SERIES, AND ATKIN S ORTHOGONAL POLYNOMIALS. M. Kaneko and D. Zagier

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1 SUPERSINGULAR j-invariants, HYPERGEOMETRIC SERIES, AND ATKIN S ORTHOGONAL POLYNOMIALS M. Kaneko and D. Zagier 1. Introduction. An elliptic curve E over a field K of characteristic p > 0 is called supersingular if the group E K has no p-torsion. This condition depends only on the j-invariant of E and it is well known cf. for a review that there are only finitely any supersingular j-invariants in F p. We are interested in coputing the polynoial ss p j = E/ F p E supersingular j je F p [j]. The polynoial describing supersingularity in ters of the λ-invariant of E defined by writing E over K in Legendre for y = xx 1x λ has a well-known and siple explicit expression, but a convenient expression for the polynoial expressing the condition of supersingularity directly in ters of the j-invariant i.e., in ters of a Weierstrass odel over K, without nubering the -torsion points over K is less easy to find. In this partially expository paper, we will describe several different ways of constructing canonical polynoials in Q[j] whose reductions odulo p give ss p j. These will be of three kinds: A. polynoials coing fro special odular fors of weight p 1, B. the Atkin orthogonal polynoials, and C. other orthogonal polynoials coing fro hypergeoetric series. In the rest of this introduction, we will describe in ore detail these various ways of getting the supersingular polynoials. A. For any even integer k >, let M k denote the space of odular fors of weight k on Γ = P SL, Z. We can write k uniquely in the for k = 1 + 4δ + 6ε with Z 0, δ {0, 1, }, ε {0, 1}, 1 and then di M k = + 1 and any odular for in M k can be written uniquely as fτ = τ E 4 τ δ E 6 τ ε f jτ for soe polynoial f of degree in jτ, the coefficient of j in f being equal to the constant ter of the Fourier expansion of f. Here, E 4, E 6 and j have 1

2 their standard eanings, recalled in 3. On the other hand, if k = p 1 for a prie nuber p 5, then deg ss p = + δ + ε and the polynoial ss p j is divisible by j δ j 178 ε. We will describe for each k four three if k od 3 odular fors E k, F k, G k, and H k of weight k such that if k = p 1 then the corresponding polynoial in j, ultiplied by j δ j 178 ε, reduces odulo p to the supersingular polynoial. These fors are defined as follows: E k is the noralized Eisenstein series of weight k ; G k is the coefficient of X k in 1 3E 4 τx 4 + E 6 τ X 6 1/ ; H k is the coefficient of X k in 1 3E 4 τx 4 + E 6 τ X 6 k/ ; F k for k od 3 is the unique noralized solution in M k of the differential equation ϑ k+ ϑ k F k = kk+ 144 E 4 F k. Here ϑ k : M k M k+ is the derivation f f ke f/1 where f = πi 1 df/dτ = q df/dq and E = / = 1 4q is the nearly odular Eisenstein series of weight ; the existence and uniqueness of F k will be shown in 3. The first result is then: Theore 1. Let k = p 1 where p 5 is prie and let f be any of the four odular fors E k, F k, G k, H k described above. Then the coefficients of the associated polynoial f are p-integral and ss p j ±j δ j 178 ε fj od p. Of these four descriptions of ss p j, the ones in ters of H k and E k are wellknown, the forer being a classical result of Hasse and Deuring and the latter a result apparently first noticed by Deligne cf. [8]. We will give self-contained and eleentary proofs of all four in 3. We will also give alternate descriptions of G k as the residue at 0 of the k+1 -th power of the Weierstrass -function and of F k as a hypergeoetric function. As a nuerical exaple, for k = 8 the polynoials fj, related to the corresponding odular fors by fτ = τ E 4 τ fjτ, are given by Ẽ k j = j j , G k j = j j , H k j = j j , F k j = j 1144 j and with p = 9 we indeed find Ẽ k j F k j G k j H k j j + j + 1 ss p j/j od p. B. The second, and even ore beautiful, description of the supersingular polynoials was found about ten years ago by Atkin, who was inspired by a paper of

3 Rankin [7] on the zeros of Eisenstein series. However, Atkin s proofs were apparently never published and are not well-known. One ain purpose of this paper is to popularize and to provide sipler proofs of his discoveries. Atkin defines a sequence of polynoials A n j Q[j], one in each degree n, as the orthogonal polynoials with respect to a special scalar product. Recall that, if V is the space of polynoials in one variable over a field K, and φ : V K a linear functional, then one can consider the scalar product on V defined by f, g = φfg and the faily which for generic φ exists and unique of onic polynoials which are utually orthogonal with respect to it. The study of such polynoials, which we will review briefly in 4, is an old subject and is iportant in any parts of atheatics. In our context, we take V to be the space of polynoials in j; thinking of j as the odular invariant jτ = q , we can identify V with the space of holoorphic Γ-invariant functions in the upper half-plane H which are eroorphic at infinity i.e. f aτ+b a b cτ+d = fτ for all c d Γ and f has a Laurent series expansion fτ = c n q n. n Theore Atkin. There is a unique functional φ on V up to a scalar ultiple for which all Hecke operators T n : V V n N are self-adjoint with respect to the associated scalar product f, g = φfg, and a unique faily of onic polynoials A n j of degree n = 0, 1,,... which are orthogonal with respect to this scalar product. We will prove this theore, and at the sae tie give several explicit descriptions of the scalar product, in 5. We ention only one here. Take the weight 1 cusp for τ = q 4q + 5q 3 + rather than q or j 1 as a local paraeter for H/Γ at infinity. Then f, g = constant ter of fτ gτ as a Laurent series in τ. 3 The polynoials A n can be found by the Gra-Schidt orthogonalization procedure or by the explicit forulas given in Theore 4 below. The first few are A 0 j = 1, A 1 j = j 70, A j = j 1640j , A 3 j = j j j , 5 A 4 j = j j j j The coefficients of A n are rational nubers in general, but they are p-integral for pries p > n. In particular, if n p p/1 is the degree of the supersingular polynoial ss p, then A np has p-integral coefficients, and we have: Theore 3 Atkin. Let p be a prie nuber. Then ss p j A np j od p. The for of this theore is a little surprising: unlike the descriptions in Theore 1, where the polynoials depended separately on, δ and ε and we therefore had four 3

