SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS

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1 SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS DAVID HANSEN AND YUSRA NAQVI Abstract. In 1973, Shimura [8] introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi s theta function and level one Hecke eigenforms. Cipra [1] generalized Selberg s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra s results further to cover theta functions of arbitrary Dirichlet characters multiplied by more general Hecke eigenforms, and we use these explicit formulas to compute optimal levels for these lifts without appealing to Shimura s deeper arguments. 1. Introduction and statement of results Let SL 2 (Z denote the set of all 2-by-2 matrices with integer entries and determinant 1, and let k be a positive integer. We say that f(z is a modular form of weight k on the congruence subgroup Γ 0 (N with multiplier ψ if f(z is a holomorphic function on the upper half of the complex plane which satisfies f(γz = (cz + d k ψ(df(z for all γ = ( a b c d SL 2 (Z with c 0 (mod N. Let M k (N, ψ denote the finite-dimensional vector space of modular forms of weight k on Γ 0 (N with multiplier ψ, where ψ is a Dirichlet character of modulus N. A modular form is called a cusp form if it vanishes at all rational points and at infinity. We let S k (N, ψ denote the subspace of M k (N, ψ consisting only of cusp forms. For k 2 a positive even integer, we define the Eisenstein series of weight k by (1.1 E k (z := 1 2k B k σ k 1 (nq n, where B n in the nth Bernoulli number and q := e 2πinz. These functions represent the simplest modular forms of weight k, and they lie in M k (1, 1. More general Eisenstein series can be defined as follows. Let χ 1 (mod a 1 and χ 2 (mod a 2 be primitive Dirichlet characters of conductors a 1, a 2, not both trivial, and set a = a 1 a 2. The character χ = χ 1 χ 2 has modulus a. If k is positive integer with χ( 1 = ( 1 k, then set (1.2 E k (χ 1, χ 2 ; z := C(k, χ 1, χ 2 + ( χ 1 (n/dχ 2 (dd k 1 q n, 1 d n

2 2 DAVID HANSEN AND YUSRA NAQVI where C(k, χ 1, χ 2 is zero unless a 1 = 1, in which case C(k, χ 1, χ 2 = 1 2L(1 k, χ, where L(s, χ = χ(nn s is the Dirichlet L-function of the character χ. We have E k (χ 1, χ 2 ; z M k (a, χ; see Chapter 4 of [4]. In a classic paper [8], Shimura invented the modern theory of modular forms of halfintegral weight. Briefly, let N, k be positive integers with ψ a Dirichlet character of modulus 4N. We say that f is a modular form of weight k + 1/2 with multiplier ψ if ( c 2k+1ɛ 2k 1 (1.3 f(γz = ψ(d d d (cz + d k+1/2 f(z for all γ Γ 0 (4N, where ɛ d is 1 or i for odd d according to whether d 1 (mod 4 or d 3 (mod 4, respectively, and ( c d is Shimura s extension of the Jacobi symbol. As above, M k+1/2 (N, ψ denotes the finite-dimensional vector space of weight k + 1/2 modular forms, and S k+1/2 (N, ψ denotes its subspace of cusp forms. Define theta functions (1.4 θ(χ; z := n Z χ(nn ν q n2 M 1/2+ν (4r 2, χχ ν 4 for χ a Dirichlet character of modulus r, where ν = 0, 1 is chosen such that χ( 1 = ( 1 ν. These functions are the simplest examples of modular forms of half-integral weight, and for k = 1/2, the space is spanned by them (c.f. [7]. For a good introduction to this material, see [6]. Shimura also established a family of nontrivial maps between modular forms of halfintegral weight and modular forms of even integer weight. These maps, known as the Shimura lifts, can be stated as follows. Theorem (Shimura. Let t be a positive squarefree integer, and suppose that f(z = b(nqn S k+1/2 (4N, ψ, where k is a positive integer. If numbers A(n are defined by (1.5 A(nn s := L(s k + 1, ψχ k 4χ t b(tn 2 n s, where χ t = ( t is the usual Kronecker character modulo t, then St (f(z := A(nqn M 2k (2N, ψ 2. Moreover, if k > 1, then S t (f(z is a cusp form. Shimura lifts play an important role in several areas of modern number theory, including Tunnell s famous work [9] on the ancient congruent number problem, and recent work by Ono [5] on congruences for the partition function. Moreover, in these particular applications, the relevant half-integral weight forms can be written as products of integer weight forms and theta functions. In light of these facts, it is desirable to have explicit formulas for the Shimura lifts in these cases. It turns out that much earlier, in unpublished work, Selberg worked out such an explicit formula. Briefly, for certain modular forms f(z M k (1, 1, Selberg found that f(4zθ(1; z M k+1/2 (4, 1 lifts to f(z 2 2 k 1 f(2z 2 M 2k (2, 1. Later on, Cipra [1] generalized Selberg s work by proving the following result.

