THE DIMENSION OF THE SPACE OF CUSP FORMS OF WEIGHT ONE

W. Duke^{∗}

November 14, 1994

1. Introduction

It is a basic problem to determine the dimension of the space of cusp forms of a given type. For classical holomorphic forms of integral weight larger than one, the dimension is well understood by means of either the Riemann–Roch theorem or the Selberg trace formula. The case of weight one, however, remains mysterious. From the point of view of spectral theory this is because these forms belong to an eigenvalue of the Laplacian which is not isolated; the difficulty of estimating nontrivially the multiplicity of such an eigenvalue is well known.

Suppose, for example, that q is a prime and that S_{1}(q) is the space of holomorphic cusp forms for Γ_{0}(q) of weight 1 with character (\frac{·}{q} ), the Legendre symbol. No nonzero cusp forms may exist unless (\frac{−1}{q}) = −1, so assume q ≡ 3 (mod 4). Hecke discovered that the existence of such cusp forms is tied up with the class number h of Q(\sqrt{−q}); if χ is any nontrivial (hence non–real) class character then

(1)  ∑_{𝔞} χ(𝔞)e(N(𝔞) z) ∈ S_{1}(q) ,

where 𝔞 runs over all nonzero integral ideals of Q(\sqrt{−q}). There are (h − 1)/2 independent forms of this type so by Siegel’s theorem we have the (ineffective) lower bound

dim S_{1}(q) ≫_{ε} q^{1/2−ε}

for all ε > 0.

∗Research supported in part by NSF Grants DMS–9202022 and DMS–9022140

In general, S_{1}(q) is not spanned by forms of Hecke’s type. The construction of specific examples demonstrating this is an active area of research (see [F] and its references). Such exotic forms seem quite rare, however, and it appears reasonable to expect that in fact

dim S_{1}(q) = \frac{1}{2} (h − 1) + O(q^{ε}) .

In particular, this would imply that dim S_{1}(q) ≪ q^{1/2} log q. Rather less, however, is actually known. Serre has shown [S] that if q = 24m − 1 or 24m + 7 then dim S_{1}(q) ≤ m − (h − 1)/2, while if q = 24m + 11 or 24m + 19 then dim S_{1}(q) ≤ m − h + 1. Sarnak has informed me that the Selberg trace formula for weight one [Hej, Chapter 9] with a suitably chosen test function yields the bound

dim S_{1}(q) ≪ \frac{q}{log q}

which, for large values of q, is currently the best known.^{1} The main object here is to improve this estimate. Theorem 1. For q prime

dim S_{1}(q) ≪ q^{11/12} log^{4} q ,

with an absolute implied constant.

Roughly speaking, the idea of the proof is to exploit two conflicting properties of the Fourier coefficients of newforms in S_{1}(q) not of Hecke’s type (1): their approximate orthogonality and the finiteness of the number of their possible values at primes. The first property is a consequence of their belonging to automorphic forms while the second is a consequence of Deligne–Serre theorem. Taken together, these properties limit the number of possible newforms which may exist.

Following Serre [S], Theorem 1 has an application to the quotient X_{0}^{∗}(q) of the modular curve X_{0}(q) by the Fricke involution z ↦ −1/qz when q = 24m − 1 is prime. In this case the genus of X_{0}^{∗}(q) is m − (h − 1)/2.

Corollary. For q = 24m − 1 prime, the space of differential forms of the first kind on X_{0}^{∗}(q) with a zero of order at least m at the cusp has dimension which is O(q^{11/12} log^{4} q).

A different kind of application of these ideas is to bound the number m_{4}(−q) of quartic number fields of discriminant −q.

1In an unpublished note J.-M. Deshouillers and H. Iwaniec obtained in this way the bound O(q log^{−3} q)

for the multiplicity of the eigenvalue 1/4 in the case of weight zero Maass cusp forms for Γ_{0}(q) with trivial character.

Theorem 2. For q prime

m_{4}(−q) ≪ q^{7/8} log^{4} q ,

with an absolute implied constant.