4 polynoials of each degree, here the polynoial depends only on n = +δ+ε. Thus a single Atkin polynoial A n ay have to do duty for as any as four different supersingular polynoials, if the four nubers 1n 13, 1n 7, 1n 5 and 1n + 1 are all prie, e.g. the supersingular polynoial for each of the four prie nubers p = 3, 9, 31, 37 is the od p reduction of the sae polynoial A 3 j. For instance, for p = 9 we find A 3 j j 3 +j +1j od p, in accordance with the nuerical exaples given in A above. Theore 4. The polynoials A n are deterined in each of the following ways: i Recursion relation: A n+1 j = 144n 9 j 4 An j n + 1n 1 1n 131n 71n 51n nn 1n 1 A n 1 j 4 for n, with initial values A 0, A 1, A as given above; ii Closed forula: A n j = n [ i 1 3i i i i=0 =0 n n 7 1 n 1 1 ] j n i iii Differential equation: j j c n j 144 A n + jj c [6n j 14436n + 7j + c /3] A n [n 4 7n j 3 487n 4 45n 30j 4c40n + 413j + 30c ] A n [n 4 n j 47n 4 13n 1j + c19n 107] A n + [n 6 j 418n 4 n ] A n = 0 where c = 178, and A n is the unique onic polynoial solution of this equation. We will give a proof of Theore 3 fro the point of view of odular fors theory in 6 and a second proof, fro the point of view of the theory of hypergeoetric functions and with the recurrence 4 as the definition of the Atkin polynoials, in 7. Atkin s original proof also used odular fors and hypergeoetric functions, but involved higher hypergeoetric functions p F q and was considerably ore coplicated. 7 also contains the proof of Theore 4 and of other explicit forulas for A n in ters of truncated hypergeoetric series. C. In 8 we will show that the polynoials F k j attached to the odular fors F k k, k od 3 defined in part A have beautiful expressions as hypergeoetric polynoials and also enjoy properties like those of the Atkin polynoials: not only do their reductions odulo p give the supersingular polynoials, but they are also orthogonal with respect to a suitable scalar product on a space of odular functions on P SL, Z. This scalar product does not have the nice property of 4

5 aking the Hecke operators self-adjoint, but is in other respects uch sipler than the Atkin scalar product the scalar product of two onoials in j and j 178 is given by a very siple forula. Moreover, the polynoials F k are given by forulas siilar to those in Theore 4, but rather sipler: they satisfy a recursion of alost exactly the sae for as 4 ore precisely, four recursions, one for each of the residue classes odulo 1 which occur, but are given by a uch sipler closed forula and satisfy a uch sipler differential equation, of order rather than 4. The detailed stateent will be given as Theore 5 in 8 when we have established ore notation. The last two sections of the paper contain a few copleentary results. As we already entioned, the recursion 4 iplies that A n j has rational coefficients and is p-integral for p > n. But the recursion shows only that its denoinator divides n!n! n 1 n!, while fro nuerical exaples we find that the denoinators are in fact far saller. For instance, the denoinator of A 9 j is only 34, and only three of its coefficients are non-integral, and the previous A n j have even fewer non-integral coefficients. In 9 we will use the closed forula in Theore 4 to study this phenoenon and soe related congruences. Finally, in 10 we will describe an eleentary arguent relating the properties of the classical odular polynoial Φ p X, Y Z[X, Y ] to the od p reduction of the polynoial Ẽp 1j, and thus the supersingular polynoial. This yields at the sae tie easy proofs of soe properties of supersingular polynoials which were entioned in the text and a partial answer to a question of E. de Shalit [9].. Supersingular elliptic curves The definition of supersingular elliptic curves over a field of characteristic p was given at the beginning of the paper. We begin by recalling the stateent and proof of the standard criterion for deciding whether a given curve over a finite field F q q = p r, p odd is supersingular or not. Proposition 1. Let E be the elliptic curve over F q defined by the equation y = fx f F q [x] of degree 3, and a p the coefficient of x p 1 in fx p 1/. Then EF q 1 N Fq /F p a p od p. Corollary. E is supersingular if and only if a p = 0. Proof. For x F q the nuber of solutions in F q of y = fx is equal to 1 + fx q 1/ naely, to 0, 1, or for fx / F q, fx = 0, or fx F q, respectively. Counting also the point at infinity, we find EF q = 1 + q fx x F q in F q. Since the su over x F q of x j equals 1 for j = q 1 and 0 for all other j in the range 0 j 3q 1/, this gives EF q = 1 a q in F q, 5

6 where a q denotes the coefficient of x q 1 in fx q 1/. In particular, a q belongs to F p and not erely to F q. But fro the expansion fx q 1 = fx p 1 1+p+ +pr 1 = fx p 1 f p x p p 1 f pr 1 x pr 1 p 1, where f pj is the polynoial obtained fro f by raising all its coefficients to the p j - th power, we see that a q = ap 1+p+ +pr 1 = N Fq /F p a p. This proves the proposition. To prove the corollary, note that if a p = 0, then EF q n 1 0 od p for all n, so E has no p-torsion over F p. Conversely, if a p 0, then EF q n 1 N Fq /F p a p n is divisible by p for n divisible by the order of N Fq /F p a p odulo p, so E F p does contain p-torsion. We now write out the contents of the corollary explicitly in ters of the standard Weierstrass equation. This will give the proof of Theore 1 in the case f = H k. Suppose that p 5. Then any elliptic curve over a field K of characteristic p can be written in a Weierstrass for E : y = x 3 3 Q x + R, 5 where the factors 3 and have been included to coincide with traditional notations. If Q and R are replaced by the Eisenstein series E 4 τ and E 6 τ then 5 is an equation of the elliptic curve C/Zτ + Z over C with j-invariant jτ. The j- invariant of E is equal to Q 3 /, where = Q 3 R /178. We define a graded hoogeneous polynoial H p 1 Q, R of degree p 1 in Q and R where Q and R have degrees 4 and 6, respectively, the Hasse polynoial, as the coefficient of x p 1 in x 3 3Qx + R p 1/, so that the odular for H p 1 E 4 τ, E 6 τ is the sae as the odular for of weight p 1 denoted H p 1 τ in the introduction. As explained there, we can write this polynoial in the for H p 1 Q, R = Q δ R ε Hp 1 j for soe polynoial H p 1 Z[j], where, δ and ε are the nubers defined by 1 with k = p 1. They are given explicitly by = [ { p ] 0 if p 1 od 3,, δ = 1 1 if p od 3, { 0 if p 1 od 4, ε = 1 if p 3 od 4. 6 It now follows fro the corollary above that, at least in the case K F p, the curve E is supersingular if and only if j δ j 178 ε Hp 1 j = 0 and hence that ss p j j δ j 178 ε Hp 1 j. The fact that the two polynoials agree up to a constant, as claied in Theore 1, therefore follows fro the well-known forula n p := deg ss p j = + δ + ε, 7 6

