ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS MATTHEW BOYLAN Abstract. In recent work, Bringmann and Ono [4] show that Ramanujan s fq) mock theta function is the holomorhic rojection of a harmonic weak Maass form of weight 1/2. In this aer, we extend work of Ono in [13]. In articular, we study holomorhic rojections of certain integer weight harmonic weak Maass forms on SL 2 Z) using Hecke oerators and the differential theta-oerator. 1. Introduction and statement of results. If k is an integer, we denote by M k resectively, S k ) the sace of holomorhic modular forms resectively, cus forms) of weight k on SL 2 Z). Let z h, the comlex uer halflane, and let q := e 2πiz. For integers n 1 and j 0, define σ j n) := d n dj, and let B j denote the jth Bernoulli number. Then the Delta-function and the normalized Eisenstein series for SL 2 Z) of even weight k 2 are given by 1.1) 1.2) z) = τn)q n = q E k z) = 1 2k 1 q m ) 24 S 12, m=1 σ k 1 n)q n. If k 4, then E k z) M k. We also set E 0 z) := 1. If the dimension of S k is one, then k {12, 16, 18, 20, 22, 26}. We note that the corresonding weight k cus forms, 1.3) f k z) := a fk n)q n = z)e k 12 z), have integer coefficients. In [13], Ono defined a harmonic weak Maass form related to the Delta-function and studied its holomorhic rojection. In this aer, we extend Ono s work by defining harmonic weak Maass forms related to the forms f k z) and by studying their holomorhic rojections. We denote saces of harmonic weak Maass forms of weight j on SL 2 Z) by H j. In Section 2, we will use Poincaré series to define, as in [5] and [13], harmonic weak Maass forms R fk z) H 2 k connected to the forms f k z) S k. Such forms may be written R fk z) = M fk z) + N fk z), where M fk z) is holomorhic on h and N fk z) is not holomorhic on h. We will find that the non-holomorhic contribution is a normalization of the eriod integral for f k z), Date: July 2, 2008. 2000 Mathematics Subject Classification. 11F03, 11F11. The author thanks the National Science Foundation for its suort through grant DMS-0600400. 1

2 MATTHEW BOYLAN given by 1.4) N fk z) = ik 1)2π) k 1 c k i where 1.5) c k := z k 2)! 4π) k 1 f k z) 2 R. f k τ) iτ + z)) 2 kdτ, Here, denotes the Petersson norm. We will also find that the function M fk z), the holomorhic rojection of R fk z), has coefficients c + f k n), and is given by 1.6) M fk z) = c + f k n)q n = k 1)!q 1 + k 1)! 2k + n= 1 See Theorem 2.1 for recise formulas for the coefficients c + f k n) as series with summands in terms of Kloosterman sums and values of modified Bessel functions of the first kind. The theory of harmonic weak Maass forms and mock theta functions is of great current interest. See, for examle, the works [3], [4], [5], [6], [7], [8], [9], [13]. Let λ {1/2, 3/2}. In [3] and [6], Bringmann, Ono, and Folsom define a mock theta function to be a function which arises as the holomorhic rojection of a harmonic weak Maass form of weight 2 λ whose non-holomorhic art is a eriod integral of a linear combination of weight λ theta-series. This definition encomasses the mock theta functions of Ramanujan, of which fq) := q n2 1 + q) 2 1 + q 2 ) 2 1 + q n ) 2 is an imortant examle. See [4] for its recise relation to Maass forms. In this sense, one may view the functions M fk z) as analogues of mock theta functions which we call mock modular forms. An imortant asect of the study of mock modular forms concerns questions on the rationality algebraicity) of their q-exansion coefficients. Mock modular forms arising from theta functions, such as fq), are known to have rational algebraic) q-exansion coefficients. In [8] and [9], the authors initiated a study of the algebraicity of coefficients of mock modular forms Mz) = c + n)q n which do not arise from theta functions. In [8], Bruinier and Ono relate the algebraicity of the coefficients of certain weight 1/2 mock modular forms to the vanishing of derivatives of quadratic twists of central critical values of weight two modular L-functions. In [9], Bruinier, Ono, and Rhoades give conditions which guarantee the algebraicity of coefficients of integer weight mock modular forms. In the roof of their result, they show that the coefficients c + n) lie in the field Qc + 1)). Part 1 of Corollary 1.3 below recovers this result for the coefficients c + f k n). However, the conditions in [9] guaranteeing the algebraicity of mock modular form coefficients do not aly in the resent setting. As such, it is not known whether any c + f k 1) is algebraic. In [13], Ono conjectures, for all ositive integers n, that the mock Delta coefficients, c + n), are irrational. This conjecture has the following consequence. If c + 1) or any coefficient c + n)) is irrational, then Lehmer s Conjecture on the non-vanishing of the tau function is true. Similarly, as a consequence of the work in this aer, we observe that if c + f k 1) or any c + f k n)) is irrational, then for all ositive integers n, we have a fk n) 0, which is the analogue of Lehmer s conjecture for the forms f k. This is art 2 of Corollary 1.3 below.

