CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS

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1 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS HANNAH LARSON AND GEOFFREY SMITH Abstract. In their work, Serre and Swinnerton-Dyer study the congruence roerties of the Fourier coefficients of modular forms. We examine similar congruence roerties, but for the coefficients a modified Taylor exansion about a CM oint. These coefficients can be shown to be the roduct of a ower of a constant transcendental factor and an algebraic integer. In our work, we give conditions on and a rime number that, if satisfied, imly that m divides the algebraic art of all the Taylor coefficients of f of sufficiently high degree. We also give effective bounds on the largest n such that m does not divide the algebraic art of the n th Taylor coefficient of f at that are shar under certain additional hyotheses. 1. Introduction and statement of results Let f = a n q n be a (holomorhic) modular form of weight k on SL 2 (Z) with integral Fourier coefficients, where q = e 2πiz. It is well known that the derivative of a modular form is not generally a modular form. However, it is ossible to define a non-holomorhic derivative which reserves modularity but not holomorhicity. Furthermore, this derivative gives rise to a Taylor series exansion, (1) (1 w) k f ( z zw 1 w ) = n=0 ( n f)(z) (4πIm(z)w)n n! ( w < 1), that converges for w < 1 and thereby gives a well-defined descrition of f on the uer half of the comlex lane (see, for examle, Secion 5.1 of [9].) Remark. In this last resect, equation (1) is a more useful exansion than the standard Taylor series f (n) (z) (w z)n n!, which only converges in a disk. Congruences of Fourier coefficients have been studied extensively. Ramanujan famously observed that σ 11 (n) (n) (mod 691), and since then Deligne and others have constructed a dee theory of congruence roerties of Fourier series using Galois reresentations [4], [6], [7]. In fact, these ideas lay a central role in Wiles roof of Fermat s Last Theorem [8]. We will instead study the congruence roerties of the Taylor coefficients, relying on the theory of differential oerators mod as exlored by Swinnerton-Dyer in 1

2 2 HANNAH LARSON AND GEOFFREY SMITH [7], rather than Galois reresentations. In general, the Taylor series coefficients are transcendental. However, for a modular form with integral Fourier coefficients and a CM oint, we can exress ( n f)() as (2) ( n f)() = t f (; n)ω 2n+k, where t f (; n) is integral over Z 1 := Z[ 1] and Ω 6 6 is a transcendental factor deending only on. The choice of Ω is not canonical. However, all choices for Ω are algebraic multiles of the canonical Ω d which is given by the Chowla-Selberg formula ([9], Section 6.3): h( d)/w( d) d 1 (3) Ω 1 ( ) χ d (j) d := j Γ, 2π d d j=1 where d is the discriminant of the quadratic extension containing, Γ is the Gamma-function, χ d ( ) = ( ) d is the Kronecker character of Q( d), h( d) is the class number, and w( d) is the number of units in the ring of integers O d. Remark. In general, choosing Ω d for Ω is not ideal for our uroses because f()/(ω d )k is not in general an algebraic integer. However, since the ring of almost holomorhic modular forms, within which the image of the ring of modular forms under n is a subsace, is a finitely generated ring, there is some algebraic number a such that we can set Ω = Ω d /a. This rocess can roduce infinitely many different Ω, and our results are true for all of them; however, our results are most interesting for those Ω such that the algebraic integer t g (; 0) has zero -adic valuation for some almost holomorhic modular form g. We give an examle of this at the start of Section 2. In view of (1) and (2), congruences of the t f (; n) translate into meaningful statements about the Taylor coefficients. Our first result shows that such Taylor coefficients become increasingly divisible by owers of for half of the rimes. Theorem 1.1. Suose that f is a holomorhic modular form of weight k with integer Fourier coefficients, and suose is a CM oint in Q( ( ) d). If 5 is a rime such that {0, 1}, then d t f (; n) 0 (mod m ) for all integers m > 1 and n (m 1) 2. It turns out that when m k 2, we have the following better bound. Theorem 1.2. Assume the hyotheses in Theorem 1.1. If m k 2 and 2k 2, then t f (; n) 0 (mod m )