7 but to ake the paper self-contained we give a direct proof. It suffices to show that the polynoial j δ j 178 ε Hp 1 j has no ultiple roots, since we have already shown that it has the sae zeros as ss p, which is square-free by definition. We first treat the roots 0 and 178. Fro the expansion we find x 3 3Qx + R p 1 = x 3 + R p 1 3 p 1 Qx x 3 + R p 3 + OQ H p 1 Q, R = p 1 p 1 p 1 R 6 + OQ if p 1 od p 1 p 3 p 5 p R 6 Q + OQ if p od 3 3 and hence in both cases that H p od p. A siilar arguent works for j = 178. That the other nubers in F p cannot be ultiple zeros of Hp 1 j follows fro the fact that this polynoial satisfies a second-order linear differential equation with polynoial coefficients and with leading coefficient jj 178. We will show in 3 that H p 1 and F p 1 agree odulo p, and the differential equation for the latter is a siple translation of the definition of F k, given explicitly in 8. This iplies that any coon zero in F p \ {0, 178} of H p 1 and its first derivative would be a zero of all higher derivatives and hence have infinite order, which is ipossible. Note that since we are in characteristic p, this arguent would fail if H p 1 were a polynoial in j p, but this is not the case since deg H p 1 = < p. Reark. We have proved in particular that there are only finitely any supersingular invariants in F p, as entioned in the introduction. In fact, one knows that these are the only supersingular j-invariants i.e., the j-invariant of a supersingular elliptic curve over any field K of characteristic p lies in F p, and also that they all lie in F p equivalently, ss p j factors into linear and quadratic polynoials in F p [j]. For proofs we refer the reader to [3] and [1], which are the two basic references for the theory of supersingular elliptic curves. In 10 we will give an eleentary proof of the fact that all roots of ss p j lie in F p using the forula ss p j = j δ j 178 ε Ẽ p 1 j, which will be established in the next section, as well as another proof of the siplicity of the zeros 0, 178 of H p 1 j od p. 3. Modular fors and supersingular polynoials Our object now is to prove Theore 1 of the Introduction. We begin by giving the definitions of the special odular fors E k, F k, G k, and H k in ore detail. For k even and positive we denote by B k the kth Bernoulli nuber and by E k τ the kth Eisenstein series E k τ = 1 k B k d k 1 q n q = e πiτ. n=1 d n 7

8 It is a odular for of weight k for k 4 and for k = is nearly odular : aτ + b E = cτ + d E τ + 6 cτ + d πi ccτ + d a b Γ. 8 c d We also define = E 3 4 E 6/178 M 1 and jτ = E 4 τ 3 / τ, the odular invariant. Fro 8 it follows easily that E = E E 4, E 4 = E E 4 E E 6 = E E 6 E 4, = E 9 and ore generally ϑ k f = f k 1 E f M k+ for any f M k, where denotes differentiation with respect to πiτ. If k is represented as in 1, then any f M k has a zero of ultiplicity δ at τ = e πi/3 and a zero of ultiplicity ε at τ = e πi/ and hence is divisible by E4E δ 6, ε and it follows that f has a representation as in with f a polynoial of degree at ost. If k+4 0 od 3, then every eleent of M k+4 is divisible by E 4, so we have an endoorphis φ k of M k defined by φ k f = E4 1 ϑ k+ϑ k f. Since the constant ter of φ k f is κ k := kk+/144 ties the constant ter of f, this ap preserves the codiension 1 subspace of cusp fors and induces on the quotient space the ap ultiplication by κ k. It follows that κ k is an eigenvalue of φ k. If we pick soe corresponding eigenvector F k, then the other eigenvectors are the odular fors i F k 1i 1 i with eigenvalues κ k 1i κ k because ϑ k i = i ϑ k 1i and di M k = + 1, so F k is unique up to a noralizing factor, which we will fix later when we give natural forulas for F k in ters of hypergeoetric series. Together with the definitions of H k and G k given in 1, this defines the four odular fors E k, F k, G k, H k M k. The letters E, F an H are eant to suggest Eisenstein series, hypergeoetric function, and Hasse invariant, while the reaining letter fills the gap. We will prove that for any prie nuber p 5 and Ẽ p 1 j F p 1 j 1 δ+ε Gp 1 j 1 δ+ε Hp 1 j od p 10 ss p j = 1 δ+ε j δ j 178 ε Hp 1 j od p, 11 which together for a slightly ore precise version of Theore 1. Here and in what follows, δ and ε are defined by 6. Equation 11 was already proved in the last section, except for the value of the constant 1 δ+ε. Since ss p is by definition onic, we only have to copute the leading coefficient of H p 1. The leading coefficient of the polynoial fj for any odular for f M k is just the constant ter of the Fourier expansion of f, i.e. the liiting value of f as q 0. Since E 4 and E 6 have the value 1 at q = 0, this nuber for H p 1 is just the coefficient of X p 1 in 1 3X 4 + X 6 p 1/ = 1 X p X p 1/, and fro 1 X p X p 1 1 Xp [ 1 + X ] 1 X p 1 3 p p 1 1 X p polynoial of degree p X p p 1 X od p 8

9 we find that this coefficient is congruent to 3 p = 1 δ+ε odulo p, as claied. We reark that the evaluation of the constant in 11 would also follow fro equation 10 which we will prove independently, since both ss p and Ẽp 1 are onic. We now proceed to equation 10. The congruence G p 1 H p 1 od p is obvious, since the generating functions of the odular fors G k and H k differ by a factor 1 3E 4 X 4 + E 6 X 6 p/ 1 + OX p od p. For the congruence between G p 1 and Ẽp 1 we again use a generating function. Consider the elliptic curve over C with Weierstrass odel 5, where Q = E 4 τ, R = E 6 τ. It can be paraetrized in a well-known way by the Weierstrass -function here renoralized to keep coefficients rational, naely by x = P u, y = 1 P u where P u = u n 4, n even 1 n/ B n nn! E nτ u n B n = nth Bernoulli nuber. Therefore the change of variables X = P u 1/ = u +... gives dx G k = Res X=0 X k+1 1 3E 4 X 4 + E 6 X 6 = Res u=0 P u k+1 du = coeff. of u k in 1 n 4 even 1 n/ k+1 B n nn! E n u n. Now take k = p 1 and observe that B n E n /n! for n < p 1 is a polynoial in E 4 and E 6 with p-integral coefficients, while p B p 1 /p 1! 1 od p, so 1 n 4 even 1 n/ B n nn! E n u n p p 1 E p 1 u p 1 + Ou p od p. Using 1 p 1/ 1 = 1 δ+ε, we obtain the desired congruence for G p 1. p Finally, we have to consider F k. This function was defined up to a constant as the unique odular for annihilated by the operator ϑ k+1 ϑ k κ k E 4. Since for k = p 1 the eigenvalues κ k 1r 0 r k/1 of the operator E4 1 ϑ k+1ϑ k reain distinct after reduction odulo p, this characterization reains valid also in characteristic p. By a odular for of weight k odulo p we ean a polynoial in E 4 and E 6 of the right degree with coefficients in F p. But using forulas 9 we find that ϑk+ ϑ k κ k E 4 f = f k + 1 E f kk E f 6 1 and this certainly vanishes odulo p if k = p 1 and f is E p 1, since then the Fourier expansion of f reduces to 1 od p. The proportionality of E p 1 and F p 1 9

