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of correspondences between modular forms of half-integral weight and modular forms of even integral weight Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi s theta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 Hecke eigenforms 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 (11 E k (z := 1 2k B k σ k 1 (nq n, where B n in the nth Bernoulli number and q := e 2πinz For k 4, 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 (mod a and χ (mod a be primitive Dirichlet characters of conductors a, a, not both trivial The product character χχ has modulus aa If k is positive integer with χ( 1 = ( 1 k, then set ( (12 E k (χ, χ ; z := C(k, χ, χ + χ(n/dχ (dd k 1 q n, where C(k, χ, χ is zero unless a = 1, in which case C(k, χ, χ = 1 2 L(1 k, χχ, where L(s, χ = χ(nn s is the Dirichlet L-function of the character χ We have E k (χ, χ ; z M k (a, χ; see Chapter 4 of [4] 1 d n

2 2 DAVID HANSEN AND YUSRA NAQVI 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 (13 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 (14 θ(χ r ; z := n Z χ r (nn ν q n2 M 1/2+ν (4r 2, χ r χ ν 4, where now and in the sequel we fix χ r to be Dirichlet character of modulus r, where ν = 0, 1 is chosen such that χ( 1 = ( 1 ν (χ is called even or odd according to whether ν = 0 or ν = 1, respectively Here χ 4 is the real nonprincipal character of modulus 4 These theta functions are the simplest examples of modular forms of half-integral weight, and for weight 1/2 the space is spanned by them (cf [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 (15 A(nn s := L(s k + 1, ψχ k 4χ t b(tn 2 n s, where Re(s is large and χ t = ( t is the real nonprincipal character modulo t, then S t (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 and other 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 (16 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 11 Let χ r be an even Dirichlet character modulo r, and write χ r = χ p α 1 1 χ p α 2 2 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 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 (17 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, (18 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 12 Let χ r be an odd Dirichlet character modulo r, and write χ r = χ p α 1 1 χ p α 2 2 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 (19 g(z := 1 πi where χ c = p α c χ p α, then we have c r gcd(c,r/c=1 cf (czf(rz/cχ c ( 1, (110 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 11 and 12 directly, without appealing to the machinery of converse theorems 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 11 and 12, 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 Acknowledgments We extend our deep gratitude to Ken Ono and Karl Mahlburg for many helpful comments and discussions We would also like to thank the NSF for funding the REU at which this paper was written, and the referee for a careful reading and helpful comments 2 Multiplicativity Properties of Modular Form Coefficients Let f(z = n=0 a(nqn M k (N, χ There exists a sequence of operators, due to Hecke, which act as linear endomorphisms of M k (N, χ Furthermore, this space is spanned by functions which are simultaneous eigenfunctions of all the Hecke operators (once the socalled oldforms are eliminated; see chapter 6 of [2] for details For these functions, the following useful proposition holds Proposition 21 If f(z = n=0 a(nqn M k (N, χ is a simultaneous eigenfunction of all the Hecke operators with a(1 = 1, then for any positive integers m, n, we have a(ma(n = d (m,n χ(dd k 1 a(mn/d 2 Furthermore, we have an inverse of Proposition 22, which we shall refer to as Selberg inversion Proposition 22 If f(z = n=0 a(nqn M k (N, χ is a Hecke eigenform with a(1 = 1, then we have a(mn = for any positive integers m, n d (m,n µ(dχ(dd k 1 a(m/da(n/d,

5 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 5 Proof We have that d (m,n = d (m,n = dδ (m,n = D (m,n = a(mn 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 3 Proofs of Theorems 11 and 12 We begin by presenting the proof of the formula for the lift in Theorem 11 From the definition of F (z, we have F (z = n=0 b(nqn with (31 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 11 As above, the Shimura lift is given by (32 S 1 (F = A(nq n with the coefficients A(n defined by (33 A(nn s = L(s k + 1, χ r ψχ 2k 4 b(n 2 n s We also need the coefficients defined by (34 c d (nq n := f(dzf(rz/d = n=0 ( 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 (31 it is easy to see that (35 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

6 6 DAVID HANSEN AND YUSRA NAQVI 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, (36 (n m(n + m 4r Also, since m n (mod d and m n χ r (m = χ d (mχ r/d (m = m (n + dm r/d (mod r/d, we have that = χ d (nχ r/d ( n = χ r/d ( 1χ d (nχ r/d (n = χ r/d ( 1χ r (n, where the characters are as defined in the statement of Theorem 11 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, (35 becomes (37 b(n 2 = χ r (n χ d ( 1 ( m(n dm a m Z gcd(d,r/d=1 We now apply Proposition 22 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 (38 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 (39 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 (310 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