3 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 3 for all n m 2 2. We conjecture that some additional hyotheses of Theorem 1.2 are unnecessary. More secifically, we conjecture the following: Conjecture. Suose that f is a holomorhic modular form of weight k with integer Fourier coefficients, and suose is a CM oint in Q( ( ) d). If 5 is a rime satisfying {0, 1} and k, then d t f (; n) 0 (mod m ) for all integers m > 1 and n m 2 2. Remark. In [2], another work about the -adic behavior of the Taylor coefficients of modular forms, Datskovsky and Guerzhoy give interesting relations between the Taylor series coefficients of a modular form about a CM oint ζ Q( d) and the coefficients about ζ/ and ζ/ 2. Their relations, however, require that ζ be a ( so-called suitable oint with resect to, which in turn requires that ( ) whereas our results require that 1. d d ) = 1, Examle. Consider the Eisenstein series of weight 4, E 4. Using equation (3), we find that Ω 4 = and E 4 (i)/(ω 4) 4 = 12. The Taylor series exansion is then given by (1 w) k E 4 ( i + iw 1 w ) = 12(Ω 4) (Ω 4) 8 (4πw) ! + t E4 (i; 50)(Ω 104 (4πw)50 4) ! Comutation shows t E4 (i, 50) = is a multile of 7 2 as exected by our conjecture. Also, we have that t E4 (i; 170) = is a multile of 7 6, giving an examle of the conjecture with m = 3 and = 7. It is also divisible by 11 2, and so is an examle of the conjecture ( ) with m = 1 and = 11. Now consider = 13, a rime which does not satisfy {0, 1}. We observe that t E4 (i; 170) is not divisible by 13 even though 170 > This aer is organized as follows. In Section 2, we introduce the machinery that will be needed to rove our results. In Section 3, we rove a number of lemmas about differential oerators mod and mod 2. In Section 4, we introduce and rove several roerties of a new valuation v that encodes certain useful divisibility roerties d

4 4 HANNAH LARSON AND GEOFFREY SMITH of a modular form. The key to roving Theorems 1.1 and 1.2 deends on the results in Sections 2, 3, and 4 in an indirect way. We accumulate owers of by keeing careful track of the owers of E 1 the Eisenstein series of weight 1 that factor into n f. Lemma 5.1 is the main device that allows us to translate these factors of E 1 to factors of dividing the t f (; n). Section 5 includes this lemma and its roof and concludes with the roof of Theorems 1.1 and 1.2. Acknowledgments: The authors would like to thank Professor Ken Ono for suggesting the toic and for advice and guidance throughout the rocess, and an anonymous referee for useful comments on a draft of this aer. We also would like to thank Jesse Silliman, Isabel Vogt, the other articiants at the 2013 Emory REU, and Alexander Smith for useful conversations. Both authors are also grateful to NSF for its suort. 2. Preliminaries The Eisenstein series E k of weight k are defined by (4) E k := 1 2k σ k 1 (n)q n, B k where B k is the k th Bernoulli number and σ k 1 is the (k 1) th divisor function. For even k 4, the E k are modular forms of weight k. Following Ramanujan, we write P = E 2, Q = E 4, and R = E 6. Note that P is not a modular form, but (5) P 3 := P πim(z) transforms like a modular form of weight 2; that is, it satisfies P ( 1/z) = z 2 P (z) and P (z + 1) = P (z). It is well known (see, for instance, [9] Proosition 4) that f is exressible as a olynomial in Q and R with coefficients in Q. Since f has weight k, every term a b,c Q b R c of this olynomial will satisfy k = 4b + 6c, which we call the weight of the monomial. We then can consider modular forms to be olynomials in Q, R all of whose monomials have the same weight. By declaring the weight of P and P to be 2, quasimodular forms are defined as those holomorhic functions on the uer half lane exressible as olynomials in P, Q, R in which every monomial has the same weight. In addition, the almost holomorhic modular forms are defined as those functions exressible as olynomials in P, Q, R in which every monomial has the same weight. Now that we have fixed our notation, we give an examle of when the canonical transcendental factor Ω d is not an ideal choice of Ω and find a suitable algebraic multile of Ω d. n=1

5 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 5 Examle. Let = Consider when = 7. Since P () = 2 (Ω 3 7 )2 7 is not an algebraic integer, the canonical transcendental factor Ω 7 is not an ideal choice of Ω. However, choosing Ω = Ω 7 ensures that the t 7 1/4 f (; n) = ( n f)()/ω 2n+k are algebraic integers for all n because P () = 3, Q() = 105 and R() = 1323 are algebraic in Ω 2 Ω 4 Ω 6 fact, rational integers. For examle, the Taylor series of the discriminant at is given by ( ) w (1 w) k = 343Ω Ω 14 (2π 7w) 343Ω 16 (2π 7w) 2 1 w 2! Comutation shows that Ω 18 (2π 7w) 3 3! t f (; 50)Ω 112 (2π 7w) ! t f (; 50) = As redicted by Theorem 1.1, t f (; 50) 0 (mod 7 2 ) Modular forms mod m. For the rest of the aer, we will take to be a fixed rime number satisfying 5, and fix f to be a modular form of weight k with integral Fourier coefficients. Given a quasimodular form g with integer Fourier coefficients, we let g (Z/Z)[[q]] be the image of its Fourier series under reduction mod. By the famous results of Von-Staudt Clausen and Kummer, we have the following congruences (see [4], Chater 10, Theorem 7.1). Lemma 2.1. We have that E 1 1 (mod ) and E +1 P (mod ). Let G Z () [P, Q, R] be the exression for g as a olynomial in P, Q, R, where Z () is the ring of integers localized at. We denote by G the image of G in (Z/ m Z)[P, Q, R], well-defined since there is a canonical ma Z () Z/ m Z for all m. Remark. While it is true that G = H imlies that g = h, it is not the case that g = h imlies that G = H. For examle, Q and QR have ower series that are congruent mod 7, but Q QR as olynomials. Because of this imortant distinction, throughout this aer, we will be careful to kee track of which ring we are working in.