To put this result in context it may be of interest to show what follows from algebraic number theory and trivial bounds for class numbers. By means of class field theory Heilbronn [Hei] showed that

m_{4}(−q) = \frac{4}{3} ∑_{k} h_{2}(k) ,

where k runs over all cubic number fields of discriminant −q. Here, for any ℓ and any number field k, h_{ℓ}(k) denotes the number of ideal classes of k of (exact) order ℓ. Furthermore, by [Ha], the number of cubic fields in the sum is \frac{3}{2} h_{3}(Q(\sqrt{−q})) . For the class number h(k) of any number field k of degree n > 1 and discriminant D we have the bound

h(k) ≪ |D|^{1/2} log^{n−1} |D|

where the implied constant depends only on n (see [N, p.153.]). Since h_{ℓ}(k) ≤ h(k) we deduce that

m_{4}(−q) ≪ q log^{3} q

with an absolute implied constant. The improvement of this given in Theorem 2 requires both the classification of quartic fields of discriminant −q by odd octahedral Galois representations of conductor q given in [S] and the proof in this case of the Artin conjecture given in [T]. If we assume the Artin conjecture for icosahedral representations then similarly we can deduce that the number of non-real quintic fields of discriminant q^{2} whose normal closure has Galois group A_{5} is O(q^{11/12} log^{4} q).

2. Approximate orthogonality of Fourier Coefficients

For q ∈ Z^{+} and ε an odd Dirichlet character mod q, let S_{1}(q) = S_{1}(q, ε) be the set of all holomorphic cusp forms for Γ_{0}(q) of weight 1 with character ε. Thus f ∈ S_{1}(q) satisfies for γ ∈ \binom{a b}{c d} ∈ Γ_{0}(q)

f(γz) = ε(d) (cz + d) f(z)

and (Im z)^{1/2} |f(z)| is uniformly bounded on the upper half–plane H. The vector space S_{1}(q) is finite dimensional and has an inner product

⟨f, g⟩ = ∫_{Γ_{0}(q)\H} f(z)ḡ(z) y^{−1}dxdy .

Each f ∈ S_{1}(q) has the Fourier expansion at ∞

f(z) = ∑_{n≥1} a_{f}(n) e(nz) .

The object of this section is to establish the following mean value result which expresses the approximate orthogonality of the a_{f}(n) over any fixed orthonormal basis B for S_{1}(q).

Proposition 1. For arbitrary c_{n} ∈ C with 1 ≤ n ≤ N we have

∑_{f∈B} | ∑_{n≤N} c_{n} a_{f}(n)|^{2} ≪ (1 + \frac{N}{q} ) ∑_{n≤N} |c_{n}|^{2}

with an absolute implied constant.

The proof we give of this uses the following duality principle.

Lemma 1. Suppose that V is a finite dimensional inner product space over C with an orthonormal basis B. Let {v_{1}, …, v_{N}} ⊂ V be a fixed set of vectors and Δ be a positive number. Then the inequality

(2)  ∑_{n≤N} |⟨u, v_{n}⟩|^{2} ≤ Δ ⟨u, u⟩

holds for all u ∈ V if and only if

(3)  ∑_{f∈B} | ∑_{n≤N} c_{n} ⟨f, v_{n}⟩|^{2} ≤ Δ ∑_{n≤N} |c_{n}|^{2}

holds for all c_{n} ∈ C.

Proof: By Parseval’s equality

(4)  \begin{aligned}∑_{f∈B} | ∑_{n≤N} c_{n} ⟨f, v_{n}⟩|^{2} & = ∑ ∑_{n,m≤N} c_{n}c̄_{m} ∑_{f∈B}⟨f, v_{n}⟩ \overline{⟨f, v_{m}⟩} \\ & = ∑ ∑_{n,m≤N} c_{n}c̄_{m}⟨v_{n}, v_{m}⟩ . \end{aligned}

Now the Lemma follows from Boas’ generalization of Bessel’s inequality as given in [D, p. 151]. For the sake of completeness we will include the rest of the proof that (2) ⇒ (3) here since this is the part which we need and refer to [D] for the converse.

Take u = ∑_{m≤N} c_{m}v_{m} in (2); then

(5)  \begin{aligned}⟨u, u⟩ & = ∑ ∑_{n,m≤N} c_{n}c̄_{m}⟨v_{n}, v_{m}⟩ = ∑_{n≤N} c_{n}⟨v_{n}, u⟩ \\ & ≤ ( ∑_{n≤N} |⟨u, v_{n}⟩|^{2})^{1/2} ( ∑_{n≤N} |c_{n}|^{2})^{1/2} \\ & ≤ Δ^{1/2}⟨u, u⟩^{1/2} (∑ |c_{n}|^{2})^{1/2} \end{aligned}

by (2). Hence

⟨u, u⟩ ≤ Δ ∑_{n≤N} |c_{n}|^{2}

and (3) follows by (4) and (5). □

Taking v_{n} = ∑_{f∈B} ā_{f}(n) f, we see that Proposition 1 is reduced to the following Lemma. Lemma 2. For any f ∈ S_{1}(q)

