HAL Id: hal-01278411
https://hal.archives-ouvertes.fr/hal-01278411
Submitted on 24 Feb 2016
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Fourier coecients of cusp forms of half-integral weight
W Kohnen, Y.-K Lau, J Wu
To cite this version:
W Kohnen, Y.-K Lau, J Wu. Fourier coecients of cusp forms of half-integral weight. Mathematische
Zeitschrift, Springer, 2013, 273 (1-2), pp.29-41. �10.1007/s00209-012-0994-z�. �hal-01278411�
FOURIER COEFFICIENTS OF CUSP FORMS OF
HALF-INTEGRAL WEIGHT
W. KOHNEN, Y.-K. LAU & J. WU
Abstract. Let f be a cusp form of half integral weight whose Fourier coefficients
a
f
(n) are all real. We study the sign change problem of a
f
(n), when n runs
over some specific sets of integers. Lower bounds of the best possible order of
magnitude are established for the number of those coefficients that have the same
signs. These give an improvement on some recent results of Bruinier & Kohnen
[2] and Kohnen [13].
1. Introduction
Owing to different reasons, the problem of sign changes of Hecke eigenvalues of
integral weight cusp forms has attracted many attentions [10, 15, 7, 14, 19, 17]. One
motivation is to delve the analogy with (real) Dirichlet characters. Real Dirichlet
characters admit only ±1 other than 0; however these eigenvalues (when properly
normalized) vary in the range from −2 to 2. A reasonable parallel one may consider
is the pattern of the signs. Such an investigation has a long history in the case of
real characters, like the problem of the least quadratic non-residue. The work [17]
provides a comprehensive discussion in the context of modular forms. In light of the
theory of half integral weight forms in Shimura [23], Waldspurger [24], Kohnen-
Zagier [16] and Kohnen [11, 12], etc, the half integral weight forms are closely
related to integral weight cusp forms and hence it is naturally important to study
the analogous sign-change problems. The case of half integral weight cusp forms,
although looking like a formal extension, is somehow more subtle. A reason is that
the Fourier coefficients of a half integral weight cuspidal Hecke eigenform in general
are not plainly multiplicative (cf. [4, page 783]). In [2], Bruinier & Kohnen studied
the sign changes of the Fourier coefficient a
f
(n) of a half integral weight cusp form
f for specific sequences of integers n, which also stimulates this work.
Throughout we let k > 1 be an integer and assume N > 4 to be divisible by 4.
Fix any Dirichlet character χ modulo N. We write S
k+1/2
(N, χ) for the space of
cusp forms of weight k + 1/2 for the group Γ
0
(N) with character χ (cf. [23]). The
space S
3/2
(N, χ) contains unary theta functions. Let S
∗
3/2
(N, χ) be the orthogonal
complement with respect to the Petersson scalar product of the subspace generated
by these theta functions (cf. [23, Section 4] and [3, Section 4]). For convenience, we
put S
∗
k+1/2
(N, χ) = S
k+1/2
(N, χ) when k > 2. Each f ∈ S
∗
k+1/2
(N, χ) has a Fourier
Date: April 28, 2015.
2000 Mathematics Subject Classification. 11F37, 11F30, 11N25.
Key words and phrases. Forms of half-integer weight, Fourier coefficients of automorphic forms,
B-free numbers.
1
2 W. KOHNEN, Y.-K. LAU & J. WU
expansion
(1.1) f(z) =
X
n>1
a
f
(n)e
2πinz
(z ∈ H),
on the complex upper half plane H. Let t > 1 be a squarefree integer. The Shimura
correspondance [23] lifts f to a cusp form f
t
of weight 2k for the group Γ
0
(N/2) with
character χ
2
. Also it gives a vital relation between their Fourier coefficients,
(1.2) a
f
t
(n) :=
X
d|n
χ
t,N
(d)d
k−1
a
f
t
n
2
d
2
,
where χ
t,N
denotes the character
χ
t,N
(d) := χ(d)
(−1)
k
t
d
and
(1.3) f
t
(z) :=
X
n>1
a
f
t
(n)e
2πinz
(z ∈ H).
(f
t
is called the Shimura lift of f associated to t.) Furthermore if f is a Hecke
eigenform, then so is the Shimura lift. In fact, in this case
f
t
= a
f
(t)f(1.4)
where f is a normalized (a
f
(1) = 1) Hecke eigenform independent of t.
