of suersingular curves. Theorem 2.1. Two ellitic curves E 1, E 2 defined over a finite field F q are isogenous over F q if and only if E 1 (F q ) = E 2 (F q ). In articular, a suersingular curve defined over the rime field F has E(F ) = 1 +, hence all such curves are isogenous over F. For a more detailed analysis of the ossible isogeny classes, see [16, 9]. For suersingular ellitic curves over finite fields, it turns out that there are some interesting isogeny invariants. In [6], Gross and Landweber in effect showed for two such curves E 1, E 2 defined and searably isogenous over F 2, ( ) ( ) 1 1 (E 1 ) (2 1)/12 = (E 2 ) (2 1)/12. This follows from the facts that these two quantities are actually in F and by (0.2) can be identified with the coefficients of the leading terms T 2 in the []-series of the isomorhic canonical formal grou laws associated to the local arameter 2x/y.

5 Suersingular congruence 95 In order to identify another isogeny invariant, we will need Théorème B/ Lemme 7 of Robert [8]. Lemma 2.2. Let ϕ : E 1 E 2 be a searable isogeny between suersingular ellitic curves. Then if ϕ ω E2 = λω E1, Corollary 2.3. B(E 2 ) = λ (+1) deg ϕb(e 1 ). B(E 2 ) 1 = λ (2 1) B(E 1 ) 1. In articular, if E 1, E 2 and ϕ are all defined over F 2, then B(E 2 ) 1 = B(E 1 ) 1. Using this corollary, together with the fact that for a suersingular curve E over a finite field F d, j(e) F 2 and there is suersingular curve E defined over F 2 and an isomorhism E = E defined over F d, we can reduce the roof of our main theorem to the case of curves defined over F Constructing suersingular curves over the rime field. For comleteness, in this section we outline details of a construction which seems to be well known but whose full details are not so readily found in the literature. A nice account of some asects of this can be found in Cox [2]. Let K = Q( ) and O K be its ring of integers which is its unique maximal order. Theorem 3.1. For any rime > 11, there are suersingular ellitic curves E defined over F and with j(e) 0, 1728 mod () and having O K End E. Now O K is a lattice in C, hence we can define the torus C/O K which has a rojective embedding as a Weierstraß cubic E K. Since O K is an O K -module, E K admits comlex multilication by O K. Proosition The j-invariant j(o K ) = E K is an algebraic integer. 2. The extension field L = K(j(O K )) is the Hilbert class field of K. 3. The ellitic curve E K is defined over L. A roerty of the Hilbert class field is that it is unramified at every rincial rime ideal in O K. In articular, if is a rime in O L lying above the rime ( ) in O K, then the residue field is Hence, O L / = F. (3.1) j(o K ) mod F.

6 96 A. Baker Since the curve E K can be defined over L, we can assume that it has the Weierstraß form E K : y 2 = 4x 3 ax b where a, b O L. Unfortunately, this might have discriminant lying in some rime ideal over ( ). To overcome this roblem we ass to -adic comletions K ( ) L which are comlete local fields with maximal discrete valuation rings O K,( ) O L,. We may ass to some finite extension L /L in which is totally ramified and the rincial rime ideal = (π) O L satisfies = λπ 12k for some integer k 0 and unit λ O L. The curve E : y 2 = 4x 3 π 4k ax π 6k b is now defined over O L L and isomorhic to E K over L. Moreover, its discriminant is λ, which reduces to a non-zero element of O L /(π), hence the reduced curve Ẽ is non-singular and so ellitic. We also have j(ẽ ) = j(e ) mod (π) = j(o K ) mod (π) with the latter lying in F. Hence, Ẽ is isomorhic over F to an ellitic curve E defined over F. The endomorhism ring of E is at least as big as O K. Notice that it cannot contain the comlex numbers i or ω since it would then have a commutative endomorhism ring of rank greater than 2. Thus we must have j(o K ) 0, 1728 mod (), and using a straightforward change of variables, can actually assume that E has the form E : y 2 = 4x 3 27j(O K) j(o K ) 1728 x 27j(O K) j(o K ) In fact, End E is non-commutative since E is suersingular. To see this, notice that from general considerations of [15, 16, 17] the action of agrees with that of the Frobenius ma. Alying Fr to the Tate module T l E for any rime l, we easily see that But this imlies that Tr Fr = = 0. E(F ) = + 1 Tr Fr = + 1, or equivalently that E is suersingular by standard results of [3, 13]. 4. The case of suersingular curves over the rime field. In [6], Gross and Landweber roved that for a suersingular ellitic curve E defined