∑_{n≤N} |a_{f}(n)|^{2} ≪ (1 + \frac{N}{q} ) ⟨f, f⟩

with an absolute implied constant. Proof: We employ a technique of Iwaniec (unpublished) to bound such sums uniformly. For any y > 0 we have

∑_{n≥1} e^{−4πny} |a_{f}(n)|^{2} = ∫_{0}^{1} |f(x + iy)|^{2} dx.

Thus, for any Y > 0,

(6)  ∑_{n≤N} F_{Y} (n) |a_{f}(n)|^{2} ≤ ∫_{Y}^{∞} ∫_{0}^{1} |f(z)|^{2}y^{−1} dxdy ,

where

(7)  F_{Y} (n) = ∫_{1}^{∞} e^{−4πnY y}y^{−1} dy ≥ ∫_{1}^{∞} e^{−4πNY y} y^{−1} dy

for n ≤ N. Setting

P (Y) = {z ∈ H ; 0 < Re z ≤ 1, Im z > Y }

we have

(8)  ∫_{Y}^{∞} ∫_{0}^{1} |f(z)|^{2} y^{−1} dxdy ≤ max_{z∈P (Y)} #{γ ∈ Γ_{0}(q)/{±1}; γz ∈ P (Y)} ⟨f, f⟩ .

Now for fixed z = x + iy ∈ H the condition that γz = (az + b)/(cqz + d) ∈ P (Y) is imposed by requiring that

(9)  Im ((az + b)/(cqz + d)) = y|cqz + d|^{−2} > Y

and that

(10)  0 < Re ((az + b)/(cqz + d)) ≤ 1.

For a fixed c and d satisfying (9) a and b are determined by (10). It follows that for z ∈ H

\begin{aligned}#\{γ ∈ Γ_{0}(q)/\{±1\}; γz ∈ P (Y)\} & ≤ 1 + #\{c > 0, d; |cqz + d|^{2} < yY^{−1}\} \\ & ≤ 1 + #\{c, d; 0 < c < q^{−1}(yY)^{−1/2} and |cqx + d|^{2} < yY^{−1}\} \end{aligned}

by taking real and imaginary parts. Hence for z ∈ P (Y)

#{γ ∈ Γ_{0}(q)/{±1}}; γz ∈ P (Y)} ≤ 1 + q^{−1}(yY)^{−1/2} (1 + 2y^{1/2}Y^{−1/2})
= 1 + q^{−1}(yY)^{−1/2} + 2q^{−1}Y^{−1}
≤ 1 + 3q^{−1}Y^{−1}.

Thus from (6) and (8) we get

∑_{n≤N} F_{Y} (n) |a_{f}(n)|^{2} ≤ (1 + 3q^{−1}Y^{−1}) ⟨f, f⟩ .

Choosing Y = 3N^{−1} and using (7) we get Lemma 2 (with 10^{18} for the absolute constant). □

3. Consequences of the Deligne–Serre theorem

Let N_{1}(q, ε) ⊂ S_{1}(q, ε) be the set of normalized newforms. For f ∈ N_{1}(q, ε) the associated L–function is an Euler product

(11)  L_{f}(s) = ∑_{n≥1} a_{f}(n) n^{−s} = ∏_{p} (1 − a_{f}(p) p^{−s} + ε(p) p^{−2s})^{−1} .

The Deligne–Serre theorem [DS] states that L_{f}(s) is the Artin L–function of an irreducible two–dimensional Galois representation ρ of conductor q with det ρ = ε (via the Artin map). We shall use two consequences of this result. The first is that a_{f}(n) satisfies the Ramanujan bound

(12)  |a_{f}(n)| ≤ d(n),

where d(n) is the divisor function. The second is that f may be classified as being of dihedral, tetrahedral, octahedral, or icosahedral type according to whether the image of ρ in P GL(2, C) is D_{h}, A_{4}, S_{4} or A_{5}.