6 6 HANNAH LARSON AND GEOFFREY SMITH Let A (Q, R) = E 1 and B (Q, R) = E +1 in the olynomial ring Z () [Q, R]. Because we have fixed, we will dro the subscrits and write A = A and B = B. For f (Z/ m Z)[[q]], we define the filtration w(f) to be the least integer k such that there exists a modular form g of weight k with f = g. By a result of Swinnerton-Dyer (see [7], Theorem 2), we have the following lemma. Lemma 2.2. We have that w(f) < k if and only if A m 1 divides F. Examle. In the revious remark, we saw that QR has ower series congruent to Q mod 7, so it has filtration less than its weight. This is imlied by the above lemma as A = R divides QR Differential Oerators mod m. The derivative D is defined by (6) Df := 1 d 2πi dz f = q d dq f = na n q n, 1 where the factor of is used to maintain integrality of Fourier coefficients. 2πi Remark. A key roerty of D that we will use is the following: as ower series, D m f D m 1 f (mod m ) for all ositive integers m. This follows immediately from Euler s theorem, for if n then the n th Fourier coefficient is multilied by n φ(m) = n m 1 ( 1) 1 (mod m ) between D m 1 f and D m f, and if n then the n th Fourier coefficient of D m 1 f is a multile of m 1 and hence vanishes mod m. By a result of Ramanujan, D is a derivation on the ring of quasimodular forms, Z () [P, Q, R], that satisfies (7) DP = P 2 Q, DQ = P Q R, DR = P R Q That is, Z () [P, Q, R] is closed under differentiation by D. The non-holomorhic derivative is defined by k (8) k := Df 4πIm(z), and sends almost holomorhic modular forms of weight k to almost holomorhic modular forms of weight k + 2. The following lemma gives information about the relationshi between these two differential oerators. Lemma 2.3. If F (P, Q, R) = D n f is a olynomial for D n f in P, Q, R, then n f = F (P, Q, R). Proof. We induct on n. When n = 0 there is nothing to rove. It is easy to show that the differential oerator is a derivation that sends Q to P Q R, R to P R Q 2, and 3 2 P to P 2 Q. Let φ be the ma that sends P to P. By (7), we have φ D = φ. 12

7 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 7 Now suose n f = φd n f. This imlies that n+1 f = n f = φd n f = φd n+1 f. Hence, the lemma is valid for n + 1, and inducting is true for all n. Another imortant relationshi between D and is given by the following equation (see [9], Section 5.1, Equation 56): For all nonnegative integers n, we have (9) n f = n r=0 ( ) r ( ) 1 n (k + n 1)! 4πIm(z) r (k + n r 1)! Dn r f. We define a differential oerator θ in the ring of modular forms by: θf := BQ AR 3 f BR AQ2 + f Q 2 R, where is the (formal) artial derivative in the olynomial ring of quasimodular forms. It sends modular forms of weight k to modular forms of weight k Lemma 2.1 and (7) together show that θf has ower series congruent to Df mod. The following result of Serre (see [6], Section 2.2, Lemme 1) describes the filtration of modular forms mod under the action of θ. Lemma 2.4. For f in the ring of modular forms mod, we have the following results: (1) If w(f) 0 (mod ), then w(θ(f)) = w(f) (2) If w(f) 0 (mod ), then w(θ(f)) w(f) Rankin-Cohen brackets. Although the derivative of a modular form is not generally modular, we have seen that the obstruction to modularity can be corrected by the non-holomorhic derivative oerator. More generally, in Section 7 of [1] Cohen defines the Rankin-Cohen brackets [, ] n as bilinear forms on the sace of modular forms that, given modular forms of weight k and k, return modular forms of weight k + k + 2n. More recisely, he roves the following theorem as Corollary 7.2: Theorem 2.5 (Rankin-Cohen). If f and g are modular forms of weight k and k resectively, then the n th Rankin-Cohen bracket [f, g] n, defined by [f, g] n = ( )( ) k + n 1 k ( 1) r + n 1 (D r f)(d s g), s r r,s 0 r+s=n is a modular form of weight k + k + 2n.

