Shimura lifts of half-integral weight modular forms - Department of ...

2 DAVID HANSEN AND YUSRA NAQVI
where C(k, χ1, χ2) is zero unless a1 = 1, in which case C(k, χ1, χ2) = 1
2L(1 − k, χ),
where L(s, χ) = ∞ n=1 χ(n)n−s is the Dirichlet L-function of the character χ. We have
Ek(χ1, χ2; z) ∈ Mk(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+1 (1.3) f(γz) = ψ(d) ɛ
d
−2k−1
d (cz + d) k+1/2 f(z)
for all γ ∈ Γ0(4N), where ɛd 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,
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(χ; z) :=
χ(n)n ν q n2
∈ M1/2+ν(4r 2 , χχ ν 4)
n∈Z
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) =
∞
n=1 b(n)qn ∈ Sk+1/2(4N, ψ), where k is a positive integer. If numbers A(n) are defined by
∞
(1.5)
A(n)n −s := L(s − k + 1, ψχ k ∞
4χt) b(tn 2 )n −s ,
n=1
where χt =
t
• is the usual Kronecker character modulo t, then St(f)(z) := ∞ n=1 A(n)qn ∈
M2k(2N, ψ2 ). 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 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It turns out that much earlier, in unpublished work, Selberg worked out such an explicit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) ∈ Mk(1, 1), Selberg found that
f(4z)θ(1; z) ∈ M k+1/2(4, 1) lifts to f(z) 2 − 2 k−1 f(2z) 2 ∈ M2k(2, 1). Later on, Cipra [1]
generalized Selberg’s work by proving the following result.
n=1