10 odulo p follows. To get the exact constant of proportionality in 10 we still have to noralize F k, which we have not done. For reasons to be explained in 8, we will do this by k 5 constant ter of the Fourier expansion of F k τ = 1 6, 1 where is defined as in equation 1. For k = p 1 the right-hand side of 1 is clearly congruent to 1 odulo p. This copletes the proof of Theore Orthogonal polynoials In this section we review what we will need fro the theory of orthogonal polynoials. Let V be the vector space of polynoials in one variable over a field K and, a scalar product on V of the for f, g = φfg where φ : V K is a linear functional. For the classical orthogonal polynoials, and also for the ones we shall be considering, K is a subfield of R and φ has the for φf = b fx wx dx a for soe real nubers a < b and soe positive function w on a, b. Applying the Gra-Schidt process to the basis {X n } n 0 of V, we obtain a unique basis of orthogonal onic polynoials P n by the recursion P n X = X n n 1 =0 X n, P P, P P X, provided that at each stage the scalar product P n, P n is not zero. This condition is satisfied generically and is autoatic for φ of the special for entioned above, since then f, f > 0 for all f 0. The next proposition describes the recursive calculation of the polynoials P n X, assuing this non-degeneracy condition. Proposition. i The polynoials P n satisfy a three-ter recursion of the for P n+1 X = X a n P n X b n P n 1 X n 1 13 for soe constants a n, b n K, b n = P n, P n P n 1, P n 1 0. ii Define a second sequence of polynoials {Q n } n 0 in K[X] by the sae recurrence as in i, but with initial values Q 0 = 0, Q 1 X = φ1. Then where ΦX = Q n X P n X = ΦX + OX n 1 K[[X 1 ]] 14 g n X n 1 K[[X 1 ]], g n = X n, 1 = φx n. 15 n=0 This property characterizes P n assued to be onic of degree n and Q n uniquely. 10

11 iii Define nubers λ n K n 1 by the continued fraction expansion g 0 + g 1 x + g x + = 1 g 0 λ 1 x 1 λ x 1... K[[x]]. 16 Then all λ n are non-zero and a n = λ n + λ n+1, b n = λ n 1 λ n for n 1. Proof. i Since the P n X are onic, we have XP n = P n+1 + a nn P n + + a n0 P 0 for soe constants a n K. The orthogonality of the P and the fact that the scalar product of two polynoials depends only on their product iply that { 0 if n, a n P, P = XP n, P = P n, XP = P n, P n if = n 1. This proves the assertion with a n = a nn, b n = a n n 1. ii Fro the definitions, f, g is just the coefficient of X 1 in the Laurent series fxgxφx. In particular, the fact that P n is orthogonal to all onoials of degree < n says that the coefficient of X r 1 in P n XΦX vanishes for 0 r n 1, i.e., that we have P n X ΦX = Q n X + OX n 1 K[X, X 1 ]] 17 for soe polynoial Q n X of degree n 1. Here K[X, X 1 ]] denotes the ring of Laurent series in X 1, i.e. sus of polynoials in X and power series in X 1, where X is to be thought of as large. Reversing the arguent shows that property 17, which is obviously the sae as 14, is equivalent to the orthogonality of P n with all lower degree polynoials and hence characterizes the onic polynoial P n copletely, as asserted. Finally, fro 17 and the recursion 13 we find that Q n+1 X X a n Q n X+b n Q n 1 X = OX n, so that this expression, which is a polynoial, ust vanish for n 1. Hence the Q n satisfy the sae recursion relation as the P n, with Q 0 = 0, Q 1 = g 0. iii Define another vector space V as K[Y ] with scalar product given by f, g = ψfg, where ψy n is 0 for n odd and g n/ for n even. Then we get a faily of orthogonal polynoials PnY by the sae construction as before. Since odd and even polynoials in Y are orthogonal to each other, it is obvious by induction that Pn has parity n for all n i.e., the even- and odd-index polynoials are even and odd, respectively, so the first part of the proposition applied to V, ψ gives a recursion of the for Pn+1Y = Y PnY λ n Pn 1 for soe non-zero constants λ n = Pn, Pn/P n 1, Pn 1 K. The arguent of ii gives a sequence of copanion polynoials Q n to the Pn of degree one less, and hence of opposite parity for which the rational functions Q ny /PnY are the best possible approxiations at infinity to g k Y k 1, and they satisfy the sae recursion Q n+1 = Y Q n λ n Q n 1 as the P s. Fro this one gets by induction on n the forula Q n+1 Q n = P n+1 P n g0 0 Y 1 Y 1 λ Y 1 λ n 0.

12 This translates by a standard calculation into the continued fraction 1 g 0 Y 1 λ 1 Y λ Y λn Y = Q n+1y Pn+1 Y = g 0 Y + g 1 Y 3 + g n Y n+1 + O 1. Y n+3 Setting x = Y and letting n tend to infinity, we get 16. Finally, the recurrence satisfied by the P n iplies the recurrence P n+ = Y λ n λ n+1 P n λ n 1 λ n P n for the P n of a given parity. But V can be identified via X = Y with the even part of V, with copatible scalar products, so P ny = P n Y. The relation asserted in the proposition between the coefficients a n and b n and the nubers λ n follows. Reark. Fro the proof we see that the necessary and sufficient condition for the scalar product defined by φ to be non-degenerate in the sense that P n, P n 0 for all n is that the power series g n x n has a continued fraction expansion as in 16 with g 0 and all λ n different fro 0. We also see that if, is positive definite then the nubers b n = λ n 1 λ n are always positive, and if f, g = b a fxgxwxdx with w 0 and a 0 then all λ n are positive. The condition a 0 is equivalent to the scalar product on V being positive definite, and then λ n = Pn, Pn/P n 1, Pn 1 > The Atkin scalar product and the Atkin polynoials We gave one definition of Atkin s scalar product in 1. The following result gives several alternate descriptions. Recall that V is the set of Γ-invariant holoorphic functions on H which grow at ost like q N at infinity for soe N, that V coincides with the set of polynoials in j, and that one can take q, j 1 or as a local paraeter at infinity, where q = e πiτ, j = jτ = q q + is the odular invariant, and = q 4q + 5q 3 + is the discriinant function. Proposition 3. The following four definitions of a scalar product on V coincide: i f, g = constant ter of fg as a Laurent series in ; ii f, g = constant ter of fge E 4 /E 6 as a Laurent series in j 1 ; iii f, g = constant ter of fge as a Laurent series in q ; iv f, g = 6 π/ π π/3 feiθ ge iθ dθ. Corollary. The scalar product, is positive definite on V R = R[j]. Proof. The equivalence of the first three forulas is iediate by writing the constant ters as 1/πi ties the corresponding residues and using the forulas d τ τ = πi E τ dτ = E τ dq q = E τe 4 τ E 6 τ djτ jτ. For the fourth, we use the global residue forula: Let F a denote the standard fundaental doain of Γ, truncated at soe height a > 1 i.e., the doain x 1, 1