3 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 3 Theorem (Cipra. If f(z S k (N, ψ is a newform, and θ(χ r ; z is the theta function of an even Dirichlet character of prime power modulus r = p m, then if we define g(z := f(zf(p µ z, the Shimura lift S 1 of f(4p µ zθ(χ r ; z is (1.6 g χr (z 2 k 1 χ r (2ψ(2g χr (2z, where µ is any integer with µ m. Cipra also proves a similar statement for theta functions with odd characters. However, Cipra s class of eligible forms f(z is limited to newforms, and his use of theta functions with characters to prime power moduli is a highly restrictive condition. We prove the following two theorems, generalizing these results. Theorem 1.1. Let χ r be an even Dirichlet character modulo r, and write χ r = χ p α 1 1 χ p α χ p α as the factorization of χ r into Dirichlet characters modulo prime powers p α 1 1, pα 2 2,..., pα with p α 1 1 pα pα = r. Let f(z M k (N, ψ be a Hecke eigenform, and set F (z := θ(χ r ; zf(4rz M k+1/2 (4N r 2, ψχ r χ k 4 with N = N/ gcd(n, r. If (1.7 g(z := where χ d = p α d χ p α, then we have gcd(d,r/d=1 f(dzf(rz/dχ d ( 1, (1.8 S 1 (F (z = g χr (z 2 k 1 χ r (2ψ(2g χr (2z M 2k (2N r 2, ψ 2 χ 2 r. Here g χ is the χ-twist of g. For the case of odd characters, the theorem is slightly different, due to the fact that the relevant theta functions now have weight 3/2. Theorem 1.2. Let χ r be an odd Dirichlet character modulo r, and write χ r = χ p α 1 1 χ p α χ p α as the factorization of χ r into Dirichlet characters modulo prime powers p α 1 1, pα 2 2,, pα with p α 1 1 pα 2 2 pα = r. If F (z := θ(χ r ; zf(4rz M k+3/2 (4N r 2, ψχ r χ k+1 4, where f(z M k (N, ψ is a Hecke eigenform, and (1.9 g(z := 1 πi where χ d = p α d χ p α, then we have gcd(d,r/d=1 df (dzf(rz/dχ d ( 1, (1.10 S 1 (F (z = g χr (z 2 k χ r (2ψ(2g χr (2z M 2k+2 (2N r 2, ψ 2 χ 2 r, where g χ is the χ-twist of g. The proofs of our theorems, like those of Selberg and Cipra, are entirely combinatorial, using only elementary properties of Dirichlet series and a multiplicativity relation for the coefficients of our starting form f(z. This multiplicativity is conditional on f(z being a Hecke eigenform. However, since any given modular form can be written as a linear combination of eigenforms, our theorems can be applied to more general products of modular

