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 Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to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.
1. Introduction and statement of results
Let SL2 (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 Γ0 (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(d)f (z) for all γ = ac db ∈
SL2 (Z) with c ≡ 0 (mod N ). Let Mk (N, ψ) denote the finite-dimensional vector space of
modular forms of weight k on Γ0 (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 Sk (N, ψ) denote the subspace of Mk (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)
Ek (z) := 1 −
∞
2k X
σk−1 (n)q n ,
Bk n=1
where Bn in the nth Bernoulli number and q := e2πinz . For k ≥ 4, these functions represent
the simplest modular forms of weight k, and they lie in Mk (1, 1). More general Eisenstein
series can be defined as follows. Let χ (mod a) and χ0 (mod a0 ) be primitive Dirichlet
characters of conductors a, a0 , not both trivial. The product character χχ0 has modulus aa0 .
If k is positive integer with χ(−1) = (−1)k , then set
(1.2)
Ek (χ, χ0 ; z) := C(k, χ, χ0 ) +
∞ X
X
n=1
χ(n/d)χ0 (d)dk−1 q n ,
d|n
where C(k, χ, χ0 ) is zero unless a = 1, in which case C(k, χ, χ0 ) = 21 L(1 − k, χχ0 ), where
P
−s is the Dirichlet L-function of the character χ. We have E (χ, χ0 ; z) ∈
L(s, χ) = ∞
k
n=1 χ(n)n
0
0
Mk (aa , χχ ); see Chapter 4 of [4].
1
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
(1.3)
f (γz) = ψ(d)
2k+1
c
d
−2k−1
(cz + d)k+1/2 f (z)
d
for all γ ∈ Γ0 (4N ), where d is 1 or
i for odd d according to whether d ≡ 1 (mod 4) or
d ≡ 3 (mod 4), respectively, and dc is Shimura’s extension of the Jacobi symbol. As above,
Mk+1/2 (N, ψ) denotes the finite-dimensional vector space of weight k + 1/2 modular forms,
and Sk+1/2 (N, ψ) denotes its subspace of cusp forms.
Define theta functions
(1.4)
θ(χr ; z) :=
X
2
χr (n)nν q n ∈ M1/2+ν (4r2 , χr χν4 ),
n∈Z
where now and in the sequel we fix χr 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 χ4 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 (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) =
Sk+1/2 (4N, ψ), where k is a positive integer. If numbers A(n) are defined by
(1.5)
∞
X
A(n)n−s := L(s − k + 1, ψχk4 χt )
∞
X
P∞
n=1 b(n)q
b(tn2 )n−s ,
n=1
n=1
t
•
where Re(s) is large and χt =
is the real nonprincipal character modulo t, then
P∞
n
St (f )(z) := n=1 A(n)q ∈ M2k (2N, ψ 2 ). Moreover, if k > 1, then St (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) ∈ Mk (1, 1), Selberg found that
f (4z)θ(1; z) ∈ Mk+1/2 (4, 1) lifts to f (z)2 − 2k−1 f (2z)2 ∈ M2k (2, 1). Later on, Cipra [1]
generalized Selberg’s work by proving the following result.
n
∈
SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS
3
Theorem (Cipra). If f (z) ∈ Sk (N, ψ) is a newform, and θ(χr ; z) is the theta function of
an even Dirichlet character of prime power modulus r = pm , then if we define g(z) :=
f (z)f (pµ z), the Shimura lift S1 of f (4pµ z)θ(χr ; z) is
gχr (z) − 2k−1 χr (2)ψ(2)gχr (2z),
(1.6)
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 χr be an even Dirichlet character modulo r, and write χr = χpα1 χpα2 ...χpαj
1
2
j
α
as the factorization of χr into Dirichlet characters modulo prime powers pα1 1 , pα2 2 , ..., pj j
α
with pα1 1 pα2 2 ...pj j = r. Let f (z) ∈ Mk (N, ψ) be a Hecke eigenform, and set F (z) :=
θ(χr ; z)f (4rz) ∈ Mk+1/2 (4N 0 r2 , ψχr χk4 ) with N 0 = N/ gcd(N, r). If
g(z) :=
(1.7)
X
f (dz)f (rz/d)χd (−1),
d|r
gcd(d,r/d)=1
where χd =
(1.8)
Q
α
pj j ||d
χpαj , then we have
j
S1 (F )(z) = gχr (z) − 2k−1 χr (2)ψ(2)gχr (2z) ∈ M2k (2N 0 r2 , ψ 2 χ2r ).