We now restrict attention to the case that q is prime and ε(·) = (\frac{·}{q} ) and write N_{1}(q) for N_{1}(q, ε). Also, let N_{dih}, N_{oct}, N_{ico} be the forms in N_{1}(q) of dihedral, octahedral and icosahedral type, respectively. It is shown in [S, p. 343 of Collected Papers] that f is of dihedral type exactly when it is of Hecke’s type (1). Proposition 2. Suppose that f ∈ N_{1}(q) for q prime. Then

(a) ⟨f, f⟩ ≪ q log^{3} q with an absolute implied constant.

(b) If f is not of dihedral type then it is either of octahedral or icosahedral type. If f ∈ N_{oct} then

(13)  a_{f}(p^{8}) − a_{f}(p^{4}) − (\frac{p}{q}) a_{f}(p^{2}) = 1 ,

while if f ∈ N_{ico} then

(14)  a(p^{12}) − a_{f}(p^{8}) − (\frac{p}{q}) a_{f}(p^{2}) = 1

for all primes p ≠ q. Proof: (a) The Rankin–Selberg convolution

(15)  φ(s) = ∑_{n≥1} b(n) n^{−s} = (1 + q^{−s}) ζ(2s) ∑_{n≥1} |a_{f}(n)|^{2}n^{−s}

is entire except for a simple pole at s = 1 with

Res_{s=1} φ(s) = \frac{2π^{2}}{q} ⟨f, f⟩

and satisfies the functional equation

(16)  Φ(s) = (\frac{q}{4π^{2}} )^{s} Γ(s)^{2}φ(s) = Φ(1 − s)

by [L]. Suppose that F ∈ C_{c}^{∞}(0, ∞) has ∫_{0}^{∞} F (x) dx = 1. Then the Mellin transform

F̂(s) = ∫_{0}^{∞} F (x) x^{s} \frac{dx}{x}

is entire, of rapid decay in vertical strips and F̂(1) = 1. By Mellin inversion,

F (x) = \frac{1}{2πi} ∫_{Re s=2} F̂(s)x^{−s}ds.

for x > 0. Thus

(17)  \begin{aligned}∑_{n≥1} b(n)F (n/q^{2}) & = \frac{1}{2πi} ∫_{Re s=2} F̂(s)φ(s)q^{2s}ds \\ & = 2π^{2}q ⟨f, f⟩ + \frac{1}{2πi} ∫_{Re s=−1} F̂(s)φ(s)q^{2s}ds. \end{aligned}

It follows from (16), (15) and (12) that

|φ(−1 + it)| ≤ (4π^{2})^{−3}q^{3} \frac{|Γ(2 − it)|^{2}}{|Γ(−1 + it)|^{2}} (1 + q^{−2})|ζ(4 − 2it)| ∑_{n≥1} d(n)^{2}n^{−2}

and by Stirling’s formula this is ≪ q^{3}(|t| + 1)^{6} with an absolute constant. Hence

(18)  \frac{1}{2πi} ∫_{Re (s)=−1} F̂(s)φ(s)q^{2s}ds ≪ q ∫_{−∞}^{∞} |F̂(−1 + it)|(|t| + 1)^{6}dt ≪ q

since F̂(−1 + it) ≪ (1 + |t|)^{−8}. By (17) and (18) we get

⟨f, f⟩ = \frac{1}{2π^{2}q} ∑_{n≥1} b(n) F (n/q^{2}) + O(1)

where the implied constant depends only on F. Now, by (15) and (12)

0 ≤ b(n) ≤ 2 ∑_{n=mℓ^{2}} d^{2}(m) = 2d_{4}(n)

where d_{4}(n) is the number of factorizations of n into 4 factors. Thus

⟨f, f⟩ ≪ \frac{1}{q} ∑_{n≥1} d_{4}(n) F (n/q^{2}) ≪ q log^{3} q.

Fixing F gives the result with an absolute implied constant. □

(b) The fact that f cannot be tetrahedral is shown in [S, Theorem 7(c)]. It is also observed in [S, p. 362] that if p ≠ q and f ∈ N_{oct} then

(19)  (\frac{p}{q}) a(p^{2}) ∈ \{−1, 0, 1, 3\} ,

while if f ∈ N_{ico} then

(20)  (\frac{p}{q}) a(p^{2}) ∈ \{−1, 0, 3, \frac{1 + \sqrt{5}}{2}, \frac{1 − \sqrt{5}}{2}\} .