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 3 In this aer, we use the Hecke oerators and theta-oerator to create certain weakly holomorhic modular forms which are modifications of the mock modular forms M fk z). We then give exlicit formulas which allow us to rove results on rationality and congruence roerties of the coefficients of these weakly holomorhic forms. Our formulas also enable us to rove results on the values of these functions in the uer half-lane. To state our results, we require facts on weakly holomorhic modular forms on SL 2 Z), forms which are holomorhic on h, but which may have a ole at the cus infinity. We denote the sace of such forms of integer weight k by M k! and note that M k M k!. For more information on holomorhic and weakly holomorhic modular forms, see [12], for examle. An imortant examle of a weakly holomorhic form is the ellitic modular invariant, jz) := E 4z) 3 z) = n= 1 cn)q n = q 1 + 744 + 196884q + M! 0. We also require certain oerators on q-exansions. If is rime, then the th Hecke oerator in level one and integer weight k acts by 1.7) an)q n T k ) := )) n an) + k 1 a q n, where a n ) = 0 if n. For integers m 1, one defines Hecke oerators T k m) in terms of the Hecke oerators of rime index in 1.7). Furthermore, the Hecke oerators reserve the saces M k and M k!. Next, we define the differential theta-oerator 1.8) θ := 1 2πi d dz = q d dq and observe that θ an)q n) = nan)q n. Alying a normalization of the Hecke oerators to jz), we obtain an imortant sequence of modular forms. We set j 0 z) := 1, and for all m 1, we set j 1 z) := jz) 744 = q 1 + 196884q +, 1.9) j m z) := mj 1 z) T 0 m)) = q m + c m n)q n M 0!. The forms j m z) have integer coefficients and satisfy interesting roerties. For examle, let τ h and define H τ z) := m=0 j mτ)q m. We will need the fact, roved in [2], that H τ z) is a weight two meromorhic modular form on SL 2 Z) and that 1.10) H τ z) = m=0 j m τ)q m θjz) jτ)) = jz) jτ) = E 14z) z) 1 jz) jτ).

4 MATTHEW BOYLAN To state our main result, we require further definitions. Let be rime. The first function we study in connection with M fk z) is L fk,z), defined by 1.11) k 1 L fk,z) : = k 1)! M f k z) T 2 k ) 1 k a fk )M fk z)) 1 )) n = k 1 c + f k 1)! k n) a fk )c + f k n) + c + f k q n. n= We then define a n) by Lfk, n= a L fk, n)qn := L fk,z). To study the functions L fk,z), we introduce auxiliary functions. For k {12, 16, 18, 20, 22, 26}, let F k be any form in M k 2 and define 1.12) F k z) := a Fk n)q n. For all ositive integers t, define 1.13) A Fk,tz) := t m=0 n=0 a Fk m)j t m z) + a Fk 0) 2k σ k 1 t) with j n z) as in 1.9). The second function we study in connection with M fk z) is 1.14) θ k 1 M fk z)) = k 1)!q 1 + n k 1 c + f k n)q n. Its study also requires the introduction of auxiliary functions. Let k be the least ositive residue of k modulo 12. We define G k z) by E 12 k z) if k 10 mod 12) z) k k 12 +1 1.15) G k z) := if k 10 mod 12). 12 +2 E 14 z) z) k 10 We note that G k z) M! k Qq)), and define a G k n) C and n k Z by 1.16) n= n k a Gk n)q n := G k z) = q nk + For fixed k and integers t n k with n k as in 1.16)), we define n k 1.17) B Gk,tz) := a Gk n)j n t z) with j n z) as in 1.9). Our main result gives exact formulas for L fk,z) and θ k 1 M fk z)). Theorem 1.1. Let k {12, 16, 18, 20, 22, 26}, and let be rime. n=t