Here gχ 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 χr be an odd Dirichlet character modulo r, and write χr = χpα1 χpα2 ...χpαj
1
2
j
α
as the factorization of χr into Dirichlet characters modulo prime powers pα1 1 , pα2 2 , · · · , pj j
α
with pα1 1 pα2 2 · · · pj j = r. If F (z) := θ(χr ; z)f (4rz) ∈ Mk+3/2 (4N 0 r2 , ψχr χk+1
4 ), where f (z) ∈
Mk (N, ψ) is a Hecke eigenform, and
X
1
cf 0 (cz)f (rz/c)χc (−1),
(1.9)
g(z) :=
πi
c|r
gcd(c,r/c)=1
where χc =
(1.10)
Q
α
pj j ||c
χpαj , then we have
j
S1 (F )(z) = gχr (z) − 2k χr (2)ψ(2)gχr (2z) ∈ M2k+2 (2N 0 r2 , ψ 2 χ2r ),
where gχ 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
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 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
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.
Acknowledgments
We extend our gratitude to Ken Ono, Sharon Garthwaite and Karl Mahlburg for many
illuminating 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
P∞
Let f (z) = n=0 a(n)q n ∈ Mk (N, χ). There exists a sequence of operators, due to Hecke,
which act as linear endomorphisms of Mk (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.
n
Proposition 2.1. If f (z) = ∞
n=0 a(n)q ∈ Mk (N, χ) is a simultaneous eigenfunction of all
the Hecke operators with a(1) = 1, then for any positive integers m, n, we have
P
X
a(m)a(n) =
χ(d)dk−1 a(mn/d2 ).
d|(m,n)
Furthermore, we have an “inverse" of Proposition 2.2, which we shall refer to as Selberg
inversion.
Proposition 2.2. If f (z) =
then we have
P∞
n=0 a(n)q
a(mn) =
n
X
d|(m,n)
for any positive integers m, n.
∈ Mk (N, χ) is a Hecke eigenform with a(1) = 1,
µ(d)χ(d)dk−1 a(m/d)a(n/d),
SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS
5
Proof. We have that
X
µ(d)χ(d)dk−1 a(m/d)a(n/d)
d|(m,n)
=
X
d|(m,n)
=
X
µ(d)χ(d)dk−1
χ(δ)δ k−1 a(mn/(dδ)2 )
δ|(m/d,n/d)
X
µ(d)χ(dδ)(dδ)k−1 a(mn/(dδ)2 )
dδ|(m,n)
=
X
X
D|(m,n)
µ(d) χ(D)Dk−1 a(mn/D2 )
d|D
= a(mn).
3. Proofs of Theorems 1.1 and 1.2
We begin by presenting the proofPof the formula for the lift in Theorem 1.1. From the
n
definition of F (z), we have F (z) = ∞
n=0 b(n)q with
b(n) =
(3.1)
X
χr (m)a
m∈Z
P∞
where f (z) =
lift is given by
n=0 a(n)q
n
n − m2 4r
,
is as in the statement of Theorem 1.1. As above, the Shimura
S1 (F ) =
(3.2)
∞
X
A(n)q n
n=1
with the coefficients A(n) defined by
∞
X
(3.3)
A(n)n
−s
= L(s − k +
1, χr ψχ2k
4 )
∞
X
b(n2 )n−s .
n=1
n=1
We also need the coefficients defined by
(3.4)
∞
X
n
cd (n)q := f (dz)f (rz/d) =
n=0
∞ X
X
a(m)a
n − dm n=0 m∈Z
r/d
qn.
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(n2 ) =
X
m∈Z
χr (m)a
(n − m)(n + m) 4r
.