13 x + y 1, y a, where τ = x + iy. The integral of fτgτe τdτ over the top edge of this doain equals f, g by forula iii, so the holoorphy of fge iplies that f, g is also given by the su of the integrals over the rest of the boundary, taken with the appropriate signs. The integrals along the vertical edges x = ± 1 cancel because fge is periodic of period 1. Replacing τ by 1/τ on the left half of the botto edge and noting that f and g are invariant under this transforation, we find that f, g equals the integral along the arc fro e πi/3 to e πi/ of [E τ τ E 1/τ]fτgτdτ. But the expression in square brackets equals 6i/πτ by the transforation law 8 of E, and this, with τ = e iθ, gives forula iv. The corollary follows iediately fro iv, since je iθ is real for θ [π/3, π/] and consequently π/ π/3 feiθ dθ > 0 for fτ any non-zero polynoial in jτ with real or a fortiori, rational coefficients. Note that forula iv can be rewritten in the for f, g = fj gj wj dj, wj = 6 π θ j here and fro now on we will coit the standard abuse of notation of using the sae letter to denote an eleent of V thought of as a function of τ, q, j or, indicating by one of these letters the arguent intended, where θ : [0, 178] [π/3, π/] is the inverse to the onotone increasing function θ je iθ. A graph of the function wj is given in Figure 1. By the results of the last section we now deduce that i there is a unique sequence of onic orthogonal polynoials A n j of degree n; ii the scalar product of two onoials j n and j equals g n+, where g n is the coefficient of jτ n 1 in Φτ = E τe 4 τ E 6 τjτ = q 4q q 3 + = 1 jτ + 70 jτ + ; iii the A n are the denoinators of the best rational-function approxiations to Φ; and iv they satisfy a recursion of the for A n+1 j = j λ n + λ n+1 A n j λ n 1 λ n A n 1 j 18 where the λ n are positive rational nubers defined by the continued fraction expansion of Φ with respect to 1/j. Coputing nuerically, we find that the first few values of g n = j n, 1 and λ n are given by g 0 = 1, g 1 = 70, g = 91150, g 3 = , g 4 = , λ 1 = 70, λ = 546, λ 3 = 374, λ 4 = 475, λ 5 = In 7, in connection with hypergeoetric functions, we will show that 70 if n = 1, λ n = n 1 n 6 + if n > 1, n 1 n

14 giving the explicit recurrence written in part i of Theore 4 of the Introduction. However, this recurrence is not needed for the proofs of the two ost iportant properties of the Atkin polynoials, their Hecke invariance and their relationship to the supersingular polynoials in characteristic p, to which we now turn. Proof of Theore. For k Z let V k denote the space of holoorphic functions in H which transfor like odular fors of weight k and have at ost exponential growth at infinity i.e., V k is the degree k part of the graded ring C[E 4, E 6, 1 ]. Note that V = V 0 and that V coincides with the set of derivatives of functions in V. We define Hecke operators on V k by f k T n τ = n k/ a b Γ\M c d n 1 cτ + d k f aτ + b cτ + d n N, f V k, where M n denotes the set of atrices with integral coefficients and deterinant n. This is not the standard noralization of T n unless k =, but will turn out to be a ore convenient noralization when we study both positive and negative weights. Notice that this forula akes sense only for f V k, since the expression cτ + d k f aτ+b cτ+d will not be independent of the choice of representative in Γ\Mn if f is not odular, but the Hecke operator at infinity f k T n τ = n k/ ad=n a, d>0 b od d d k f aτ + b d akes sense for any 1-periodic function f and agrees with k T n if f V k because a b the atrices with 0 b < d = n 0 d a are a set of representatives for Γ\M n. We clai that and Res f k T n h = Res f h k T n f, h C[q 1, q]] 0 ge T n = g 0 T n E od V g V 0, 1 where Res F for a 1-periodic holoorphic function F on H denotes the residue at infinity of πif τdτ, i.e., the constant ter of F as a Laurent series in q. Theore then follows using description iii of the Atkin scalar product and the fact that V V V and that Res vanishes on V : f 0 T n, g = Res f 0 T n g E = Res f ge Tn = Res f g 0 T n E = f, g 0 T n f, g V. To prove 0, we check that Tn acts on Fourier series by A r q r k Tn = n k/ A rd q ar r d 1 k ad=n r 14

15 and consequently that Res f k T n h = n k/ ad=n = n 1 k/ d 1 k A dr B ar r ad=n a 1+k s B as A ds = Res f h k T n for f = A r q r, h = B s q s. For 1, we use the well-known fact, equivalent to the transforation equation 8 in 3, that E τ = Eτ + 3, where y = Iτ πy and the non-holoorphic function Eτ transfors like a odular for of weight. Denoting by V the space of functions with the last property, and observing that V V V and that T n preserves V, we have g E T n g 0 T n E 3 π g y 1 T n g 0 T n y 1 od V. The right-hand side of this forula vanishes by virtue of the calculation g y 1 T n τ = ad=n b od d n d g aτ + b d aτ + b 1 I = y 1 g 0 T n τ, d so the left-hand side, which is holoorphic, belongs to V as claied. The uniqueness clause in the theore is easy to prove. Any functional φ : V C as in Theore annihilates the polynoials h n = j T n 1 j 1 T n n and h = j T j j j T. But these polynoials span a codiension 1 subspace of V, since deg h n = n and h is not a linear cobination of the h n s, so φ is unique. Reark. One can prove in the sae way the ore general adjunction forula f k T n, g = f, g k T n f V k, g V k, where the pairing, : V k V k C is defined by f, g = Res f g E. 6. Modular proof of Theore 3 We saw in the last section that the Atkin polynoials are the denoinators of the best rational approxiations to the function Φ = E E 4 /E 6 j, considered as a function of j near infinity. If p 5 is a prie, then we know that the Eisenstein series E p 1 and E p+1 are congruent as power series in q and therefore also as power series in 1/j to the constant function 1 and to the Eisenstein series E, respectively, so we can replace Φ odulo p by the odular for E p+1 E 4 /E p 1 E 6 j, which has weight 0 and hence is itself a rational function of j. This rational function is then a perfect, and hence certainly a best possible, approxiation to itself, so its denoinator ust be the od p reduction of the corresponding Atkin polynoial. But this denoinator is essentially Ẽp 1, and hence essentially the supersingular polynoial for p by the result of 3. We now give the details of this arguent. 15