4 4 DAVID HANSEN AND YUSRA NAQVI forms and theta functions by the linearity of the Shimura lift. Furthermore, we compute the levels of the lifts in Theorems 1.1 and 1.2 directly, without appealing to any of Shimura s results. In fact, our theorems are completely independent of Shimura s work. In Section 2, we define and explain the notion of a Hecke eigenform and the associated multiplicativity relations for its coefficients. In Section 3, we present proofs of Theorems 1.1 and 1.2, and we discuss a method of determining the cuspidality of the lifts given by these theorems. We also show how to obtain the optimal level for the lifted forms. Section 4 contains a discussion of examples and applications. 2. Multiplicativity Properties of Modular Form Coefficients Let f(z = n=0 a(nqn M k (N, ψ. The action of the nth Hecke operator T ψ n of weight k on f(z is given by (2.1 f(z T ψ n = ( m=0 d (m,n ψ(dd k 1 a(mn/d 2 q m. It is known, by work of Hecke, that these operators are linear endomorphisms on M k (N, ψ. They also map cusp forms to cusp forms. Furthermore, there exist modular forms f(z M k (N, ψ which are simultaneous eigenfunctions of all the Hecke operators; in other words, they satisfy (2.2 f(z T ψ n = λ(nf(z for all positive integers n, where the λ(n are complex numbers. If f(z satisfies these conditions, we generally refer to it as a Hecke eigenform. In this case, by combining (2.1 and (2.2, we easily get (2.3 λ(na(m = d (m,n ψ(dd k 1 a(mn/d 2. If a(1 = 1, then this reveals that in fact λ(n = a(n, and we can then reformulate (2.3 as follows. Proposition 2.1. If f(z = n=0 a(nqn M k (N, χ is a simultaneous eigenfunction of all the Hecke operators Tn ψ with a(1 = 1, then for any positive integers m, n, we have a(ma(n = d (m,n ψ(dd k 1 a(mn/d 2.

5 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 5 For the Eisenstein series defined in the introduction (see 1.2, we can prove this directly as follows: d (m,n = d (m,n = d (m,n d 2 δ mn = d (m,n d 2 δ mn = d m δ n χ(dd k 1 a(mn/d 2 χ(dd k 1 δ mn/d 2 χ 1 (mn/(d 2 δχ 2 (δδ k 1 χ 1 (dχ 2 (dχ 1 (mn/(d 2 δχ 2 (δd k 1 δ k 1 χ 1 (mn/(dδχ 2 (dδ(dδ k 1 χ 1 (mn/(dδχ 2 (dδ(dδ k 1 = a(ma(n. Furthermore, we have an inverse of Proposition 2.2, which we shall refer to as Selberg inversion. Proposition 2.2. If f(z = n=0 a(nqn M k (N, χ is a Hecke eigenform with a(1 = 1, then we have a(mn = µ(dχ(dd k 1 a(m/da(n/d, for any positive integers m, n. Proof. We have that d (m,n = d (m,n = dδ (m,n = D (m,n = a(mn. d (m,n µ(dχ(dd k 1 a(m/da(n/d µ(dχ(dd k 1 δ (m/d,n/d µ(dχ(dδ(dδ k 1 a(mn/(dδ 2 ( d D µ(d χ(dd k 1 a(mn/d 2 χ(δδ k 1 a(mn/(dδ 2

6 6 DAVID HANSEN AND YUSRA NAQVI 3. Proofs of Theorems 1.1 and 1.2 We begin by presenting the proof of the formula for the lift in Theorem 1.1. From the definition of F (z, we have F (z = n=0 b(nqn with (3.1 b(n = ( n m 2 χ r (ma, 4r m Z where f(z = n=0 a(nqn is as in the statement of Theorem 1.1. As above, the Shimura lift is given by (3.2 S 1 (F = A(nq n with the coefficients A(n defined by (3.3 A(nn s = L(s k + 1, χ r ψχ 2k 4 We also need the coefficients defined by (3.4 c d (nq n := f(dzf(rz/d = n=0 b(n 2 n s. ( n dm a(ma r/d n=0 m Z q n. Throughout the proof, we use the convention that a modular form coefficient is zero if its argument is negative or not integral. From (3.1 it is easy to see that (3.5 b(n 2 = ( (n m(n + m χ r (ma. 4r m Z This is a finite sum with non-zero coefficients whenever (n m(n + m/(4r N. Note that n + m and n m must both be even for n and m to be integers with 4 (n 2 m 2. Let gcd((n m/2, r = d. Thus, m n (mod 2d and m n (mod 2r/d. Now suppose gcd(d, r/d = d > 1. This implies that m n n 0 (mod 2d, so d gcd(m, r and so χ r (m = 0. Therefore, we only consider the cases in which gcd(d, r/d = 1. We have m = n + 2dm for some m Z, so n m = 2dm and n + m = 2n + 2dm. Thus, (3.6 (n m(n + m 4r Also, since m n (mod d and m n = χ d (mχ r/d (m = χ r/d ( 1χ d (nχ r/d (n = m (n + dm. r/d (mod r/d, we have that χ r (m = χ d (nχ r/d ( n = χ r/d ( 1χ r (n,