7 Suersingular congruence 97 over F 2 the following identity holds whenever j(e) 0, 1728 mod (): { (4.1) (E) (2 1)/12 1 if Fr = 2 = [( 1/)] E (Case A), 1 if Fr 2 = [ ( 1/)] E (Case B). Here Fr 2 : E E (2) = E is the relative Frobenius ma and the stated ossibilities are the only ones that can occur. They also observe that if Case A holds, then E[4] E(F 2). Since E[4] = 16, this means that E(F 2) 0 mod (16). In Case B, a modification of their discussion shows that none of the elements of order 4 can be in F 2. On the other hand, in all cases, E[2] E(F 2). Let us now consider the case of such a curve actually defined over the rime F. Then it is well known that the number of oints over F is E(F ) = 1 +. Using the form of the zeta function over F given by the Weil Conjectures, we easily find that { E(F 2) = = (1 + ) 2 4 mod (8) if 1 mod (4), 0 mod (8) if 3 mod (4). Hence, for such a curve, we have Case A holds 3 mod (4), Case B holds 1 mod (4). Since B(E) F, B(E) 1 = 1. In Case A, we have ( 1/) = 1, and so by (4.1), (E) (2 1)/12 = 1 = B(E) 1. In Case B, ( 1/) = 1, and by (4.1), (E) (2 1)/12 = 1 = B(E) 1. Hence we have roved the following: Theorem 4.1. For a suersingular ellitic curve E defined over F and satisfying j(e) 0, 1728 mod (), B(E) 1 = (E) (2 1)/ The case of suersingular curves over the field of order 2. Having dealt with suersingular curves over the rime field, we now turn

8 98 A. Baker to those defined over F 2. For > 11, choose a suersingular ellitic curve E 0 defined over F with j(e) 0, 1728 mod () this is always ossible courtesy of Theorem 3.1. Let E be a suersingular ellitic curve defined over F 2 and with j(e) 0, 1728 mod (). By [16], E(F 2) = 1 ± = (1 ± ) 2. By the Weil Conjectures, any curve defined over F has E(F 2) = (1 + ) 2. By Theorem 2.1, if E(F 2) = (1 + ) 2, there is an isogeny E 0 E defined over F 2. There is a unique factorization of the form ϕ = s ϕ Fr k, where Fr k : E 0 E (k ) 0 = E 0 is the k-fold iterated Frobenius ma, and s ϕ : E 0 E is searable. Hence we might as well assume that ϕ itself is searable. Now alying Corollary 2.3 we may deduce that B(E) 1 = B(E 0 ) 1 = 1. Notice that if 1 mod (4) then Case B of Section 4 alies to E, while if 3 mod (4) then Case A alies. Thus we find that B(E) 1 = 1 = (2 1)/12. If E(F 2) = (1 ) 2, we may twist by any non-square u in F 2 to obtain a curve E u : y 2 = 4x 3 au 2 x bu 3 which can easily be seen to have E u (F 2) = (1+) 2. If v F 4 with v 2 = u, (x, y) (v 2 x, v 3 ) defines an isomorhism ϕ v : E = E u over F 4, and we have ϕ vω E u = ω E. By the above result for E u together with Corollary 2.3, and the fact that (E u ) = u 6 (E u ), we now see that ( ) ( ) 1 1 B(E) 1 = B(E u ) 1 = (E u ) (2 1)/12 = (E) (2 1)/12. Similar arguments allow our identity to be roved directly for suersingular curves with j(e) 0, 1728 mod (). Hence for rimes 1 mod (12) we can avoid the use of Theorem 3.1, but when 1 mod (12), we do require this result. 6. Relations with other work. In [4], Kaneko and Zagier discuss the suersingular olynomial ss (X) = E (X j(e)),