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|(n2 − m2 ). Let
gcd((n − m)/2, r) = d. Thus, m ≡ n (mod 2d) and m ≡ −n (mod 2r/d). Now suppose
gcd(d, r/d) = d0 > 1. This implies that m ≡ n ≡ −n ≡ 0 (mod 2d0 ), so d| gcd(m, r) and
6
DAVID HANSEN AND YUSRA NAQVI
so χr (m) = 0. Therefore, we only consider the cases in which gcd(d, r/d) = 1. We have
m = n + 2dm0 for some m0 ∈ Z, so n − m = −2dm0 and n + m = 2n + 2dm0 . Thus,
(n − m)(n + m)
−m0 (n + dm0 )
=
.
4r
r/d
(3.6)
Also, since m ≡ n (mod d) and m ≡ −n (mod r/d), we have that
χr (m) = χd (m)χr/d (m)
= χd (n)χr/d (−n) = χr/d (−1)χd (n)χr/d (n)
= χr/d (−1)χr (n),
where the characters are as defined in the statement of Theorem 1.1. Since χr (−1) =
χr/d (−1)χd (−1) = 1, we have χd (−1) = χr/d (−1) = ±1. Thus, by changing the variable
m0 to −m, (3.5) becomes
X
b(n2 ) = χr (n)
(3.7)
χd (−1)
X m(n − dm) a
r/d
m∈Z
d|r
gcd(d,r/d)=1
.
We now apply Proposition 2.2 to get
X
b(n2 ) = χr (n)
χd (−1)
X
χd (−1)
X
µ(δ)ψ(δ)δ k−1 a
X
µ(δ)ψ(δ)δ k−1
χd (−1)
d|r
gcd(d,r/d)=1
X
δr/d
a
δr/d
X m n/δ − dm a
r/d
m∈Z
δ|n
d|r
gcd(d,r/d)=1
= χr (n)
m n − dm X
m∈Z δ|(m,n)
d|r
gcd(d,r/d)=1
= χr (n)
X
a
r/d
µ(δ)ψ(δ)δ k−1 cd (n/δ).
δ|n
Rewriting these formulas as Dirichlet series immediately gives
(3.8)
∞
X
b(n2 )n−s =
n=1
X
d|r
gcd(d,r/d)=1
χd (−1)
∞ X
X
µ(δ)ψ(δ)χr (δ)δ k−1 χr (n/δ)cd (n/δ)n−s ,
n=1 δ|n
and we can easily pull out the reciprocal of a Dirichlet L-function to produce
(3.9)
∞
X
b(n2 )n−s =
n=1
1
L(s − k + 1, ψχr )
X
χd (−1)
d|r
gcd(d,r/d)=1
∞
X
χr (n)cd (n)n−s .
n=1
2
Multiplying by L(s − k + 1, χr ψχ2k
4 ) = L(s − k + 1, χr ψχ4 ), as in the definition of the
Shimura lift, we have
(3.10)
∞
X
n=1
A(n)n−s =
L(s − k + 1, χr ψχ24 )
L(s − k + 1, χr ψ)
X
d|r
gcd(d,r/d)=1
χd (−1)
∞
X
n=1
χr (n)cd (n)n−s .
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 −
2k−1−s χr (2)ψ(2). 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)
X
b(n2 ) = χr (n)
χr/d (−1)
X m(n − dm) a
m∈Z
d|r
gcd(d,r/d)=1
r/d
(n − 2dm).
Recall that χr is odd here, so χr (−1) = χr/d (−1)χd (−1) = −1, and so we have that
χd (−1) = −χr/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 (see Sec. 2.3 of [6]).
Proposition 3.1. Let f (z) be a modular form of weight k on some subgroup of SL2 (Z). Then
1 d
˜
2πi dz f (z) = (f (z) + kE2 (z)f (z))/12, where E2 (z) is the Eisenstein series defined in (1.1)
and f˜(z) is a modular form of weight k + 2.
Note that E2 is not a modular form. Using this proposition, we easily obtain
X
1
χc (−1)cf 0 (cz)f (rz/c)
g(z) =
πi
c|r
gcd(c,r/c)=1
=
=
=
1
2π 2 i2
1
12πi
1
12πi
X
χc (−1)f (rz/c)
c|r
gcd(c,r/c)=1
X
∂
f (cz)
∂z
χc (−1)f (rz/c)(f˜c (z) + kE2 (z)f (cz))
c|r
gcd(c,r/c)=1
X
χc (−1)f (rz/c)f˜c (z),
c|r
gcd(c,r/c)=1
where f˜c (z) is a modular form of weight k +2 and level cN . The sum involving E2 ’s vanishes
due to cancellation in characters, namely χc (−1) = −χr/c (−1).