7 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 7 where the characters are as defined in the statement of Theorem 1.1. Since χ r ( 1 = χ r/d ( 1χ d ( 1 = 1, we have χ d ( 1 = χ r/d ( 1 = ±1. Thus, by changing the variable m to m, (3.5 becomes (3.7 b(n 2 = χ r (n χ d ( 1 ( m(n dm a. m Z gcd(d,r/d=1 We now apply Proposition 2.2 to get b(n 2 = χ r (n χ d ( 1 m Z = χ r (n = χ r (n gcd(d,r/d=1 gcd(d,r/d=1 gcd(d,r/d=1 δ (m,n r/d ( m ( n dm µ(δψ(δδ k 1 a a δr/d δr/d χ d ( 1 µ(δψ(δδ ( m ( n/δ dm k 1 a a r/d r/d δ n m Z χ d ( 1 δ n µ(δψ(δδ k 1 c d (n/δ. Rewriting these formulas as Dirichlet series immediately gives (3.8 b(n 2 n s = χ d ( 1 µ(δψ(δχ r (δδ k 1 χ r (n/δc d (n/δn s, gcd(d,r/d=1 δ n and we can easily pull out the reciprocal of a Dirichlet L-function to produce (3.9 b(n 2 n s 1 = χ d ( 1 χ r (nc d (nn s. L(s k + 1, ψχ r gcd(d,r/d=1 Multiplying by L(s k + 1, χ r ψχ 2k 4 = L(s k + 1, χ rψχ 2 4, as in the definition of the Shimura lift, we have (3.10 A(nn s = L(s k + 1, χ rψχ 2 4 χ d ( 1 χ r (nc d (nn s. L(s k + 1, χ r ψ gcd(d,r/d=1 By an easy consideration of Euler products, the quotient of the L-functions simplifies to 1 2 k 1 s χ r (2ψ(2, and rewriting into q-series completes the proof of the identity for the lift. The proof of the equation for the lift in Theorem 1.2 follows along the same lines, with appropriate changes due to the slightly different expression for the theta function. The congruence condition reasoning following (3.5 does not change, and (3.7 becomes (3.11 b(n 2 = χ r (n χ r/d ( 1 ( m(n dm a (n 2dm. r/d m Z gcd(d,r/d=1

8 8 DAVID HANSEN AND YUSRA NAQVI Recall that χ r is odd here, so χ r ( 1 = χ r/d ( 1χ d ( 1 = 1, and so we have that χ d ( 1 = χ r/d ( 1 = ±1. Selberg inversion applies again, and the derivatives of modular forms appearing in the definition of g(z arise naturally from the linear form in m and n appearing in (3.11. In the odd case, it is not immediately clear that g(z is in fact a modular form, since it contains derivatives of modular forms. However, it is in fact easy to prove modularity by employing the following useful fact. Proposition 3.1. Let f(z be a modular form of weight k on some subgroup of SL 2 (Z. Then 1 d 2πi dz f(z = ( f(z + ke 2 (zf(z/12, where E 2 (z is the Eisenstein series defined in (1.1 and f(z is a modular form of weight k + 2. Note that E 2 is not a modular form (see [6]. Using this proposition, we easily obtain g(z = 1 χ d ( 1df (dzf(rz/d πi = 1 2π 2 i 2 = 1 12πi = 1 12πi gcd(d,r/d=1 gcd(d,r/d=1 gcd(d,r/d=1 gcd(d,r/d=1 χ d ( 1f(rz/d z f(dz χ d ( 1f(rz/d( f d (z + ke 2 (zf(dz χ d ( 1f(rz/d f d (z, where f d (z is a modular form of weight k+2 and level dn. The sum involving E 2 s vanishes due to cancellation in characters, namely χ d ( 1 = χ r/d ( 1. To complete the proofs of Theorems 1.1 and 1.2, it suffies to compute the levels of the relevant Shimura lifts. Because g(z lies in the space M 2k (N r, ψ 2, it is easy to see by the general theory of twists (see [6], Sec. 2.2 that g χr (z M 2k (N r 3, χ 2 rψ 2. However, we can in fact show that g χr (z lies in the space M 2k (N r 2, χ 2 rψ 2. To do this, we demonstrate the invariance of g χr (z under a complete set of representatives of right cosets of Γ 0 (N r 3 in Γ 0 (N r 2. By Proposition 2.5 of [2], we have that [Γ 0 (N r 2 : Γ 0 (N r 3 ] = r, so such a set of representatives is given by ( 1 0 (3.12 α := N r 2 1 for = 0, 1, 2,..., r 1. For convenience, we define the slash operator for γ GL + 2 (Q by (3.13 f(z k γ := f(γz(cz + d k (det γ k/2. With this notation, we need to show g χr (z k α = g χr (z for = 0, 1, 2,..., r 1. Using Proposition 17 in Sec. 3.3 of [3] and defining τ(χ r := r 1 m=0 χ r(me 2πim/r, we first write