Let f ∈ S
∗
k+1/2
(N, χ
0
) be a cusp form with trivial character χ
0
, squarefree level and
real coefficients a
f
(n). Suppose that f lies in the plus space, that is, a
f
(n) = 0 when
(−1)
k
n ≡ 2, 3 (mod 4), see [16, 12]. Bruinier & Kohnen [2] gave the conjectures
(1.5) lim
x→∞
|{n 6 x : a
f
(n) ≷ 0}|
|{n 6 x : a
f
(n) 6= 0}|
=
1
2
and
(1.6) lim
x→∞
|{|d| 6 x : d fundamental discriminant, a
f
(|d|) ≷ 0}|
|{|d| 6 x : d fundamental discriminant, a
f
(|d|) 6= 0}|
=
1
2
with empirical evidence, which may be, however, out of the present reach. Alter-
natively, they considered the change in signs of a
f
(n) when n runs over specific sets
of integers, such as {tn
2
}
n∈N
, {tp
2ν
}
ν∈N
and {tn
2
t
}
t squarefrees
. Here t is a positive
squarefree integer such that a
f
(t) 6= 0, p denotes any fixed prime and n
t
is an integer
determined by t (cf. [2, Theorems 2.1 and 2.2]). Amongst other things, their ap-
proach comprises a well-known robust analytic tool - Landau’s theorem on Dirichlet
series.
Our first result gives an improvement to [2, Theorem 2.1] and [13, Theorem],
exploiting tools in analytic number theory in connection with Rankin-Selberg L-
functions.
Theorem 1. Let k > 1 be an integer, N > 4 an integer divisible by 4 and χ
be a Dirichlet character modulo N. Suppose that f ∈ S
∗
k+1/2
(N, χ) and t > 1 is
a squarefree integer such that a
f
(t) 6= 0. Assume that the sequence {a
f
(tn
2
)}
n∈N
is
real. Then {a
f
(tn
2
)}
n∈N
has infinitely many sign changes. More specifically there is a
small constant α = α(f, t) > 0 such that for all sufficiently large x, i.e. x > x
0
(f, t),
FOURIER COEFFICIENTS OF CUSP FORMS OF HALF-INTEGRAL WEIGHT 3
a
f
(tp
2
) has (at least) one sign-change when p runs through primes in the interval
[x
α
, x].
Our proof shows an alternative (other than [13]) to remove the hypothesis on the
non-vanishing of L(s, χ
t,N
) on (0, 1) (Chowla’s conjecture if χ
t,N
is quadratic), see
Theorem 2.1 of [2]. This conjecture asserts that L(s, χ
t,N
) has no zeros in the interval
(0, 1). Kohnen [13] removed the hypothesis by refining the argument of [2]. However
as in [2], the method did not produce a quantitative result. In this regard Theorem
1 goes further and in fact, the proof here applies to the finer sequence of primes, that
is, we narrow down to the infinitely many sign changes in {a
f
(tp
2
)}
p primes
(instead
of {a
f
(tn
2
)}
n∈N
).
The form f in Theorem 1 is not assumed to be a Hecke eigenform. Imposing this
assumption, if the Shimura lift f
t
, or equivalently f in (1.4) when a
f
(t) 6= 0, is not
of CM type (see Remark 1), we can tell more in the next theorems. A salience
now is the retrieve of multiplicativity. More precisely, for any fixed (squarefree) t
and Hecke eigenform f, the arithmetic function n 7→ a
f
(tn
2
) is multiplicative in the
following sense (cf. [23, (1.18)]):
(1.7) a
f
(tm
2
)a
f
(tn
2
) = a
f
(t)a
f
(tm
2
n
2
) if (m, n) = 1.
The condition of a Hecke eigenform f is indispensable in our argument, as we start
with (1.7). These results are clearly the best possible in order of magnitude.
Theorem 2. Let k > 1 be an integer, N > 4 an integer divisible by 4 and χ be a real
Dirichlet character modulo N. Suppose that f ∈ S
∗
k+1/2
(N, χ) is a Hecke eigenform
and t > 1 is a squarefree integer such that a
f
(t) 6= 0. Assume that its Shimura lift
is not of CM type. Then we have
(1.8)
X
n6x, n is squarefree
(n,Nt)=1, a
f
(tn
2
)≷ 0
1
f,t
x
for x > x
0
(f, t). If N/2 is squarefree, the assumption of a non-CM Shimura lift will
automatically hold and hence can be omitted.
Remark 1. A Hecke eigenform f is of CM type if there exists a non-trivial Dirichlet
character ϕ such that λ
f
(p) = ϕ(p)λ
f
(p) for all primes p in a set of primes of density
1 (see [22, Section 7.2]). Here and in the sequel the Vinogradov symbol f(x) g(x)
(or g(x) f(x)) means |f (x)| 6 Cg(x) for all sufficiently large x > x
0
, where C is
a positive constant. We also write
∗
or
∗
to stress the dependence of the implied
constants on ∗.