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 5 1) In the notation above 1.3), 1.11), 1.12), 1.13)), we have L fk,z) = A F k,z) a fk )A Fk,1z) F k z) M! 2 k. 2) In the notation above 1.6), 1.14), 1.16), 1.17)), we have nk θ k 1 m=1 M fk z)) = mk 1 c + f k m)b Gk,mz) k 1)!B Gk, 1z) M! G k z) k. Before roceeding, we illustrate Theorem 1.1 by alying it to f 12 z) = z). Let be rime. With F 12 z) = E 10 z) in art 1 of the theorem, we find that L, z) = j z) 264 m=1 σ 9m)j m z) 65520σ 691 11) τ) ) j 1 z) 247944 691. E 10 z) This is the main result in [13]. Using art 2 of the theorem, we obtain 1.18) θ 11 M z)) = z)c + 1) 11!j 2z) + 24j 1 z) + 324)). On examining the right-hand sides in Theorem 1.1, we deduce roerties of the forms L fk,z) and θ k 1 M fk z)). First, we find that the coefficients of L fk,z) are integers. Corollary 1.2. Let k {12, 16, 18, 20, 22, 26}, and let be rime. Then for all integers n, we have )) a n) = 1 n Lfk k 1 c +, f k 1)! k n) a fk )c + f k n) + c + f k Z. A consequence of Corollary 1.2 is the following. Corollary 1.3. Let k {12, 16, 18, 20, 22, 26}. 1) For all integers n, we have c + f k n) Qc + f k 1)). 2) If c + f k 1) or any c + f k n)) is irrational, then for all integers n, we have a fk n) 0. As an examle of art 1, for all rimes, Corollary 1.2 gives 1.19) c + f k ) = k 1)! a L fk, 1) k 1 + a f k ) k 1 c + f k 1). The general case follows by induction. As remarked above, the conclusion of art 2 of the corollary is the analogue of Lehmer s conjecture for the forms f k. It is roved in Section 3. Theorem 1.1 and Corollary 1.2 ermit us to aly well-known congruence roerties for coefficients of the Eisenstein series E k z) and the cus forms f k z) to obtain congruences for L fk,z). Corollary 1.4. Let be rime, and let τ) = a f12 ) as in 1.1). 1) If k {12, 16, 18, 20, 22, 26}, then we have j z) mod 24) if = 2 or 23 mod 24) L fk,z) j z) + 12j 1 z) mod 24) if = 3 or 11 mod 24) j z) + 1)j 1 z) mod 24) if 5.

6 MATTHEW BOYLAN 2) If k {18, 22, 26}, then we have j z) 2j 1 z) mod 5) if 1 mod 5) L fk,z) j z) + + 1)j 1 z) + 2) mod 5) if 2, 3 mod 5) j z) mod 5) if 4 mod 5). 3) If k {20, 26}, then we have j z) 2j 1 z) mod 7) if 1 mod 7) j z) + 2 + 1)j 1 z) + 1) mod 7) if 2, 4 mod 7) L fk,z) j z) + 3 + 1) mod 7) if 3, 5 mod 7) j z) mod 7) if 6 mod 7). 4) If k {12, 22}, then we have 5) We also have L fk,z) j z) τ)j 1 z) + 2 + 1 τ)) mod 11). L f26,z) j z) τ)j 1 z) 2 + 1 τ)) mod 13) L f18,z) j z) a f18 )j 1 z) + 7 + 1 a f18 )) mod 17) L f20,z) j z) a f20 )j 1 z) + 5 + 1 a f20 )) mod 19). Remark. In [13], Ono roved art 1 of Theorem 1.1 and Corollaries 1.2, 1.3, and 1.4 for f 12 z) = z) S 12. We also remark that Guerzhoy has roved related congruences. See Theorem 2 and Corollary 1 of [10]. Next, we study values of L fk,z) and θ k 1 M fk z)) at oints in the uer half-lane. If gz) is a function on h and τ h, then we define v τ gz)) to be the order of vanishing of gz) at τ. Recalling 1.2), 1.10), and 1.12), we define a meromorhic modular form on SL 2 Z) of weight k with coefficients b Fk,τn) C by b Fk,τn)q n := H τ z)f k z) a Fk 0)E k z). n=0 Similarly, recalling 1.10) and 1.15), we define a meromorhic modular form on SL 2 Z) of weight 2 k with coefficients β Gk,τn) C by n= n k β Gk,τn)q n := H τ z)g k z). Corollary 1.5. Let k {12, 16, 18, 20, 22, 26}, let τ h, and let be rime. 1) We have and θ k 1 M fk τ)) = L fk,τ) = b F k,τ) a fk )b Fk,τ1) F k τ) nk m=1 mk 1 c + f k m)β Gk,τ m) k 1)!β Gk,τ1). G k τ)