8 8 HANNAH LARSON AND GEOFFREY SMITH 3. Differential oerators mod and mod Differential oerators mod. We now develo several results about the action of differential oerators on modular forms and quasimodular forms mod. In doing so, we will connect our two notions of derivative, first as a formal derivation on our olynomial rings in P, Q, R, and second as a oeration on formal ower series mod. The imortant result in this section are Lemma 3.3, which shows that the oerator D reserves modularity mod, and Lemma 3.6, which gives useful divisibility roerties of modular forms under certain reeated alications of D. Excet where otherwise noted, all of the following lemmas aly in the ring (Z/Z)[P, Q, R]. Proosition 3.1. Given and k, let n be the unique integer such that 0 n k + 1 <. Then D n k+1 f is congruent mod to a modular form of weight 2n k + 2. Proof. Evaluating the Rankin-Cohen bracket, we find [f, B] n k+1 = r,s 0 r+s=n k+1 ( n 0 ( 1) r ( n s )( (n + 1) k + 1 )( (n + 1) k + 1 n k + 1 r ) (D r f)(d s B) ) (D n k+1 f)b (mod ). The left-hand side is a modular form and B is a modular form. Since the second binomial coefficient is a unit mod (the ustairs term is between and 2, so it is not divisible by ), we must have that D n k+1 f is congruent mod to a modular form. Corollary 3.2. We have D 2 A 0 (mod ). Proof. The second derivative D 2 A is a modular form mod by Proosition 3.1 and has ower series congruent to zero mod by Lemma 2.1. Lemma 3.3. The th derivative D f is a modular form mod. Proof. We consider the cases when k 1 (mod ) and when k 1 (mod ) searately. First suose k 1 (mod ). Pick n such that 0 n k + 1 <. By Proosition 3.1, we have D n k+1 f is congruent mod to a modular form of weight 2n k + 2. Now ick m such that 0 m (2n k + 2) + 1 <. Proosition 3.1 imlies D m (2n k+2)+1 (D n k+1 f) = D f is a modular form mod.

9 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 9 Now suose k 1 (mod ). We evaluate the Rankin-Cohen bracket [f, A] = ( )( ) k ( 1) r (D r f)(d s A) s r r,s 0 r+s= ( )( ) k (D f)a (mod ). 0 The left-hand side is a modular form and A is a modular form, so D f must be a modular form mod. We now use the modularity of D f mod and D n k+1 f mod to rove results about divisibility by A mod by finding modular forms of different weights with congruent ower series (see Lemma 2.2). In the following roositions and subsequent lemma, these modular forms of different weights will come from alying the θ oerator. Proosition 3.4. We have D r f A r θ r f (mod ). Proof. We use induction on r. Let r = 1. We have D f Df as ower series by Fermat s Little Theorem. Furthermore, Df θf as ower series by the definition of θ and the fact that A has ower series congruent to 1 mod. Since D f has weight 2 + k and θf has wight k + + 1, we have that A divides D f mod, and in fact, D f Aθf (mod ). Now suose D (r 1) f A r 1 θ r 1 f (mod ). Then D r f D (A r 1 θ r 1 f) A r 1 D (θ r 1 f) A r θ r f (mod ). Proosition 3.5. Given and k, let n be the unique integer such that 0 n k + 1 <. Then θ n k+1 f is an element of the ideal (A n k+1, ) in Z () [P, Q, R]. Proof. We have that the weight of θ n k+1 f = k + ( + 1)(n k + 1). Because D n k+1 is a modular form mod, we have D n k+1 f θ n k+1 f mod. That is, there exists a modular form of weight 2n k + 2 which is congruent to θ n k+1 mod. Since 2n k + 2 < k + ( + 1)(n k + 1) we have that A divides θ k+1 with multilicity given by k + ( + 1)(n k + 1) (2n k + 2) = n k Lemma 3.6. For all i k 1, we have D 2 i f is in the ideal (A 2 k+1 i, ) within Z () [P, Q, R]. Proof. Working mod, Proosition 3.4 gives D 2 i f A i θ i f A i θ k i 1 θ k+1 f (mod ),

10 10 HANNAH LARSON AND GEOFFREY SMITH and Proosition 3.5 imlies A 2 k+1 i divides D 2 i mod. Our above lemmas are only concerned with modular forms. The following roosition and corollary instead rove some useful results about differentiation of our simlest quasimodular form, P. Proosition 3.7. The form D P is modular mod. Proof. Recall (see (7)) that Q = P 2 12DP. So, working mod, we have D Q = 2P D P 12D +1 P is a modular form. Let XP i be the leading term in D P mod as a olynomial in P. Then the leading term of D Q as a olynomial in P is 2XP i+1 (2 + 2 i)xp i+1 ixp i+1 (mod ). Because D Q is a modular form mod (by Lemma 3.3, we have i 0 (mod ). Since we cannot have i the weight of X would then be at most 2 we must have i = 0, so D P is a modular form mod. Corollary 3.8. We have D 2 P is in the ideal (A 2, ) within Z () [P, Q, R]. Proof. As a ower series, D 2 P D P (mod ), and by Proosition 3.7, both are modular forms mod. Since D 2 P has weight and D P has weight 2 + 2, we have A must divide D 2 P with multilicity at least Differential oerators mod 2. Having develoed the necessary machinery mod we now turn our attention to some results about modular forms mod 2, that is, in the ring (Z/ 2 Z)[P, Q, R]. The following roosition and the consequent Lemma 3.10 are analogous to Proosition 3.1 and Lemma 3.3 resectively. Proosition 3.9. Given and k, let n be the unique integer such that 0 n 2 k + 1 < 2. Then D n2 k+1 f is congruent to a modular form mod 2. Proof. We evaluate the Rankin-Cohen bracket: ( )( ) n [f, BA ] n 2 k+1 = ( 1) r 2 (n + 1) 2 k + 1 (D r f)(d s (BA )) s r r+s=n 2 k+1 ( )( ) n ( 1) (n i) k+1 2 (n + 1) 2 k + 1 (D (n i) k+1 f)(d i (BA )) i (n i) k + 1 0<i<n ( ) + ( 1) n2 k+1 (n + 1) 2 k + 1 (D n2 k+1 f)(ba ) (mod 2 ). n 2 k + 1 By Proosition 3.1 and Lemma 3.3, the derivatives D (n i) k+1 and D i (BA ) are modular mod. Since every term accumulates a factor of from its first binomial coefficient excet when i = 0, every term after the first is a modular form mod 2. By Theorem 2.3, we have [f, BA ] n 2 k+1 is a modular form, so D n2 k+1 must be a modular form mod 2.