9 Suersingular congruence 99 where the roduct is taken over all isomorhism classes of suersingular curves over F. Thus in the ring M(F ) we have A = Qδ R ε ss (j) m j δ (j 1728) ε, where we write = 12m +4δ+6ε+1 with δ, ε {0, 1}. Using Proosition 1.3 we obtain a formula for B in terms of the derivation. If α 0, 1728 mod () is a root of ss (X), then there is a suersingular ellitic curve E : y 2 = 4x 3 27α α 1728 x 27α α 1728 with j(e) = α. Then for some λ F, B(E) = λ αε+1 ss (α) m (α 1728) δ+ε. Combining this with Theorem 1.4 gives (6.1) ss (α) 1 = ( 1) ε 1 α 2(δ 1)( 1)/3 (α 1728) (ε 1)( 1)/2 which is the conjectured result [4], equation (40). Thus we have also roved the equivalent conjectural equation (39) of de Shalit. References [1] A. B a k e r, Isogenies of suersingular ellitic curves over finite fields and oerations in ellitic cohomology, in rearation. [2] D. A. Cox, Primes of the Form x 2 +ny 2. Fermat, Class Field Theory and Comlex Multilication, Wiley, [3] D. Husemoller, Ellitic Curves, Sringer, [4] M. Kaneko and D. Zagier, Suersingular j-invariants, hyergeometric series, and Atkin s orthogonal olynomials, rerint. [5] N. M. Katz, -adic roerties of modular schemes and modular forms, in: Lecture Notes in Math. 350, Sringer, 1973, [6] P. S. Landweber, Suersingular ellitic curves and congruences for Legendre olynomials, in: Lecture Notes in Math. 1326, Sringer, 1988, [7] J. Lubin, J.-P. Serre and J. Tate, Ellitic curves and formal grous, mimeograhed notes from the Woods Hole conference, available at htt:// edu/ voloch/lst.html. [8] G. Robert, Congruences entre séries d Eisenstein, dans le cas suersingulier, Invent. Math. 61 (1980), [9] H.-G. Rück, A note on ellitic curves over finite fields, Math. Com. 49 (1987), [10] J.-P. S e r r e, Congruences et formes modulaires (d arès H. P. F. Swinnerton-Dyer), Sém. Bourbaki 24 e Année, 1971/2, No. 416, Lecture Notes in Math. 317, Sringer, 1973, [11], Formes modulaires et fonctions zêta -adiques, Lecture Notes in Math. 350, Sringer, 1973,

10 100 A. Baker [12] E. d e S h a l i t, Kronecker s olynomial, suersingular ellitic curves, and -adic eriods of modular curves, in: Contem. Math. 165, Amer. Math. Soc., 1994, [13] J. Silverman, The Arithmetic of Ellitic Curves, Sringer, [14] J. Tate, The arithmetic of ellitic curves, Invent. Math. 23 (1974), [15], Endomorhisms of abelian varieties over finite fields, ibid. 2 (1966), [16] W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Su. (4) 2 (1969), [17] W. C. Waterhouse and J. S. Milne, Abelian varieties over finite fields, in: Proc. Symos. Pure Math. 20, Amer. Math. Soc., 1971, Deartment of Mathematics University of Glasgow Glasgow G12 8QW Scotland a.baker@maths.gla.ac.uk Received on (3328)