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 7 2) If ω = 1 + 3, then we have 2 2 and 3) We have and v ω L fk,z)) v ω θ k 1 M fk z))) { 2 if k 0 mod 6) 1 if k 4 mod 6) { 2 if k 2 mod 6) 1 if k 4 mod 6). v i L fk,z)) 1 if k 0 mod 4) v i θ k 1 M fk z))) 1 if k 2 mod 4). Remark. Setting F k z) = E k 2 z) in Theorem 1.1, art 2 of the corollary imlies, for k 0, 4 mod 6), that a fk ) = j ω) 2k 2) 2 m=1 σ k 3m)j m ω) + 2k σ k 1 ). 744 2k 2) 2 + 2k Similarly, art 3 of the corollary imlies, for k 0 mod 4), that a fk ) = j i) 2k 2) 2 m=1 σ k 3m)j m i) + 2k σ k 1 ). 984 2k 2) 2 + 2k Guerzhoy roved these identities indeendently in [10] Theorem 3). Corollary 1.3 underscores the significance of the coefficients c + f k 1). Using Corollary 1.5, we rovide alternative exressions for these coefficients. Perhas they might shed some light on questions of rationality and algebraicity. Corollary 1.6. The following are true. { ω if k 2, 4 mod 6) 1) Set τ :=. Let n k be as in 1.16). Then we have i if k 2 mod 4). c + f k 1) = k 1)!B G k, 1τ) n k m=2 mk 1 c + f k m)b Gk,mτ). B Gk,1τ) 2) There is a τ h with v τ θ 11 M z)) 1. For such τ, we have c + 1) = 11!j 2τ) + 24j 1 τ) + 324). Part 1 of the corollary follows from Theorem 1.1 and arts 2 and 3 of Corollary 1.5. Part 2 is roved in Section 3. The lan for Sections 2 and 3 of the aer is as follows. In Section 2.1, we rovide the necessary background on harmonic weak Maass forms. In Section 2.2, we construct certain Poincaré series and use them to define the functions M fk z). In Section 3, we rove Theorem 1.1 and its corollaries.

8 MATTHEW BOYLAN 2. Harmonic weak Maass forms and Poincaré Series. 2.1. Harmonic weak Maass forms. We recall the definition of integer weight harmonic weak Maass forms on SL 2 Z). Let k Z, let x, y R, and let z = x + iy h. The weight k hyerbolic Lalacian is defined by k := y 2 2 ) x + 2 2 y 2 + iky x + i ). y A harmonic weak Maass form of weight k on SL 2 Z) is a smooth function f on h satisfying the following roerties. ) a b 1) For all A = SL c d 2 Z) and all z h, we have faz) = cz + d) k fz). 2) We have k f = 0. 3) There is a olynomial P f = n 0 c+ f n)qn C[[q 1 ]] such that fz) P f z) = Oe εy ) as y for some ε > 0. Using these conditions, one can show that harmonic weak Maass forms have q-exansions of the form 2.1) fz) = c + f n)qn + c f n)γk 1, 4πny)q n, n=n 0 where n 0 Z and the incomlete Gamma function is given by Γa, x) := x e t t adt t. For more information on this and other secial functions used in this section, see, for examle, [1]. We refer to the first resectively, second) sum in 2.1) as the holomorhic resectively, non-holomorhic) rojection of fz). We also observe that M! k H k and that the Hecke oerators reserve H k. 2.2. Poincaré Series. We ) now construct Poincaré series as in [5]. Let k be an integer, let a b z h, and let A = SL c d 2 Z). For functions f : h C, we define f k A)z) := cz + d) k faz). Now, let m Z, and let ϕ m : R + C be a function which satisfies ϕ m y) = Oy α ) as y 0 for some α R. If β R, we set eβ) := e 2πiβ. In this notation, we define ϕ mz) := ϕ m y)emx). Noting that ϕ mz) is invariant under the action of Γ := the Poincaré series Pm, k, ϕ m ; z) := We will study secializations of this series. A Γ \SL 2 Z) { ) 1 n ± 0 1 ϕ m k A)z). } : n Z, we define