11 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 11 Lemma The form D 2 f is modular mod 2. Proof. We consider the cases when k 1 (mod 2 ) and when k 1 (mod 2 ) searately. First suose k 1 (mod 2 ). Let n be the unique integer such that 0 n 2 k + 1 < 2. By Proosition 3.9, we have D n2 k+1 f is congruent to a modular form of weight 2n 2 k + 2 mod 2. Now let m be the unique integer such that 0 m 2 (2n 2 k + 2) + 1 < 2. Then Lemma 3.9 imlies D m2 (2n 2 k+2)+1 (D n2 k+1 f) = D 2 f is a modular form mod 2. Now suose k 1 (mod 2 ). We have the following exansion of the Rankin- Cohen bracket: [f, A ] 2 = ( 1) r,s 0 r+s= 2 )( r( k s r ) (D r f)(d s A ). When s > 0, the first binomial coefficient is divisible by and D s A is also divisible by. So, working mod 2, we have ( )( ) k + [f, A ] 2 (D 2 f)a (mod 2 ). 0 2 By Theorem 2.3, the left-hand side is a modular form, and since A is modular form and does not divide ( ), we have D 2 f is a modular form mod The valuation v In this section, we define a function v which behaves like a valuation with resect to the ideal (A, ). The goal of this section is to understand the behavior of v under reeated alications of the differential oerator D 2. The imortant results are Lemma 4.6, which gives a lower bound on v(d m2 f) in terms of m, and Lemma 4.7, which gives a stronger lower bound under additional assumtions on, k and m. We define the function v : Z () [P, Q, R] Z by (10) v(f) = su{n f (A, ) n }. In other words, v(f) is the sum the -adic valuation of f and the suremum of the set of all nonnegative integers i such that f is exressible as f = A i G for some quasimodular form G. Note that v(df) v(f) 0 for all quasimodular forms f and v(fg) v(f)+v(g) for all quasimodular forms f and g. Remark. We have used quotation marks around the word valuation because in general we do not have the equality v(fg) = v(f) + v(g), so v is not technically a valuation.

12 12 HANNAH LARSON AND GEOFFREY SMITH Examle. Let = 5. Then A = E 4 = Q. Consider D 5 A = 35 P 5 Q + 175P 3 Q P 4 R + 25 P Q3 175P 2 QR Q2 R + 25 P 324 R2. It is a multile of 5, as exected from Corollary 3.2, but is neither a multile of A 5 = Q 5 nor of 25. So v(d 5 A) = 1 0 = v(a) Imortant Facts About v. Before we can rove our key lemmas, we need several facts about v under a single alication of the differential oerator D 2. Proosition 4.1 gives a lower bound (indeendent of f) for v under the differential oerator D 2 and Proosition 4.2 gives a lower bound on v(d 2 A ). These then allow us to bound v(d 2 f) in terms of v(f) in Proosition 4.3, which will be imortant in roving Lemma 4.6. Proosition 4.1. We have v(d 2 f) 2. Proof. It suffices to show that D 2 f A 2 +A N (mod 2 ) for some modular forms M and N. By Lemma 3.3, we can write D f = M +G(P ), where M is a olynomial in Q and R and G(P ) = i X ip i is a olynomial in P with coefficients X i that are olynomials in Q and R. We claim that A M + i X ia i B i is a modular form with ower series congruent to D f. Since A 1 (mod 2 ), the first term has ower series congruent to M; recalling that B P (mod ) and A 1 (mod ) as ower series, it is clear that X i A i B i X i P i (mod 2 ). Hence, in the ring of ower series, D 2 f D f A M + i X ia i B i (mod 2 ), showing that D 2 f has ower series congruent to a modular form of weight k +2+( 1). Since D 2 f has weight k and is a modular form mod by Lemma 3.10, we have that A divides D 2 f by Lemma 2.2. Proosition 4.2. We have v(d 2 A ) 3. Proof. Using the roduct rule we exand (11) D 2 A = j j = 2 2! j 1! j! (Dj 1 A) (D j A). We roceed in cases. Case 1: j r = 2 for some r. In this case, v((d j 1 A) (D j A)) = v(a 1 D 2 A) 2 by Proosition 4.1. Because there are exactly terms of this tye, v of their sum will be at least 3. Case 2: there exists some r such that j r 0 (mod ). We have 2 divides 2! and there exists some r such that j j 1! j! r > 1, so by Corollary 3.2, the terms 2! j 1! j (Dj 1! A) (D j A) are divisible by 3. Case 3: j r 0 (mod ) for all r and j r 2 for all r.