9 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 9 g χr (z as a sum of linear transforms, where we have set r 1 g χr (z = r 1 τ(χ r v=0 r 1 = r 1 τ(χ r (3.14 γ v := It then follows that r 1 g χr (z α = r 1 τ(χ r r 1 = r 1 τ(χ r v=0 r 1 = r 1 τ(χ r v=0 r 1 = r 1 τ(χ r = g χr (z. v=0 v=0 v=0 ( 1 v/r. 0 1 χ r (vg(z v/r χ r (vg(z γ v, χ r (vg(z γ v α k ( ( 1 v/r 1 0 χ r (vg(z k 0 1 N r 2 1 ( 1 vn χ r (vg(z r N v 2 ( 1 v/r k N r 2 vn r χ r (vg(z k γ v Note that the first matrix in the fourth line is in Γ 0 (N r with d 1 (mod N, and so it has an invariant action on g(z. Having an explicit form for the lift allows us to check its cuspidality directly, without using the analytic machinery of Shimura s theorem. If f(z is a cusp form, then it is easy to see that S 1 (F (z must also be a cusp form, since a sum of cusp forms is itself cuspidal. We now consider, as a simple example, the case in which f(z M k (1, 1 is a Hecke eigenform that is not a cusp form. Let F (z = θ(χ r ; zf(4rz, and recall that (3.15 g(z = χ δ ( 1f(δzf(rz/δ. δ r gcd(δ,r/δ=1 Also recall that if 2 r, then we have that S 1 (F (z = g χ (z. If r is odd, then we define h(z := g(z 2 k 1 g(2z, noting that in this case, the Shimura lift is h χ (z. We shall proceed by computing the Fourier expansions of g(z and h(z around a complete set of cusps. Let g γ (z denote g(z 2k γ. For any γ = ( a b c d SL2 (Z, we have ( ( ( a b (3.16 f(δz = δ k c d k/2 δ 0 a b f(z. k 0 1 c d

