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

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(n)qn ∈ Mk(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(d)d k−1 a(mn/d 2
) q m .
It is known, by work of Hecke, that these operators are linear endomorphisms on Mk(N, ψ).
They also map cusp forms to cusp forms. Furthermore, there exist modular forms f(z) ∈
Mk(N, ψ) which are simultaneous eigenfunctions of all the Hecke operators; in other words,
they satisfy
(2.2) f(z) | T ψ n = λ(n)f(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(n)a(m) =
d|(m,n)
ψ(d)d 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(n)qn ∈ Mk(N, χ) is a simultaneous eigenfunction of
all the Hecke operators T ψ n with a(1) = 1, then for any positive integers m, n, we have
a(m)a(n) =
d|(m,n)
ψ(d)d k−1 a(mn/d 2 ).