13 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 13 2! We have divides, and divides any term j 1! j! Djr A where j r is nonzero. Since there are at least two nonzero j r we have 3 2! j 1! j (Dj 1! A) (D j A). In each case, v is at least 3, so v(d 2 A ) is at least 3, as desired. Proosition 4.3. We have v(d 2 f) v(f) + 1. Proof. Write f = i v(f) i A i M i where the M i are modular forms. To rove the roosition, it suffices to show that v(d 2 (A i M i )) i + 1. We exand D 2 (A i M i ) using the roduct rule: 2 ( ) (12) D 2 (A i 2 M i ) = (D j A i )(D 2 j M i ). j j=0 When j = 0, Proosition 4.1 imlies v(d 2 M i ) 2, so v(a i D 2 M i ) i + 2. When ) ) 1 j 2 1, we have the inequalities: v 1 and v(d j A i ) v(a i ) i, j from which we conclude v( ( ) 2 j (D j A i )(D 2 j M i )) i + 1. Finally, when j = 2, we have the following equation: D 2 A i = r 1! r i! (Dr 1 A ) (D r i A ). r r i = 2 2! If some r = 2, Proosition ( 4.2 imlies v(a (i 1) D) 2 A ) i+2. Otherwise, divides 2!, and we have v 2! r 1! r i! r 1! r i! (Dr 1 A ) (D r i A ) i + 1. Hence, for all j, we have ( ( 2) ) v j (D j A i )(D 2 j M i ) i + 1, which imlies the roosition. Making some additional assumtions on v(f) and, we give stronger analogues of the revious roosition, which will be necessary to rove Lemma 4.7. ( ( 2 Proosition 4.4. If v(f) k 1 and 2k 2, then v(d 2 f) v(f) + 2. Proof. As in the roof of Proosition 4.3, it suffices to show that v(d 2 A i M i ) i+2. By the roduct rule, we have 2 ( ) (13) D 2 (A i 2 M i ) = (D j A i )(D 2 j M i ). j We considering the following cases: Case 1: j = 0. When j = 0, Proosition 4.1 imlies v(a i D 2 M i ) i + 2. Case 2: j 0 (mod ). j=0

14 14 HANNAH LARSON AND GEOFFREY SMITH In this case, 2 divides ( ) 2 j so v increases by at least 2. Case 3: j 0 (mod ) and j (k 1). We have divides ( ) 2 j, and Lemma 3.6 says A divides D 2 j M i (mod ); that is, v(d 2 j M i ) 1. Therefore, v( ( ) 2 j (D j A i )(D 2 j M i )) i + 2. Case 4: j 0 (mod ) and (k 1) < j < 2. Since m k 1, we have i < j. Hence, v(d j A i ) i + 1 because 2 divides D r A with r > 1. Because we also have a factor of from the binomial, v( ( ) 2 j (D j A i )(D 2 j M i )) i + 2. Case 5: j = 2. We have the following equation: D 2 A i = 2 r 1! r! (Dr 1 A ) (D r A ). r r i = 2 If there exists an s such that r s 0 (mod ), then 2 divides 2 r 1! r! and we are done. If there exists an s such that r s = 2, the result is immediate from Proosition 4.2. Otherwise, r s 0 (mod ) and r s 2 for all s. Therefore, since i <, there exists an s such that r s >, and hence 2 divides D rs A. In addition, divides 2, so r 1! r! v( 2 r 1! r (Dr 1! A ) (D r A )) i + 2, establishing the case. Hence, for all j we have v( ( ) 2 j (D j A i )(D 2 j M i )) i+2, roving the roosition. Proosition 4.5. If v(f) k 2 and 2k 2, then v(d 2 f) v(f) + 1. Proof. It suffices to show that v(d 2 A i M i ) i + 1. From the roduct rule we have D 2 (A i M i ) = ( ) 2 (D j A i )(D 2 j M i ). j j If j 0 (mod ), then divides the binomial coefficient and we are done. It remains to consider terms in which j 0 (mod ). If j (k 2), then Lemma 3.6 says v(d 2 j M i ) 1. If j > (k 2), then we have j > i, so v(d j A i ) i + 1. Thus, v ( ( 2 j ) (D j A i )(D 2 j M i )) i + 1 for all j, roving the roosition Key lemmas. Using the above facts, we now rove two imortant lemmas which make recise the increasing behavior of v under reeated alications of the differential oerator D 2. Lemma 4.6 is alicable in the context of Theorem 1.1. Lemma 4.7 is stronger version of Lemma 4.6 under additional assumtions which translate to the additional assumtions of Theorem 1.2.