The following result refines [2, Theorem 2.2].
Theorem 3. Let k > 1 be an integer, N > 4 an integer divisible by 4 and χ be a real
Dirichlet character modulo N. Suppose that f ∈ S
∗
k+1/2
(N, χ) is a Hecke eigenform,
and t is a positive squarefree integer for which a
f
(t) 6= 0. For any prime p - N,
define θ
f
(p) ∈ [0, π] by the relation λ
f
(p) = 2 cos θ
f
(p) where λ
f
(p)p
k−1/2
is the p-th
Fourier coefficient of f in (1.4). We have the following results where ε = 1 or −1
in Case (ii)-(iv).
Case (i). θ
f
(p) = 0. Then a
f
(tp
2ν
) has the same sign as a
f
(t), for all ν > 0.
4 W. KOHNEN, Y.-K. LAU & J. WU
Case (ii). θ
f
(p) = π. Then
(1.9)
X
ν6x
εa
f
(tp
2ν
)> 0
1 ∼
1
2
x (x → ∞).
Case (iii). θ
f
(p)/(2π) = m/n ∈ (0, 1/2) is rational with (m, n) = 1. Then
(1.10)
X
ν6x
εa
f
(t)
−1
a
f
(tp
2ν
)> (
√
3/2−1/
√
p)p
ν(k−1/2)
/ sin θ
f
(p)
1 >
1
n
x + O
f
(1) (x → ∞).
Case (iv). θ
f
(p)/(2π) ∈ (0, 1/2) is irrational. Then
(1.11)
X
ν6x
εa
f
(t)
−1
a
f
(tp
2ν
)> (c−1/
√
p)p
ν(k−1/2)
/ sin θ
f
(p)
1 >
1
2
−
arcsin c
π
x+o(x) (x → ∞)
for any c ∈ (1/
√
p, 1).
Remark 2. Cases (i) and (ii) can happen for at most finitely many primes p only.
Indeed if we let K
f
be the number field generated by all the Fourier coefficients a
f
(n)
of f, then the total number of primes p for which 0 6= cos θ
f
t
(p) ∈ Q cannot exceed
r where 2
r
k[K
f
: Q], i.e., 2
r
is the greatest power of two that divides the degree
of K
f
over Q. This follows from the proof of [2, Remark 2.3]: if 0 6= λ
f
(p) ∈ Q,
then
√
p ∈ K
f
for a
f
(p) = λ
f
(p)p
k−1/2
, whence our assertion follows by the fact
[Q(
√
p
1
, . . . ,
√
p
t
) : Q] = 2
t
for distinct primes p
1
, . . . , p
t
.
For other values α ∈ (−2, 2), the Sato-Tate conjecture suggests that λ
f
(p) = α
holds only for a zero density of primes p. When α = 0, it is shown to be true in
Serre [22]; though in this case (α = 0) and f is a non CM form, one may anticipate,
parallel to Lehmer’s conjecture in [20], that no prime p for λ
f
(p) = 0 exists. Another
resemble question is (the analogue of) Lang-Trotter conjecture - the primes for which
λ
f
(p)p
(k−1/2)
= α is of zero density, for any α.
The positive proportion of integers from {tn
2
}
n∈N
(resp. {tp
2ν
}
ν∈N
) on which
a
f
(tn
2
) (resp. a
f
(tp
2ν
)) are of the same sign, shown in Theorems 2 and 3, en-
courages our belief in Conjecture (1.5). Finally we would remark that for the
sequence {a
f
(tp
2
)}
p primes
, there is also a positive density (over the set of primes)
of sign changes.
Theorem 4. Let k > 1 be an integer, N > 4 an integer divisible by 4 and χ be a real
Dirichlet character modulo N. Suppose that f ∈ S
∗
k+1/2
(N, χ) is a Hecke eigenform
and t > 1 is squarefree such that its Shimura lift is not of CM type and
a
f
(t) 6= 0. Then we have
(1.12)
X
p6x
εa
f
(t)
−1
a
f
(tp
2
)> 1.68p
k−1/2
1
f
x
log x
for x > x
0
(f) and ε = ±1.
This is a direct application of (4.1) (with ν = 1) below and that there exists a
positive density of primes for which λ
f
(p) > 1.681 and λ
f
(p) < −1.681 respectively,
shown in [9, Theorem 4.10].