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 9 We first consider the Poincaré series P1, k, 1; z). It is well-known that P1, k, 1; z) S k. Since S k has dimension one for k {12, 16, 18, 20, 22, 26}, the form P1, k, 1; z) is a constant multile of f k z). Moreover, we observe that 2.2) P1, k, 1; z) = c k f k z), with c k as in 1.5). These facts may be found, for examle, in 3.3 of [11]. Next, we consider certain non-holomorhic Poincaré series. Let M ν,µ z) be the M-Whittaker function. For k Z and s C, define M s y) := y k 2 M k 2 sgny),s 1 2 y ), and for integers m 1, set ϕ m z) := M 1 k 4πmy). Then, for integers k > 2, we define 2 the Poincaré series R fk z) := P 1, 2 k, ϕ 1 ; z). Recalling the definition 1.4) of N fk z) and using 2.2), we have N fk z) = ik 1)2π) k 1 c k i z We now define the series M fk z) by f k τ) i iτ + z)) 2 kdτ = ik 1)2π)k 1 z M fk z) := R fk z) N fk z). P1, k, 1, τ) iτ + z)) 2 kdτ. We require further secial functions. For j Z, we denote by I j the I j -Bessel function. For m, n, c Z with c > 0, the Kloosterman sum Km, n, c) is given by 2.3) Km, n, c) := ) mv + nv, c v Z/cZ) e where v is v 1 mod c). We now state the main theorem of this section. It is a secial case of Theorem 1.1 of [5]. It asserts that M fk z) is a mock modular form for f k z) and rovides formulas for its coefficients and the coefficients of N fk z). Theorem 2.1. Let k {12, 16, 18, 20, 22, 26}. 1) In the notation above, we have R fk z) = M fk z) + N fk z) H 2 k. 2) The function M fk z) is holomorhic with q-series exansion M fk z) = k 1)!q 1 + k 1)! 2k + where for integers n 1, we have c + f k n) = 2πi k n k 1 2 c=1 c + f k n)q n, ) K 1, n, c) 4π n I k 1. c c 3) The function N fk z) is non-holomorhic with q-series exansion N fk z) = c f k m)γk 1, 4πmy)q m, m=1 where for integers m 1, we have c f k m) = k 1)c k a fk m) m k 1, with a fk m) Z as in 1.3) and c k R as in 1.5).

10 MATTHEW BOYLAN Remark 1. The functions M fk z) deend on the reresentation of f k z) as a linear combination of Poincaré series Pm, k, 1; z) S k. In this work, we choose the simlest reresentation, given by 2.2). Remark 2. We comute the constant term in the q-exansion of M fk z) using 2.3) and facts from elementary number theory. In [5], we find that the constant term is We comute that K 2 k 1, 0, c) = c=1 c + f k 0) = 2πi) k v Z/cZ) e c=1 K 2 k 1, 0, c) c k. mv ) = e c v Z/cZ) mv ) = µc), c where µ is the Möbius function. Thus, since k is even, we have K 2 k 1, 0, c) µc) = = 1 c k c k ζk) = 2 k!, 2πi) k where ζ is the Riemann zeta-function. c=1 Remark 3. We obtain the exansion for N fk z) from 1.4) as in the roof of Theorem 1.1 of [5]. 3. Proofs of Theorem 1.1 and its corollaries. We now turn to the roofs of Theorem 1.1 and its corollaries. The basic idea is as follows. We use the Hecke oerators and theta oerator to eliminate N fk z), the non-holomorhic art of the harmonic weak Maass form R fk z). This is where we use the fact that the forms f k lie in 1-dimensional saces. What remains is a modification of the mock modular form M fk z) which is a weakly holomorhic modular form with an exlicit rincial art. Since SL 2 Z) has genus one and only one cus, one can exress the weakly holomorhic modular form exlicitly in terms of certain olynomials in j Faber olynomials). To begin, we note that for all integers m and for k {12, 16, 18, 20, 22, 26}, the forms f k z) are normalized eigenforms for the Hecke oerators T k m) since the dimension of S k is one. Therefore, for all rimes and for all integers n 1, from 1.7) we have ) n 3.1) a fk n) + k 1 a fk = a fk )a fk n). Next, we comute N fk z) T 2 k ). To do so, we recall the definitions of the U) and V ) oerators on functions f : h C: fz) U) := 1 1 ) z + a 3.2) f, 3.3) a=0 fz) V ) := fz). We also recall, for all integers j, that the Hecke oerators T j ) may be written as 3.4) fz) T j ) = fz) U) + j 1 fz) V ). We now state two imortant facts. Though the first was roved in [8] for half-integral weights, see Theorem 7.4), we rovide a short roof in our setting.