10 10 DAVID HANSEN AND YUSRA NAQVI Let δ = gcd(c, δ. We have that there exists an integer y such that (δ/δ (cy + d, and so we get ( a f(δz k c Inserting this into the definition of g(z gives ( a g(z 2k c b = d = δ r gcd(δ,r/δ=1 δ r gcd(δ,r/δ=1 ( b = δ d k/2 aδ/δ δ f(z ( (ay + b δ y k c/δ δ (cy + d/δ 0 δ/δ ( δ = (δ/δ k 2 z δ y f. δ ( a χ δ ( 1f(δzf(rz/δ 2k c ( δ k ( r/δ χ δ ( 1 (δ, c (r/δ, c b d kf ( δ 2 z y δ f ( δ 2 z y δ, where δ is as before, δ = (r/δ, c and y and y are integers that depend on δ. transforms into ( r kf ( δ 2 z y ( δ 2 z y (3.17 g γ (z = χ δ ( 1 f. (r, c δ δ δ r gcd(δ,r/δ=1 This We now consider ( a (3.18 g(2z 2k c ( ( b = 2 d k 2 0 a b g(z, 2k 0 1 c d which gives us that g(2z 2k ( a b c d = gγ (2z if c is even or g γ ((z x/2 if c is odd, where x is some integer that depends on d. This yields ( a h(z 2k c b = g d γ (z 2 k 1 g γ (2z or = g γ (z 2 k 1 g γ ((z x/2. Thus, in all cases, the constant term of the Fourier expansion is a constant multiple of ( r (r,c k a(0 2 χ δ ( 1, and hence this term vanishes if and only if f is a cusp form or (3.19 χ δ ( 1 = 0. δ r gcd(δ,r/δ=1 In particular, this sum vanishes if and only if χ r decomposes into a product of Dirichlet characters to prime power moduli which are not all even. Note that by [7], this is equivalent to θ(χ r ; z being a cusp form. This same method can be applied to modular forms of higher level; however, the computations are more complicated.

11 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS Examples and Applications In this section, we present some examples illustrating Theorems 1.1 and 1.2 in which we calculate the explicit form of the Shimura lift for certain half-integral weight forms. Example 1. We begin with Theorem 1.1, by defining f(z := η(z 5 /η(5z, where η(z := q 1/24 n>0(1 q n is the Dedekind eta function. This function is in the space M 2 (5, χ 5, and in fact we have f(z = 5E 2 (1, χ 5 ; z. We compute the Shimura lift of f(48zη(24z = 1 2 f(48zθ(χ 12; z. To utilize Theorem 1.1, we factor χ 12 = χ 4 χ 3 and 12 = to obtain (4.1 g(z = f(zf(12z f(3zf(4z. Because χ 12 (2 = 0, the second term in (1.7 vanishes and we have (4.2 ( η(48z 5 η(24z ( η(z 5 η(12z 5 S 1 = η(240z η(5zη(60z η(3z5 η(4z 5 = 25q 7 +50q q q η(15zη(20z χ 12 Example 2. We now illustrate Theorem 1.2 by computing the lift of (60z 2 θ(χ 15 ; z, where (z = η(z 24 is the standard discriminant function. To apply our theorem, we must write (z 2 as a linear combination of Hecke eigenforms. The two Hecke eigenforms, say f 1 (z and f 2 (z, of weight 24 and level 1 have Fourier expansions (4.3 f i (z = q + a i q 2 + ( a i q with a 1 = and a 2 = Hence, (4.4 (z 2 = f 1(z f 2 (z To apply Theorem 1.2, we factor 15 = 3 5 and χ 15 = χ 3 χ 5 to obtain (4.5 g(z = 1 ( f πi 1(zf 1 (15z 3f 1(3zf 1 (5z + 5f 1(5zf 1 (3z 15f 1(15zf 1 (z and hence S 1 (θ(χ 15 ; zf 1 (60z(z = g χ 15 (z g χ 15 (2z. A similar formula holds for the lift of f 2 (z. References [1] B. A. Cipra. On the Shimura lift, après Selberg. J. Number Theory, 32(1:58 64, [2] H. Iwaniec. Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, [3] N. Koblitz. Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, [4] T. Miyake. Modular forms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, Translated from the 1976 Japanese original by Yoshitaka Maeda. [5] K. Ono. Distribution of the partition function modulo m. Ann. of Math. (2, 151(1: , [6] K. Ono. The web of modularity: arithmetic of the coefficients of modular forms and q-series, volume 102 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004.

12 12 DAVID HANSEN AND YUSRA NAQVI [7] J.-P. Serre and H. M. Stark. Modular forms of weight 1/2. In Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976, pages Lecture Notes in Math., Vol Springer, Berlin, [8] G. Shimura. On modular forms of half integral weight. Ann. of Math. (2, 97: , [9] J. B. Tunnell. A classical Diophantine problem and modular forms of weight 3/2. Invent. Math., 72(2: , Department of Mathematics, Brown University, Providence, RI address: david hansen@brown.edu Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA address: yusra.naqvi@gmail.com