16 Let p be a prie 5 Theore 3 is trivial for p = or 3, and define, δ, and ε by 6. They coincide with the, δ, and ε of 1 for weight k = p 1, while the corresponding nubers for weight p + 1 are + δ + ε 1, 1 δ, and 1 ε, respectively. Equation for the Eisenstein series E p 1 and E p+1 therefore becoes so E p 1 = E δ 4 E ε 6 Ẽp 1, E p+1 = +δ+ε 1 E4 δ E6 1 ε Ẽ p+1, Φ = E E 4 E 6 j E p+1e 4 E p 1 E 6 j = Ẽ p+1 j δ j 178 εẽ p 1 od p. The right-hand side here is a rational function whose denoinator is ss p j od p by the result of 3, while the nuerator is a polynoial of degree + δ + ε 1 = n p 1. On the other hand, Φ is equal to B n j/a n j + Oj n 1 for any n by equation 14, where A n is the nth Atkin polynoial and B n a certain polynoial of degree n 1. Taking n = n p and ultiplying A n and B n if necessary by a coon power of p to ake A np p-integral and priitive odulo p, we obtain Ẽ p+1 j ss p j Φ B np Ā np + O j n p 1 od p where Ān p and B np are polynoials over F p of degree n p and n p 1, respectively. Multiplying this identity by Ān p jss p j, we find that the expression B np ss p Ān p Ẽ p+1 is Oj 1 as j and hence, since it is a polynoial, vanishes. If we show that Ẽp+1j is prie to ss p j, then it follows that ss p divides Ān p j od p and hence, since A np is onic of degree n p, that A np indeed has p-integral coefficients and reduces to ss p od p. This copriality assertion is a consequence since ss p has no ultiple roots of the identity Ẽ p+1 j 1 ss pj + 8 δ ss pj j which in turn follows fro the forulas + 6 ε ss pj j 178, 1ϑ p 1 E p 1 = 1q d dq E p 1 p 1E E p 1 E E p+1 od p, δ ϑ p 1 E p 1 = j δ j 178 ε 3 j + ε j d Ẽ p 1. dj This copletes the proof. We reark that the last steps of the proof could have been siplified if we had used the forula 19, which will be proved in the next section, for then it would follow i that the coefficients in 18 are p-integral for n n p, and consequently that A np j is a polynoial with p-integral coefficients and can be reduced odulo p, and ii that the od p reductions of A np and B np are coprie because A np B np 1 B np A np 1 = ±b 1 b np 1 0 od p. It then follows by an arguent like the one above that the reduction of A np divides, and hence, since it is onic, equals, ss p. 16

17 7. Hypergeoetric aspects of A n j and ss p j Recall the definition of the classical Gauss hypergeoetric series F = F 1 : a n b n a b F a, b; c; x = x n n n = x n x < 1 c n n=0 n=0 where a n denotes the ascending factorial aa + 1 a + n 1. The inus signs in the second forula are because Gauss chose to use ascending rather than descending factorials. If c = k is a non-positive integer, then F a, b; c; x is not defined, since all ters after the kth have infinite coefficients; if instead a or b is a non-positive integer, then F a, b; c; x is a polynoial. In this section we will relate the Atkin polynoials to hypergeoetric and truncated hypergeoetric series, and use this relationship to give a second proof of Theore 3 and to prove the various forulas for A n j given in Theore 4 of the Introduction. We first define four onic polynoials Un, ε Vn δ of every degree n 0 by the forulas c n j n F 1 1, ; 1; j = U 0 n j + O1/j j n 1 j 178 F 7 1, ; 1; j = U 1 n j + O1/j j 178 n F 1 1, 7 1 ; 1; j = V 0 n j + O1/j j j 178 n 1 F 5 1, 11 1 ; 1; j = V 1 n j + O1/j as j truncated hypergeoetric functions. We will prove two results about these polynoials: that the Atkin polynoials can be expressed as linear cobinations of the, and that their reductions odulo pries give the supersingular polynoials. Together, these give a second proof of the relation between the Atkin and the supersingular polynoials. But now we will take the recursion relation 4 rather than the orthogonality property with respect to the Atkin scalar product as the defining property of A n, so that the new proof is sipler or ore coplicated than the first one we gave depending on which definition of the A n s one considers to be the ore fundaental one. Proposition 4. The functions A n j n 0 defined by the recurrence and initial conditions given in part i of Theore 4 have the following expressions in ters of the polynoials introduced above: n n + A n j = n n 1 U 0 n j, A n j = A n j = A n j = =0 n =0 n n =0 n =0 n n 7 1 n 13 1 n 5 1 n n 1 n 1 n 1 1 U 1 n j, 1 V 0 n j, 1 V 1 n j.

18 Proof. We will prove only the first of these forulas which is the sae as the forula given in part ii of Theore 4, the other cases being siilar. Denote the expression on the right of this forula by A 0 n. The equality A 0 n = A n is checked directly for n, so we ust prove the recursion A 0 n+1 = j a n A 0 n b n A 0 n 1, where a n and b n are the rational functions of n occurring in forula 4. We can rewrite the definition of A 0 n as A 0 n = n k=0 cn, ku 0 k with 1 5/1 13/1 n 1 cn, 0 = 1 3n, n n n 1 1 n n 5/1 13/1 cn, k = cn, 0 1 3k. k k k k Then, noting that ju 0 k = U 0 k+1 13k+3 1/1 k+1 5/1 k+1, we find A 0 n+1j j a n A 0 nj + b n A 0 n 1j n [ = cn + 1, k cn, k 1 + an cn, k k=0 + b n cn 1, k ] U 0 k + n k=0 1 3k+3 1/1 k + 1 5/1 k + 1 cn, k for n, where we have set cn, 1 = 0 and used that cn, n = 1, cn 1, n = 0. We can check directly that the coefficient of Uk 0 on the right equals 0 for k 1 and equals 84cn, 0/n 1 for k = 0. Substituting the value U0 0 = 1 and the values of a n, b n and cn, k and writing k = n, we then find that the right-hand side of equals 1cn, 0 n 1 n+1 =0 [ n n + 1 n + 1 1n + 1 ] n, where the extra ter = n + 1 coes fro the ultiple of U 0 0 in. This last su is just the coefficient of x n+1 in 1 + x 1 n [71 + x n+1 1n x n ] and hence vanishes for n. Reark. The forulas in Proposition 4 can be inverted, e.g., we have U 0 nj = n n n n A n j. =0 Proposition 5. Let p 5 be a prie nuber and write p as 1n 8δ 6ε + 1 with n N, δ, ε {0, 1}. Then ss p j U ε nj V δ n j od p. 18