15 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 15 Lemma 4.6. If m 1 is an integer, then v(d m2 f) m + 1. Proof. When m = 1, we have v(d 2 f) 2 by Proosition 4.1. We roceed by induction on m. Suose that for all n m, we have v(d n2 f) n+1. By equation (9) we have m 2 (14) m2 f = a r T r D m2 r f, r=0 where T = 1 and a 4πy r = ( ) m 2 (k+m 2 1)!. We claim that v(a r (k+m 2 r 1)! rd m2 r f) m + 2 for all r > 0. Let j be such that (j 1) 2 < r j 2. We will show that j+1 divides a r. Suose j = 1. If r <, then 2 divides the binomial coefficient. If r < 2, then divides the binomial once and divides the comlementary factor at least once. When r = 2, we have divides the comlementary factor at least twice. Now suose j 2. The comlementary factor is divisible by (j 1). Since (j 1) > j + 1, we have j+1 divides a r. Hence we have shown that j+1 divides a r. Now we see v(a r D m2 r f) = v(a r D j2 r (D (m j)2 f)) m j j + 1 m + 2, roving the claim. Since P = P + 12T, the maing P 0 followed by T P/12 sends m2 f to D m2 f by Lemma 2.3. We write C r for the coefficient of P r in D m2 f. By the induction hyothesis, we have v(c 0 ) m + 1 so Proosition 4.3 imlies v(d 2 C 0 ) m + 2. The claim above roves that v(c r ) m + 2 for all r > 0, so we have v(d 2 P r C r ) m + 2 for all r. Hence, we have shown that v(d (m+1)2 f) m + 2. Inducting, we have v(d m2 f) m + 1 for all m, as desired. Lemma 4.7. Suose that 2k 2. If m is an integer satisfying 1 2m k 2, then we have the following: (1) v(d m2 f) 2m 1. (2) v(d m2 f) 2m. Proof. When m = 1, the results follow from Proositions 4.5 and 4.1 resectively. We roceed by induction on m. Suose that for all n m, we have v(d n2 f) 2n 1 and v(d n2 f) 2n. We will show that v(d (m+1)2 f) 2m + 1 and v(d (m+1)2 f) 2m + 2. As in Lemma 4.6, we write m 2 (15) m2 f = a r T r D m2 r f, r=0 where T = 1 4πy and a r = ( m 2 r ) (k+m 2 1)! (k+m 2 r 1)!. We convert the above exression for m2 f into an exression for D m2 f as a olynomial in P using the ma P 0

16 16 HANNAH LARSON AND GEOFFREY SMITH and T P and write C 12 r for the coefficient of P r in D m2 f. Note that v(c r ) v(a r D m2 r f). We first give lower bounds for the v(c r ). When r = 0, the induction hyothesis imlies v(c 0 ) = v(a r D m2 f) = 2m. For 1 r k 1, it is easy to see that 2 divides a r and the induction hyothesis imlies v(c r ) = v(a r D r D m2 f) 2m + 1. For r > k 1, we claim that v(c r ) 2m + 2. We consider the following cases: Case 1: k 1 < r. We have that 3 a r, and so v(c r ) v(a r D r D m2 f) 3 + 2m 1 = 2m + 2. Case 2: < r 2. We have 4 a r, and so v(c r ) v(a r D 2 r D (m 1)2 f) 4 + 2(m 1) = 2m + 2. Case 3: r > 2. Let j be an integer such that (j 1) 2 < r j 2. It is easy to see that (+1)(j 1) divides a r. Then v(c r ) v(a r D j2 r D (m j)2 f) (+1)(j 1)+2m 2j 2m+2. Hence we have shown v(c r ) 2m + 2 for all r > k 1. We next establish v(d (m+1)2 f) 2m+1. We have D (m+1)2 f = r D2 (P r C r ). Since v(c 0 ) = 2m, Proosition 4.5 imlies D 2 (C 0 ) 2m + 1. For all r > 0, we have v(c r ) 2m + 1, so v(d (m+1)2 f) 2m + 1. Finally, we show that v(d (m+1)2 f) 2m+2. We have D (m+1)2 f = r D2 (P r C r ). By Proosition 4.4, we have v(d 2 C 0 ) 2m + 2. For 1 r k 1, we have v(c r ) 2m + 1 and we claim that v(d 2 (P r C r )) 2m + 2. We have following exansion for D 2 (P r C r ): (16) D 2 (P r C r ) = j j r+1 = 2 2 j 1! j r+1! (Dj 1 P ) (D jr P )(D j r+1 C r ). We have divides 2 j 1! j r+1, and have hence established our claim, excet for the! terms in which there exists an s such that j s = 2. Consider a term in which j s = 2. If s = r + 1, then we have v(d 2 C r ) v(c r ) + 1 2m + 2 by Proosition 4.3. Otherwise s r, and since Corollary 3.8 imlies v(d 2 P ) 1, we have ( ) 2 v j 1! j r+1! (Dj 1 P ) (D jr P )(D j r+1 C r ) 2m + 2. Finally, for r > k 1, we have v(c r ) 2m + 2, so D 2 (P r C r ) 2m + 2. We have shown that v(d (m+1)2 f) 2m + 1 and v(d (m+1)2 f) 2m + 2. Inducting, we have v(d m2 f) 2m 1 and v(d m2 f) 2m for all m, as desired.