To complete the proofs of Theorems 1.1 and 1.2, it suffies to compute the levels of the
relevant lifted forms. Because g(z) lies in the space M2k (N 0 r, ψ 2 ), it is easy to see by the
general theory of twists (see [6], Sec. 2.2) that gχr (z) ∈ M2k (N 0 r3 , χ2r ψ 2 ). However, we can
in fact show that gχr (z) lies in the space M2k (N 0 r2 , χ2r ψ 2 ). To do this, we demonstrate the
invariance of gχr (z) under a complete set of representatives of right cosets of Γ0 (N 0 r3 ) in
8
DAVID HANSEN AND YUSRA NAQVI
Γ0 (N 0 r2 ). By Proposition 2.5 of [2], we have that [Γ0 (N 0 r2 ) : Γ0 (N 0 r3 )] = r, so such a set
of representatives is given by
αj :=
(3.12)
1
0
jN 0 r2 1
for j = 0, 1, 2, ..., r − 1. For convenience, we define the slash operator for γ ∈ GL+
2 (Q) by
f (z) |k γ := f (γz)(cz + d)−k (det γ)k/2 .
(3.13)
With this notation, we need to show gχr (z) |k αj = gχr (z) for j = 0, 1, 2, ..., r − 1. Using
P
2πim/r , we first write
Proposition 17 in Sec. 3.3 of [3] and defining τ (χr ) := r−1
m=0 χr (m)e
gχr (z) as a sum of linear transforms,
gχr (z) = r−1 τ (χr )
= r−1 τ (χr )
r−1
X
v=0
r−1
X
χ̄r (v)g(z − v/r)
χ̄r (v)g(z) | γv ,
v=0
where we have set
γv :=
(3.14)
1 −v/r 0
1
.
It then follows that
gχr (z) | αj = r
−1
τ (χr )
r−1
X
χ̄r (v)g(z) γv αj
k
v=0
=r
−1
τ (χr )
= r−1 τ (χr )
= r−1 τ (χr )
r−1
X
v=0
r−1
X
v=0
r−1
X
1 −v/r
χ̄r (v)g(z) k
0
1
1 − jvN 0 r
χ̄r (v)g(z) k
jN 0 r2
1
0
0
2
jN r 1
−jN 0 v 2 1 −v/r
1
jvN 0 r + 1 0
χ̄r (v)g(z) k γv
v=0
= gχr (z).
Note that the first matrix in the fourth line is in Γ0 (N 0 r) with d ≡ 1 (mod N 0 ), 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 S1 (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) ∈ Mk (1, 1) is a Hecke eigenform
that is not a cusp form. Let F (z) = θ(χr ; z)f (4rz), and recall that
(3.15)
g(z) =
X
δ|r
gcd(δ,r/δ)=1
χδ (−1)f (δz)f (rz/δ).
SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS
9
Also recall that if 2|r, then we have that S1 (F )(z) = gχ (z). If r is odd, then we define
h(z) := g(z) − 2k−1 g(2z), noting that in this case, the Shimura lift is hχ (z). We shall
proceed by computing the Fourier expansions of g(z) and h(z) around a complete set of
cusps.
Let gγ (z) denote g(z) |2k γ. For any γ = ac db ∈ SL2 (Z), we have
b
δ 0 a b
.
= δ −k/2 f (z) c d
c d
k 0 1
a
f (δz) (3.16)
k
δ0
Let = gcd(c, δ). We have that there exists an integer y such that (δ/δ 0 )|(cy + d), and so
we get
δ 0 (ay + b) δ 0 −y b
aδ/δ 0
= δ −k/2 f (z) 0
0
c/δ δ (cy + d)/δ 0 δ/δ 0
c d
k
a
f (δz) k
= (δ/δ 0 )−k f
δ 02 z − δ 0 y δ
.