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 11 Proosition 3.1. Let be rime, and let k {12, 16, 18, 20, 22, 26}. Then we have N fk z) T 2 k ) = 1 k a fk )N fk z), with a fk n) as in 1.3). Proof. Using art 3 of Theorem 2.1, 3.1), 3.2), 3.3), and 3.4), we comute: N fk z) T 2 k ) = N fk z) U 2 k ) + 1 k N fk z) V 2 k ) a fk n) = k 1)c k n) k 1Γk 1, 4πny)q n + 1 k = k 1)c k 1 k = k 1)c k 1 k a fk ) a fk n) + k 1 a fk n a fk n) n k 1 Γk 1, 4πny)q n. )) Γk 1, 4πny) n k 1 a fk n) Γk 1, 4πny)q n nk 1 q n ) The second fact we require is a secial case of a result of Bruinier, Ono, and Rhoades [9] which follows from Bol s identity relating the theta-oerator and the Maass raising oerator. Theorem 3.2. If k 2 is an integer and fz) H 2 k has holomorhic rojection Mz) = c + n)q n, then we have θ k 1 fz)) = θ k 1 Mz)) = n k 1 c + n)q n M! k. Remark. We emhasize that in [9], the authors recisely comute the image of θ k 1 on saces of harmonic weak Maass forms of weight 2 k < 0 on subgrous of tye Γ 0 N) with nebentyus character. 3.1. Proof of Theorem 1.1. We may now rove Theorem 1.1. From art 1 of Theorem 2.1 and the fact that the Hecke oerators reserve H 2 k, we see that L fk,z) := k 1 k 1)! R f k z) T 2 k ) 1 k a fk )R fk z)) H 2 k. Then, by Proosition 3.1, art 2 of Theorem 2.1, and 1.7), we find that 3.5) L fk,z) = k 1 k 1)! M f k z) T 2 k ) 1 k a fk )M fk z)) = q a fk )q 1 + 2k σ k 1 ) a fk )) 1 )) n + k 1 c + f k 1)! k n) a fk )c + f k n) + c + f k q n. Moreover, we observe that L fk,z) M 2 k! is a weakly holomorhic modular form since it is a harmonic weak Maass form whose holomorhic rojection is zero.

12 MATTHEW BOYLAN With F k z) = n=0 a F k n)q n M k 2, we note that L fk,z)f k z) M 0!. Using 3.5), we comute L fk,z)f k z) = a Fk m)q m + a Fk 0) 2k σ k 1 ) B m=0 k a fk ) a Fk 0)q 1 + a Fk 1) + a Fk 0) 2k ) + Oq). If we define A Fk,tz) as in 1.13), we find that L fk,z)f k z) A Fk,z) a fk )A Fk,1z)) = Oq) is a modular form of weight zero which is holomorhic on h and at infinity. Hence, it is constant. This constant is zero, thus comleting the roof of art 1 of Theorem 1.1. Part 2 follows similarly. With G k z) M k! as in 1.15), we see from Theorem 3.2 that θ k 1 M fk z))g k z) M 0!. Using 1.14) and 1.16), we comute θ k 1 M fk z))g k z) = k 1)! n k n= 1 a Gk n)q n 1 + n k m=1 m k 1 c + f k m) n k n=m a Gk n)q m n + Oq). If we define B Gk,tz) as in 1.17), we find that nk ) θ k 1 M fk z))g k z) m k 1 c + f k m)b Gk,mz) k 1)!B Gk, 1z) = Oq) m=1 is a modular form of weight zero which is holomorhic on h and at infinity. The conclusion follows. 