19 Proof. Again we treat only the case of U 0 n in detail, the other cases being siilar. So assue that p 1 od 4; then we want to show that ss p j U 0 nj od p. Write p = 4l + 1. Expanding H p 1 by the trinoial theore, we have H p 1 = coefficient of X p 1 in 1 3E 4 X 4 + E 6 X 6 l = r, s 0 r+3s=l = [l/3] l 3E 4 l! r 3E4 E6 s r! s! l r s! k=0 l! 4 l 3k! k! l + k! 7 k j 178, j where k = s/ and we have used that E6/E 4 3 = j 178/j. The calculation with Un 1 in the case p 3 od 4 would be siilar but with s = k + 1, while to obtain the results for Vn δ we would instead set r = 3k + δ and then expand in powers of j/j 178. But one checks easily, either directly or by induction on k, that l! 4 k l 1 1 k 5 1 k l 3k! k! l + k! 7 l k! 1 k od p so using 11 and noting that [l/3] = = n δ and 1 δ = 3/p 3 l od p, we find writing F a, b; c; x for the hypergeoetric series truncated at degree l ss p j j δ Hp 1 j 3 3l j n F 1 [l/3] l 1, 5 1 ; 1 ; j od p. Now, taking into account that the coefficients of x k and y k in F 1 1, 5 1 ; 1; x and F 1 1, 5 1 ; 1 ; y vanish odulo p if k is in the range [l/3] < k l and that both F 1 1, 5 1 ; 1; x and F 1 1, 5 1 ; 1 ; 1 x satisfy the sae second order linear differential equation with polynoial coefficients of degree at ost, we can conclude that the polynoial on the right-hand of the last forula is a ultiple of Unj. 0 This ultiple ust then be 1 because the supersingular polynoial is onic. Hypergeoetric proof of Theore 3. The theore is trivial for p = or 3. Otherwise n p is the sae as the nuber n in the proposition. Applying Proposition 4 to U ε n or to V δ n with this value of n iediately gives the desired result, since all coefficients except the one for = 0 vanish odulo p. So we actually get two proofs. Proof of Theore 4. i We need to prove only the explicit forula 19 for the coefficients of the continued fraction expansion of Φ = E E 4 /E 6 j with respect to 1/j, since then 4 reduces to equation 18, which we have already proved. Fro the forulas 9 we iediately find that Φ = dlog /dj. On the other hand, can be expressed hypergeoetrically in ters of j by the forula = 1 j F 1 1, 5 1 ; 1; 178 j 1. It follows with the aid of Gauss s contiguous relations that 19

20 Φ = F 13 1, 5 1 ; 1; 178 j /jf 1 1, ; 1; j, and 19 now follows fro Gauss s wellknown forula for the continued fraction expansion of a quotient of contiguous hypergeoetric functions, as given in []. ii We have just shown that the Atkin polynoials are indeed the A n of Proposition 4. The closed forula given in Theore 4 is then just a rewriting of the first forula of that proposition, as already entioned. iii The closed forula of part ii is equivalent to saying that A n j can be obtained by truncating the product F 1 1, 5 1 ; 1; x F n 1 1, n ; 1 n; x 3 at x n and inverting it i.e., set x = 178/j and ultiply by j n. Notice that soe truncation is necessary, since the coefficients of the second factor in 3 becoe infinite fro degree n onwards. In fact we can truncate at x for any between n and n 1, since the coefficient of x i in 3 vanishes for n < i < n, as can be seen by letting γ 1 in the following identity, which is a consequence of Gauss s contiguous relation and two forulas of Heine [4], proved in [5]: F α, β; γ; x F n α, n + 1 β; n + γ; x + δ n x n 1 x F 1 α, 1 β; γ; x F α + n + 1, β + n; γ + n; x α β α γ β γ = polynoial of degree n n 1, n+1 n n 1 n δn = 1 γ n n. n n n 1 The differential equation for A n now follows fro this truncation arguent and the fact that the product of two hypergeoetric functions or of any two functions satisfying linear differential equations of second order satisfies a fourth order linear differential equation whose coefficients can be calculated by an explicit procedure. For the uniqueness, we observe that if the function A n j is replaced by a polynoial beginning j d for soe integer d 0, then the left-hand side of the differential equation in iii has leading ter n n d, so the differential equation can be satisfied only if d = n. As a corollary to Theore 4, we have Proposition 6 Atkin. The scalar product of A n with itself and its special values at j = 0 and j = 178 are given for n 1 by γ n A n, A n = 1 6n+1 1/1 n 5/1 n 7/1 n 13/1 n n 1! n!, A n 0 = 1 3n+1 1/1 n 5/1 n n 1!, A n 178 = 1 3n+1 1/1 n 7/1 n n 1!. Proof. The values at j = 0 and j = 178 can be checked directly fro the recursion 4, while the forula for the scalar product follows fro the recursion and part i 0

21 of the Proposition, 4. The fact that A n, A n > 0 for all n gives a second proof of the fact, already entioned in 5, that Atkin s scalar product is positive definite. Fro the explicit forula for A n, A n and Stirling s forula we see that the length of A n in the Atkin nor is asyptotically equal to 6/π 43 n as n. 8. Hypergeoetric properties of F k. Recall that the odular for F k τ for k od 3 is the unique noralized solution of the second order differential equation ϑ k+ ϑ k F k kk E 4 F k = 0, 4 the noralization being given as in 1. We introduce the following notations: ν 0 = 1 δ, ν 1 = 1 ε, ν = k + 1 ν 0 + ν 1 + ν = + 1, 3 6 X 0 = J = j 178, X 1 = 1 J, X = 1 X 0 + X 1 + X = 0, Y 0 = E4 3, Y 1 = E6, Y = 178 Y 0 + Y 1 + Y = 0, where, δ and ε are associated to k by 1 as usual. The following theore gives various explicit descriptions of the F k and their associated polynoials F k j, siilar to those given earlier for A n, G k, and H k. Theore 5. Suppose k 0, k od 3. Then we have: i Differential equation: Fk j is the unique noralized polynoial solution of jj 178 F k + {1 ν 1 j + 1 ν 0 j 178} F k + ν F k = 0. ii Closed forulas: Let σ be any perutation of {0, 1, }. Then F k j = sgnσ 178 νσ Xσ0 F, +ν σ0 ; 1 ν σ ; X σ X σ0 and F k τ = sgnσ E δ 4 E ε 6 1 l νσ0 νσ Yσ l l l Y l σ0. l=0 iii Recursion relation: The F k j satisfy + 1 ν 1 ν F k+1 ν [ 1 + ν 1 ν j 1781 ν 0 ν 0 + ν 1 + ν ] Fk ν 0 ν ν F k 1 = 0 k 1. 1