Inserting this into the definition of g(z) gives
a
g(z) 2k
b
=
c d
=
a
X
χδ (−1)f (δz)f (rz/δ)
2k
δ|r
gcd(δ,r/δ)=1
δ
X
χδ (−1)
δ|r
gcd(δ,r/δ)=1
(δ, c)
b
c d
−k r/δ
(r/δ, c)
−k δ 02 z − y 0 δ 002 z − y 00 f
δ
f
δ
,
where δ 0 is as before, δ 00 = (r/δ, c) and y 0 and y 00 are integers that depend on δ. This
transforms into
r −k δ 02 z − y 0 δ 002 z − y 00 X
f
χδ (−1)
f
.
(3.17)
gγ (z) =
(r, c)
δ
δ
δ|r
gcd(δ,r/δ)=1
We now consider
(3.18)
2 0a b b
= 2−k g(z) ,
c d
c d
2k 0 1
a
g(2z) 2k
which gives us that g(2z) |2k ac db = gγ (2z) if c is even or gγ ((z − x)/2) if c is odd, where
x is some integer that depends on d. This yields
a
h(z) 2k
b
= gγ (z) − 2k−1 gγ (2z) or
c d
= gγ (z) − 2−k−1 gγ ((z − x)/2).
Thus, in all cases,
the constant term of the Fourier expansion is a constant multiple of
P
r −k
( (r,c)
) a(0)2 χδ (−1), and hence this term vanishes if and only if f is a cusp form or
(3.19)
X
δ|r
gcd(δ,r/δ)=1
χδ (−1) = 0.
10
DAVID HANSEN AND YUSRA NAQVI
In particular, this sum vanishes if and only if χr 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(χr ; 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 1.1 and 1.2. Throughout
this section, χr will exclusively denote the real nonprincipal character of modulus r.
Example 1. We begin with Theorem 1.1, by defining f (z) := η(z)5 /η(5z), where
η(z) := q 1/24
Y
(1 − q n )
n>0
is the Dedekind eta function. This function is in the space M2 (5, χ5 ), and in fact we have
f (z) = −5E2 (1, χ5 ; z). We compute the Shimura lift of f (48z)η(24z) = 21 f (48z)θ(χ12 ; z).
To utilize Theorem 1.1, we factor χ12 = χ4 χ3 and 12 = 22 · 3 to obtain
g(z) = f (z)f (12z) − f (3z)f (4z).
(4.1)
Because χ12 (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 η(3z)5 η(4z)5 S1
=
−
= 25q 7 +50q 11 +100q 13 +150q 17 +...
η(240z)
η(5z)η(60z) η(15z)η(20z) χ12
Example 2. We now illustrate Theorem 1.2 by computing the lift of ∆(60z)2 θ(χ15 ; 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
f1 (z) and f2 (z), of weight 24 and level 1 have Fourier expansions
fi (z) = q + ai q 2 + (195660 − 48ai )q 3 + ...
√
with a1 = 540 + 12 144169 and a2 = 540 − 12 144169. Hence,
(4.3)
√
f1 (z) − f2 (z)
√
.
24 144169
To apply Theorem 1.2, we factor 15 = 3 · 5 and χ15 = χ3 χ5 to obtain
1 0
f1 (z)f1 (15z) − 3f10 (3z)f1 (5z) + 5f10 (5z)f1 (3z) − 15f10 (15z)f1 (z)
(4.5)
g(z) =
πi
and hence S1 (θ(χ15 ; z)f1 (60z))(z) = gχ15 (z)+224 gχ15 (2z). A similar formula holds for f2 (z).
(4.4)
∆(z)2 =
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):293–307, 2000.
SHIMURA LIFTS OF MODULAR FORMS WITH THETA FUNCTIONS
11
[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 27–67. Lecture Notes in Math., Vol. 627.
Springer, Berlin, 1977.
[8] G. Shimura. On modular forms of half integral weight. Ann. of Math. (2), 97:440–481, 1973.
[9] J. B. Tunnell. A classical Diophantine problem and modular forms of weight 3/2. Invent. Math.,
72(2):323–334, 1983.
Department of Mathematics, Brown University, Providence, RI 02912
E-mail address: david hansen@brown.edu
Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081
E-mail address: yusra.naqvi@gmail.com