3.2. Proof of Corollaries 1.2 and 1.3. To rove Corollary 1.2, it suffices to show that L fk,z) Zq)). We consider cases. When k {12, 16}, we aly Theorem 1.1 with F k z) = E k 2 z) M k 2 Z[[q]]. Here, we note that 2k 2) 2 Z. Furthermore, we observe that B 12 = 691, B 2730 16 = 3617, and that 510 for all rimes, σ 11 ) a f12 ) = τ) mod 691) σ 15 ) a f16 ) mod 3617). These congruences may be found, for examle, in [14]. Therefore, for k {12, 16}, we must have 2k 3.6) σ k 1 ) a fk )) Z. Hence, from 1.2), 1.13), 3.6), and Theorem 1.1, we deduce that 3.7) L fk,z) = A E k 2,z) a fk )A Ek 2,1z) E k 2 z) = j z) 2k 2) 2 m=1 σ k 3m)j m z) a fk ) has integer coefficients. E k 2 z) ) j 1 z) 2k 2) 2 + 2k σ k 1 ) a fk ))

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 13 When k {18, 20, 22}, we aly Theorem 1.1 with F k z) = z)e k 14 z) S k Z[[q]], from which we obtain m=1 L fk,z) = a F k m)j m z) a fk ) Zq)). z)e k 14 z) Lastly, when k = 26, we aly Theorem 1.1 with F 26 z) = z) 2 S 24 Z[[q], which gives m=2 L f26,z) = a F k m)j m z) Zq)). z) 2 To see art 2 of Corollary 1.3, we first remark that if a fk ) 0 for all rimes, then a fk n) 0 for all n. Therefore, we suose that c + f k m) is irrational for some m and that a fk ) = 0 for some rime. Corollary 1.2 imlies that for all n, 1 k 1)! k 1 c + f k n) + c + f k n )) Z. Setting n = 1, we see that c + f k ) Q. But then 1.19) shows that c + f k 1) Q. Part 1 of the corollary now imlies that c + f k m) Qc + f k 1)) must be rational, a contradiction. 3.3. Proof of Corollary 1.4. On alying Theorem 1.1 with F k z) = E k 2 z) M k 2, we obtain L fk,z) as in 3.7). We then use the von Staudt-Claussen congruences for the denominators of Bernoulli numbers see [12], Lemma 1.22, for examle) and congruences for the coefficients a fk ) see [14], Corollary to Theorem 4, for examle) to rove Corollary 1.4. To begin, we note that for all j 2 and even, we have 2j B j 0 mod 24), so E j z) 1 mod 24). Moreover, for k {12, 16, 18, 20, 22, 26}, we have f k z) z) mod 24). Part 1 of the corollary holds since for all rimes, we have 0 mod 24) if = 2 τ) = a f12 ) 12 mod 24) if = 3 + 1 mod 24) if 5. Next, we note that if l is rime and l 1 k 2, then we have 2k 2) 2 E k 2 z) 1 mod l). From 3.7), we see that if l 1 k 2, then 3.8) L fk,z) j z) a fk )j 1 z) + 2k + 1 a fk )) When l 5, this is the case for airs mod l). 0 mod l), so l, k) {5, 18), 5, 22), 5, 26), 7, 20), 7,26), 11, 12), 11, 22),13,26), 17, 18), 19,20)}. The remaining arts of Corollary 1.4 are secializations of 3.8) using congruences for f k z), with θ as in 1.8), and congruences for τ) when is rime. These congruences are: θ z) mod 5) if k {18, 22, 26} θ z) mod 7) if k {20, 26} 3.9) f k z) z) mod 11) if k {12, 22} θ z) mod 13) if k = 26, { + 2 mod 5) 3.10) τ) + 4 mod 7).