For general f ∈ N_{1}(q, ε) it follows from (11) that if we set x = ε̄(p) a_{f}(p^{2}) then for p ≠ q and n ≥ 0

(21)  ε̄^{n}(p) a_{f}(p^{2n}) = P_{n}(x) ,

where P_{0}(x) = 1, P_{1}(x) = x, and P_{n+1}(x) = (x − 1) P_{n}(x) − P_{n−1}(x) for n ≥ 1. Thus P_{2}(x) = x^{2} − x − 1, P_{3}(x) = x^{3} − 2x^{2} − x + 1, etc. One may check that

x(x + 1)(x − 1)(x − 3) + 1 = P_{4}(x) − P_{2}(x) − P_{1}(x) ,

while

x(x + 1)(x − 2)(x − 3)(x^{2} − x − 1) + 1 = P_{6}(x) − P_{4}(x) − P_{1}(x) .

Thus by (19)–(21) we finish the proof of (b). □ Remark: It has been checked that no linear form ∑_{ℓ=1}^{7} c_{ℓ} a_{f}(p^{ℓ}) takes positive values for all possible values of a_{f}(p), p ≠ q, when f ∈ N_{oct}. A similar remark applies to f ∈ N_{ico}.

4. Counting newforms and quartic fields

By combining Propositions 1 and 2 we may now estimate the number of newforms of octahedral or icosahedral type. Proposition 3. For q prime

#N_{oct} ≪ q^{7/8} log^{4} q (a)

#N_{ico} ≪ q^{11/12} log^{4} q (b)

with absolute implied constants. Proof: By Propositions 1 and 2(a) we have for any c_{n} ∈ C

∑_{f∈N_{1}(q)} | ∑_{n≤N} c_{n} a_{f}(n)|^{2} ≪ (q + N) log^{3} q ∑_{n≤N} |c_{n}|^{2} .

Thus we deduce by positivity the inequality

(22)  ∑_{f∈N_{oct}} | ∑_{n≤N} c_{n} a_{f}(n)|^{2} ≪ (q + N) log^{3} q ∑_{n≤N} |c_{n}|^{2}.

Choose N = q and

c_{n} = \{ \begin{matrix}1, & n = p^{8} ≤ q, \\ −1, & n = p^{4} ≤ q^{1/2}, \\ − (\frac{p}{q}) , & n = p^{2} ≤ q^{1/4}, \\ 0, & otherwise, \end{matrix}

for p prime. By the prime number theorem and (13)

∑_{n≤N} c_{n}a_{f}(n) ∼ \frac{8q^{1/8}}{log q} , for f ∈ N_{oct} ,

while

∑_{n≤N} |c_{n}|^{2} ∼ \frac{24q^{1/8}}{log q} .

Hence, by (22)

#N_{oct} ≪ q^{7/8} log^{4} q .

A similar argument works for N_{ico}. □

Theorem 1 now follows from Propositions 2(b) and 3 since

#N_{dih} = \frac{h − 1}{2} ≪ q^{1/2} log q .

For its Corollary we use the fact, shown in [S], that the required space of differentials has dimension

\frac{1}{2} dim S_{1}(q) − \frac{1}{4} (h − 1) .

Theorem 2 follows from Proposition 3(a), Theorem 8 of [S] and Tunnell’s proof [T] of the Artin conjecture for octahedral Galois representations. Here we also use the fact that the Galois closure of any quartic extension with discriminant −q has Galois group S_{4}, since otherwise the discriminant would not be square free (e.g., see [B]).

Acknowledgement. I would like to thank J-P. Serre for his many helpful comments. This paper was written while I was participating in the 1994-95 program on Automorphic Forms at MSRI. I thank the organizers and the Institute for providing financial support and a congenial atmosphere in which to work.

References

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[Hei] H. Heilbronn, On the 2–classgroup of cubic fields, in Studies in Pure Math. (L. Mirsky, ed.), Academic Press 1971, 117–119.

[Hej] D. A. Hejhal, The Selberg Trace Formula for PSL(2, R), Volume 2, Springer LNM # 1001, 1983.

[N] W. Narkiewicz, Elementary and Analytic Theory of Numbers, Springer 1990.

[L] W.–C. Li, L–series of Rankin type and their functional equations, Math. Ann. 244 (1979), 135–166.

[S] J–P. Serre, Modular forms of weight one and Galois representations, Algebraic Number Fields, ed. by A. Fröhlich, Academic Press 1977, 193–268, in Serre’s Collected Papers, III, 292–367.

[T] J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. A.M.S. 5 (1981), 173–175.

William Duke Department of Mathematics, Rutgers University New Brunswick, NJ 08903 email: duke@math.rutgers.edu

Current Address:

Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720–5070