7 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 7 All Dirichlet series appearing converge absolutely for Re(s sufficiently large, and by an easy consideration of Euler products, the quotient of the L-functions simplifies to 1 2 k 1 s χ r (2ψ(2 Rewriting into q-series completes the proof of the identity for the lift The proof of the equation for the lift in Theorem 12 follows along the same lines, with appropriate changes due to the slightly different expression for the theta function The congruence condition reasoning following (35 does not change, and (37 becomes (311 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 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 (311 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 (see Sec 23 of [6] Proposition 31 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 (11 and f(z is a modular form of weight k + 2 Note that E 2 is not a modular form Using this proposition, we easily obtain g(z = 1 χ c ( 1cf (czf(rz/c πi = 1 2π 2 i 2 = 1 12πi = 1 12πi c r gcd(c,r/c=1 c r gcd(c,r/c=1 c r gcd(c,r/c=1 c r gcd(c,r/c=1 χ c ( 1f(rz/c z f(cz χ c ( 1f(rz/c( f c (z + ke 2 (zf(cz χ c ( 1f(rz/c f c (z, where f c (z is a modular form of weight k+2 and level cn The sum involving E 2 s vanishes due to cancellation in characters, namely χ c ( 1 = χ r/c ( 1 To complete the proofs of Theorems 11 and 12, it suffies to compute the levels of the relevant lifted forms 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 22 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

8 8 DAVID HANSEN AND YUSRA NAQVI Γ 0 (N r 2 By Proposition 25 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 (312 α := N r 2 1 for = 0, 1, 2,, r 1 For convenience, we define the slash operator for γ GL + 2 (Q by (313 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 33 of [3] and defining τ(χ r := r 1 m=0 χ r(me 2πim/r, we first write g χr (z as a sum of linear transforms, where we have set r 1 g χr (z = r 1 τ(χ r r 1 = r 1 τ(χ r (314 γ v := It then follows that r 1 g χr (z α = r 1 τ(χ r r 1 = r 1 τ(χ r r 1 = r 1 τ(χ r r 1 = r 1 τ(χ r = g χr (z ( 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

9 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 9 that is not a cusp form Let F (z = θ(χ r ; zf(4rz, and recall that (315 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 (316 f(δz = δ k c d k/2 δ 0 a b f(z k 0 1 c d Let δ = gcd(c, δ We have that there exists an integer y such that (δ/δ (cy + d, and so we get ( ( a b f(δz = δ k c 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 Inserting this into the definition of g(z gives ( a b ( a b g(z = χ 2k c d δ ( 1f(δzf(rz/δ 2k c d = δ r gcd(δ,r/δ=1 δ r gcd(δ,r/δ=1 ( δ k ( r/δ χ δ ( 1 (δ, c (r/δ, c kf ( δ 2 z y δ f ( δ 2 z y 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 (317 g γ (z = χ δ ( 1 f (r, c δ δ δ r gcd(δ,r/δ=1 We now consider ( a (318 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 a b 2k 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 b h(z = g 2k c d γ (z 2 k 1 g γ (2z or = g γ (z 2 k 1 g γ ((z x/2 δ, This

10 10 DAVID HANSEN AND YUSRA NAQVI 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 (319 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 4 Examples and Applications In this section, we present some examples illustrating Theorems 11 and 12 Throughout this section, χ r will exclusively denote the real nonprincipal character of modulus r Example 1 We begin with Theorem 11, 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 11, we factor χ 12 = χ 4 χ 3 and 12 = to obtain (41 g(z = f(zf(12z f(3zf(4z Because χ 12 (2 = 0, the second term in (17 vanishes and we have (42 ( 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 17 + η(15zη(20z χ 12 Example 2 We now illustrate Theorem 12 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 (43 f i (z = q + a i q 2 + ( a i q 3 + with a 1 = and a 2 = Hence, (44 (z 2 = f 1(z f 2 (z To apply Theorem 12, we factor 15 = 3 5 and χ 15 = χ 3 χ 5 to obtain (45 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 χ15 (z+2 24 g χ15 (2z A similar formula holds for f 2 (z

11 SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS 11 References [1] B A Cipra On the Shimura lift, après Selberg J Number Theory, 32(1:58 64, 1989 [2] H Iwaniec Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics American Mathematical Society, Providence, RI, 1997 [3] N Koblitz Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics Springer-Verlag, New York, second edition, 1993 [4] T Miyake Modular forms Springer Monographs in Mathematics Springer-Verlag, Berlin, english edition, 2006 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: , 2000 [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 [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 627 Springer, Berlin, 1977 [8] G Shimura On modular forms of half integral weight Ann of Math (2, 97: , 1973 [9] J B Tunnell A classical Diophantine problem and modular forms of weight 3/2 Invent Math, 72(2: , 1983 Department of Mathematics, Brown University, Providence, RI address: david hansen@brownedu Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA address: yusranaqvi@gmailcom