14 MATTHEW BOYLAN To rove the congruences for f k z) modulo 5, 7, and 13, we note that θ z) = z)e 2 z) and that if k k mod l 1), then Kummer s congruences for Bernoulli numbers imly that E k z) E k z) mod l). For k {18, 22, 26}, 3.9) and 3.10) imly that a fk ) τ) + 2 ) + 1) mod 5), 5) where ) is the Legendre symbol. Using that 2k 1 mod 5), we obtain art 2 of the corollary. For k {20, 26}, 3.9) and 3.10) imly that a fk ) τ) + 4 ) 1 2 + mod 7). 7)) Using the fact that 2k 3 mod 7) gives art 3 of the corollary. For k {12, 22}, 3.9) together with the fact that 2k 2 mod 11) gives art 4 of the corollary. Part 5 of the corollary is similar. 3.4. Proof of Corollaries 1.5 and 1.6. Recalling 1.2), 1.10), 1.12), and 1.13), for n 1, we find that the nth coefficient of b Fk,τn)q n = H τ z)f k z) a Fk 0)E k z) is given by b Fk,τn) = n=0 n m=0 a Fk m)j n m τ) + a Fk 0) 2k σ k 1 n) = A Fk,nz). Similarly, from 1.10), 1.16), and 1.17), we find that the nth coefficient of n= n k β Gk,τn)q n = H τ z)g k z) is given by β Gk,τn) = n m= n k a Gk m)j n m τ) = n k m= n Theorem 1.1 now imlies art 1 of Corollary 1.5. Next, noting from 3.5) that we find that L fk,z) = q + M! 2 k, L fk,z) z) = z) A Fk,z) a fk )A Fk,1z)) F k z) a Gk m)j m+n τ) = B Gk, nτ). = 1 + M 12+2 k. Alying the valence formula for forms in M 12+2 k see [12], Theorem 1.29), we see that v ω L fk,z)) 2k + 1) mod 3), v i L fk,z)) k + 1 mod 2). 2

ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS 15 Since these quantities are non-negative L fk, z) is weakly holomorhic), we obtain arts 2 and 3 of the corollary. Similarly, observing from 1.14) that θ k 1 M fk z)) z) = k 1)! + M k+12, we aly the valence formula for forms in M k+12 to deduce arts 2 and 3 of the corollary for the weakly holomorhic modular form θ k 1 M fk z)). To obtain art 2 of Corollary 1.6, we first note that for all τ h we have v τ z)) = 0. Then alying the valence formula to θ 11 M z)) z) = 11! + M 24 gives 1 2 v iθ 11 M z)) + 1 3 v ωθ 11 M z)) + v P θ 11 M z)) = 2, P where the sum is over all P h inequivalent to i, ω in a fundamental domain. Since θ 11 M z)) is weakly holomorhic, for all τ h, we have v τ θ 11 M z)) 0. Hence, for some τ h, we must have v τ θ 11 M z)) 1. Since τ) 0, 1.18) imlies the result. Acknowledgment The author thanks K. Ono for helful comments and suggestions in the rearation of this aer. References [1] M. Abramowitz and I. Stegun, Handbook of mathematical functions. Dover Publications, 1965. [2] T. Asai, M. Kaneko, and H. Ninomiya, Zeros of certain modular functions and an alication. Comm. Math. Univ. Sancti Pauli 46 1997), 93-101. [3] K. Bringmann, A. Folsom, and K. Ono, q-series and weight 3/2 Maass forms. Prerint. [4] K. Bringmann and K. Ono, The fq) mock theta function conjecture and artition ranks. Invent. Math. 165 2006), 243-266. [5] K. Bringmann and K. Ono, Lifting ellitic cus forms to Maass forms with an alication to artitions. Proc. Natl. Acad. Sci. USA 104 2007), no. 10, 3725-3731. [6] K. Bringmann and K. Ono, Dyson s ranks and Maass forms. Ann. of Math., to aear. [7] K. Bringmann, K. Ono, and R.C. Rhoades, Eulerian series as modular forms. J. Amer. Math. Soc., to aear. [8] J. H. Bruinier and K. Ono, Heegner divisors, L-functions, and harmonic weak Maass forms. Prerint. [9] J. H. Bruinier, K. Ono, and R.C. Rhoades, Differential oerators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues. Math. Ann., to aear. [10] P. Guerzhoy, Hecke oerators for weakly holomorhic modular forms and suersingular congruences. Prerint. [11] H. Iwaniec, Toics in the classical theory of automorhic forms. Grad. Stud. in Math. 17, Amer. Math. Soc., Providence, RI, 1997. [12] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS 102, Amer. Math. Soc., 2004. [13] K. Ono, A mock theta function for the Delta-function. Proceedings of the 2007 Integers Conference, to aear. [14] H. P. F. Swinnerton-Dyer, On l-adic reresentations and congruences for coefficients of modular forms. Modular functions of one variable, III Proc. Internat. Summer School, Univ. Antwer, 1972), Lecture Notes in Math., Vol. 350, Sringer Berlin, 1973, 1-55. Deartment of Mathematics, University of South Carolina, Columbia, SC 29208 E-mail address: boylan@math.sc.edu