22 iv Generating function : For k Z 0 and any α denote by G k,α τ the coefficient of X k in 1 3E 4 τ X 4 + E 6 τ X 6 α. Then + ε 1 1 F k τ = 1 +δ ε 6 k G + ε k, k τ. 6 Proof. i We can easily transfor the equation 4 into the one in ters of j, by using forulas 9 and the relation F k = E4E δ 6 ε F k. The uniqueness follows fro the sae arguent used in the proof of Theore 4, iii in 7. ii The equation in i is in fact a hypergeoetric differential equation and has a polynoial solution F, + ν ; 1 ν 0 ; j/178. We have 6 polynoial solutions out of Kuer s 4 solutions to this equation. By the uniqueness, they differ only by constant factors. Making the expressions syetric, we obtain the forula in the theore. The forula for F k follows iediately. iii The first three arguents of the hypergeoetric series in the forula for F k j will change by 1 if we replace k by k ± 1. The recursion is therefore a consequence of Gauss s contiguous relations. iv Put Y α = 1 3E 4 X 4 + E 6 X 6 α. Fro the relations Y α = Y α 1 1 3E 4 X 4 + E 6 X 6, X Y α = α Y α 1 1E 4 X 3 + 1E 6 X 5, k=0 ϑ k G k,α X k = 1 πi we obtain respectively τ Y α E 1 X X Y α = α Y α 1 E 6 X 4 E 4 X 6 G k,α = G k,α 1 3 E 4 G k 4,α 1 + E 6 G k 6,α 1, 5 k G k,α = 1 α E 4 G k 4,α α E 6 G k 6,α 1, 6 ϑ k G k,α = α E 6 G k 4,α 1 α E 4 G k 6,α 1. 7 Solving 5 and 6 for E 4 G k 4,α 1 and E 6 G k 6,α 1 and substituting the expressions obtained into 7, we get ϑ k G k,α = 1 k α α k α Gk+,α α k Gk+,α. Using this repeatedly we finally obtain kk + ϑ k+ ϑ k G k,α E 4 G k,α 144 = α k [ 1 k α 3 α + 1α k α α k { k α α k α k k + α α kk α + 1 ] Gk+4,α. Gk+4,α+ } Gk+4,α+1

23 Since the right-hand side vanishes if α = k /6 we deduce that G k,k /6 τ satisfies the sae differential equation as F k τ. The constant ter of the Fourier expansion of G k,k /6 equals the coefficient of X k in 1 3X 4 + E 6 k /6 = 1 X k /3 1 + X k /6, which in turn equals k/ k 1 i k i 3 i i=0 k 6 k i = 1 k k k/ 3 k i=0 This together with 1 gives F k τ = c G k, k τ with 6 k 5 c = 1 6 / 3 k i k i k = 3 k/ 3. k k 3 + ε k 1 = 1 +δ ε 6. k + ε Rearks. 1. The syetry in the closed forula ii is the reason why we chose the noralization 1 of F k.. Part iv of the theore akes it clear why the odular fors H p 1, G p 1 and F p 1 in Theore 1 are up to scalar factors congruent to one another odulo p up to scalar factors: they are just the specializations of G p 1,α to the three values α = 1, p 1 and p 3 6, which are the sae odulo p. 3. In the notation of [10], the forula for F k can be written as F k = sgnσ E δ 4 E ε 6 H 1 ν σ, 1 ν σ0 ; Y σ, Y σ0, where H n k, l; X, Y is the polynoial r+s=n 1r n+k 1 n+l 1 s r X r Y s, which satisfies the hidden syetry that it is ±1-syetric in the three variables k, X, l, Y,, Z, where k + l + = n, X + Y + Z = 0. This polynoial is essentially the Wigner 3J-sybol of quantu echanics and arose in [10] in connection with the Cohen bracket operation on odular fors. We now interpret the F k as orthogonal polynoials. Consider the space W = C[j 1/3, j 178 1/ ] which is identified with the space of holoorphic functions on H invariant under the coutator subgroup [Γ, Γ] of Γ = P SL, Z and growing at ost like a polynoial in q 1. For each residue class r odulo 6, denote by χ r the character of the cyclic group Γ/[Γ, Γ] deterined uniquely by χ od [Γ, Γ] = e πir/3, and let W = W r be the corresponding decoposition of W, i.e., r od 6 W r is the χ r -eigenspace with respect to the action of Γ. We note that if δ {0, 1, } and ε {0, 1} are deterined by the congruence r 4δ + 6ε od 1, then W r is identified with j δ/3 j 178 ε/ C[j]. Now we define a scalar product on W by the forula f, g = f g dj, j 1/3 178 j 1/ f, g W. 8 3

24 On each subspace W r, the scalar product induced by 8 is positive or negative definite on j δ/3 j 178 ε/ R[j] depending whether ε = 0 or 1. Hence we have six failies of onic polynoials {f r } 0, f r being of degree, such that the j δ/3 j 178 ε/ f r are orthogonal with respect to this scalar product. On the other hand, for each even residue class r 4δ + 6ε od 1 and 0, put j = j F, + ν 0 ; 1 ν ; 178 j F r where ν 0 = δ and ν = δ + 6ε + 1. If r od 3 and k = 1 + 4δ + 6ε, this is nothing but F k j, renoralized to be onic. Theore 6. f r n = F r n for any r od 6 and 0. F r Proof. For fixed r, the satisfy the following recursion which is equivalent to part iii of Theore 5 if r od 3: F r +1 j = j λ + λ +1 F r j λ 1 λ F r 1 1 where λ n = n 3 6ν n 3 6ν 0 n ν 0 ν 1 n + 1 ν 0 ν 1 ν 1 = 1 ε/. By the general theory reviewed in 4, the are orthogonal with respect to the scalar product whose values g n = j n, 1 are given by the continued fraction 16. This continued fraction is equal to g 0 F 1, 1 ν 0 ; ν 0 ν 1 ; 178/j by []. Hence the j n, 1 are given by j n, 1 = g n ν 0 1 n / ν0 +ν 1 n, which is a constant ultiple of j n ν j ν 1 dj beta function. This latter integral is just 1 ε ties the inner product j δ/3 j 178 ε/ j n, j δ/3 j 178 ε/ 1 in 8. The theore follows. Reark. Up to a rescaling by 178, our polynoial is essentially a Jacobi polynoial. These are polynoials P n α,β generalizing the Chebyshev polynoials, which have paraeters 1, 1 and coe in four types even and odd, first and second kind, corresponding to the fourfold decoposition of C[x 1/, 1 x 1/ ], whereas our paraeters are 1 3, 1 and we have six failies. F r 9 The denoinators of the Atkin polynoials Fro the relation 4 we get by induction that the denoinator of A n+1 j is at ost n + 1!n!/ n n!, but this is far too large, e.g., the denoinator of A 8 j is only 5 rather than , and indeed only two of its coefficients, those of j 7 and j, have any denoinator at all. Siilarly, if we fix a prie p > 3, then 4 would lead one to expect that p would occur in the denoinator of all A n with n > p/, but instead we find, for instance, that for p = 11 all of the A n with 4