17 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS Main results At this oint, we have a number of results about various derivatives of a modular form lying in some ideal (A, ) n. To translate these results to the form of Theorems 1.1 and 1.2, for which we desire the divisibility of ( n f)() by owers of, we rove the following lemma. ( ) d Lemma 5.1. Let be a CM oint with d. If 5 is a rime such that ( ) d {0, 1}, where is the Legendre symbol, we have t E 1 (; 0) 0 (mod ), where E 1 is the normalized Eisenstein series defined in Section 2.1. Proof. From equation 2 of [3] we have that A = E 1 can be exressed uniquely as A() = () n Q() δ R() ɛ f(j()), where 1 = 12n + 4δ + 6ɛ and f is a olynomial. The suersingular olynomial is defined as ss (j) := (j j(e)). E/F E suersingular By Theorem 1 of [3], it satisfies the equivalence ss (j()) ±j() δ (j() 1728) ɛ f(j()) (mod ). From Theorem 7.25 of [5] we know that j() is a root of ss (j) whenever ( d ) = 0, 1, where d is the discriminant of Q()/Q. This imlies j() is a root of f, so long as j() 0, 1728; that is, i and ρ = e 2πi/3. Since R(i) = Q(ρ) = 0, we have A() 0 (mod ). Now, using this lemma and the roerties of v, we can rove our main results Proofs of main results. Proof of Theorem 1.1. Fix, n and m satisfying the hyotheses. Lemma 4.6 imlies D m2 f (A, ) m+1 within the ring of quasimodular forms, so we can write D n f as a sum of the form D n f = m i A i H i, 0 i m for some quasimodular forms H i. Since A is fixed under the ma φ : Z () [P, Q, R] Z () [P, Q, R] that sends P to P, Lemma 2.3 imlies n f = m i A i φ(h i ). In articular, we have that t f (; n) = n f() = m i A()i φ(h i )() = m i A()i φ(h i )(). Ω 2n+k Ω 2n+k Ω i( 1) Ω 2n+k i( 1) 0 i m 0 i m

18 18 HANNAH LARSON AND GEOFFREY SMITH By Lemma 5.1, we have A()i is an algebraic integer multile of i, and by (2) we Ω i( 1) have is an algebraic integer, so t f (; n) 0 (mod m ). φ(h i )() Ω 2n+k i( 1) Theorem 1.2 follows from a comarable argument, relying on Lemma 4.7 rather than Lemma 4.6. Proof of Theorem 1.2. Fix, n and m satisfying the hyotheses. By Lemma 4.7, we can write D n f = m i (A ) i H i, 0 i m for some quasimodular forms H i. Since A is fixed under the ma sending P P, Lemma 2.3 imlies n f can be written as m i (A ) i φ(h i ), where φ : Z () [P, Q, R] Z () [P, Q, R] is the ma that sends P P. In articular, we have that t f (; n) = n f() Ω 2n+k = 0 i m m i A()i φ(h i )() Ω 2n+k = 0 i m m i A()i Ω i( 1) φ(h i )() Ω 2n+k i( 1) By Lemma 5.1, we have A()i is an algebraic integer multile of i, and by (2) we Ω i( 1) have is an algebraic integer, so t f (; n) 0 (mod m ). φ(h i )() Ω 2n+k i( 1). References [1] H. Cohen. Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann., 217(3): , [2] B. Datskovsky and P. Guerzhoy. -adic interolation of Taylor coefficients of modular forms. Math. Ann., 340(2): , [3] M. Kaneko and D. Zagier. Suersingular j-invariants, hyergeometric series, and Atkin s orthogonal olynomials. Proceedings of the Conference on Comutational Asects of Number Theory, ages , [4] S. Lang. Introduction to Modular Forms. Graduate Texts in Mathematics. Sringer-Verlang, [5] K. Ono. The web of modularity: Arithmetic of the Coefficients of Modular Forms and q-series. American Mathematical Society, [6] J.-P. Serre. Formes modulaires et fonctions zêta -adiques. Proc. Internat. Summer School, Univ. Antwer, ages , [7] H. P. F. Swinnerton-Dyer. On l-adic reresentations and congruences for coefficients of modular forms. In W. Kuijk and J.-P. Serre, editors, Modular Functions of One Variable III. Sringer Berlin Heidelberg, [8] A. Wiles. Modular ellitic curves and Fermat s Last Theorem. Annals of Mathematics, 141(3): , 1995.

19 CONGRUENCE PROPERTIES OF TAYLOR COEFFICIENTS OF MODULAR FORMS 19 [9] D. Zagier. Ellitic modular forms and their alications. In The of modular forms, Universitext, ages Sringer, Donald St., Eugene, OR, address: hannahlarson@college.harvard.edu 1 Huckleberry Rd., Hokinton, MA, address: geoffrey.smith@yale.edu