1. Introduction The problem of estimating the dimension of the space of cusp forms of weight one, raised by Serre [Ser77], has attracted considerable attention in the recent past (see, for example, [Klu06], [BG08], [Wo99], [MV02], [Du95] ). There is an intrinsic difficulty in the case of weight one which is not present in the case of higher weights. For weight k ≥ 2, it is well-known that the dimension of the space of cusp forms of weight k and level q is of size kq (i.e., kq). It is widely speculated (see the above references) that given a level q and a fixed nebentypus χ(mod q), the dimension of the space S1 (q, χ) of the space 1 of cusp forms of weight one and nebentypus χ is Oε (q 2 +ε ). But the 6 best known bound for this is Oε (q 7 +ε ), due to Michel and Venkatesh [MV02]. The difficulty with the cusp forms of weight one, as explained in §2, stems from the fact that in the whole space of all Maass cusp forms, these span an eigenspace with eigenvalue 41 ( k2 (1 − k2 ) for general weight k), which is not isolated in the continuous spectrum of the Laplace-Beltrami operator. This makes it difficult to count those forms using analytic tools like Selberg trace fomula with a good accuracy. For a detailed exposition on how the holomorphic forms, and in particular, the weight one forms sits inside the space of all Maass cusp forms, the reader is referred to [DFI02]. Now, to understand the problem at hand in more depth, we first note that by the Deligne-Serre theorem [DS74], to every normalized, primitive form f of weight one and nebentypus χ is associated an irreducible, continuous Galois representation ρ : GQ −→ GL2 (C) such that for all primes p not dividing the level q, af (p) = Tr(ρ(Frp ))

(1)

and det(ρ(Frp )) = χ(p). (2) Here GQ is the absolute Galois group and af (n) is the n-th Hecke eigenvalue of f . From the profinite topology of the Galois group, it follows that the image of the projective representation in PGL2 (C) is finite 1

and therefore has to be isomorphic to one of the following : (a) a dihedral group D2n for some n (b) A4 , S4 , or A5 . The Galois representations and the associated forms are said to be of the dihedral, tetrahedral, octahedral, and icosahedral type according to the corresponding projective images. The dihedral Galois representations give rise to certain theta series that were studied by Hecke and they are quite well-understood (see [Ser77], §7). The number of dihe1 dral forms is known to be Oε (q 2 +ε ) and it comes from√ Siegel’s bound on the class number of the imaginary quadratic field Q( −q) (see [Ser77] or [Du95]). The forms of other kinds are believed to occur very rarely and it is expected that the number of other kinds of primitive forms (with a fixed nebentypus modulo q) should only be Oε (q ε ), and thus it is natural to conjecture, 1

dim S1 (q, χ) ε q 2 +ε . Proving this seems to be out of reach by currently available techniques of analytic number theory. However, if one assumes a conjectural large sieve inequality for the Fourier coefficients of Maass cusp forms of weight one, which is expected to hold according to the philosophy of large sieve, then this result follows. This is shown in [Gan08]. The best known unconditional bound for the dimension of the whole space 6 is O(q 7 +ε ) as proved by Michel and Venkatesh [MV02]. This comes from the bound for the subspace of icosahedral type, which gives the worst bound among all. There has been some improvements upon 4 the bound O(q 5 +ε ) of Michel and Venkatesh for the forms of octahedral type, most notably by Kl¨ uners [Klu06], who has proved that the 1 2 2 bound is O(q (log q) ) for octahderal forms of level q or q 2 for a prime q. Unlike the previous works which use analytic tools like large sieves, he proves this result by counting quartic S4 extensions of Q of a given discriminant and matching them up with octahedral Galois reprsentations. If we average over the levels, then it is possible to get much better bound, at least in the octahedral case. This has been carried out by Ellenberg [Ell03] and by Bhargava and Ghate [BG08]. In fact, Bhargava and Ghate have shown that the dimension of the space of forms of octahedral type is bounded, on average over the prime levels, by an absolute constant. 2

In this paper, we improve upon the bound of Michel and Venkatesh for the case of octahedral forms by exploiting a small obesrvation (see Proposition (6)) while proceeding along the lines of [Du95] and [MV02]. However, we make certain simplifying assumptions (as in the papers of Serre and Duke) that the level q is prime and that the nebentypus is the Legendre symbol nq . For a non-zero form to exist in this situation, we must also have q ≡ 3(mod 4) (see §of [Ser77]). Our main result is the following theorem. Theorem 1. Let q ≡ 3(mod 4) be a prime and let χ(n) = nq . Then the subspace S1oct (q, χ) of S1 (q, χ) spanned by the cusp forms of octahedral type has dimension 3

dim S1oct (q, χ) ε q 4 +ε . Even though this result is weaker than the result of Kl¨ uners quoted above, this method of proof is probably amenable for adaptation to other kinds of modular forms (such as, of icosahedral type, or, for Hilbert modular forms). Moreover, it may be possible to apply the kind of technique used in this paper to modular forms of general level and nebentypus to obtain bounds stronger than known now and the author wishes to come back to this question in the future. Wong [Wo99] has generalised Duke’s theorem to all levels but the author has not been able to understand his method. It is also to be noted that there are mistakes in the statement of Theorem 10 and remark 6 in [Wo99] which 3 claim the bound O(N 4 (log N )4 ) for octahedral forms of level N , at least in the case of prime level N , as in [Du95] and here. It appears from 5 the proof of that theorem that the bound should be O(N 6 (log N )4 ) instead. Acknowledgements Most of this work, including the main theorem, was done some time around the end of 2005 and was a part of the author’s PhD thesis submitted to the Graduate School, Rutgers University in May, 2006. He wrote the thesis under the supervision of H. Iwaniec. It is a pleasure to thank him for his generosity, patience, and support, and above all, for teaching the beautiful subject of analytic number theory to the author. The author also wishes to thank J. Tunnell, R.T. Curtis, and J.-P. Serre for patiently answeing some questions that came up during the work and while writing up this manuscript. He also thanks the Rutgers Mathematics Department, the Institute of Mathematical Sciences, Chennai, and the Tata Institute of Fundamental Research, Mumbai for providing excellent conditions for work. The author has 3

benifitted from many enlightening conversations with several colleagues and friends, in particular, with Dipendra Prasad, Eknath Ghate, C.S. Rajan, Nick Gill, and Vivek M. Mallick. It is a pleasure to acknowledge their contribution. Finally, the author wishes to thank the referee for pointing out several errors in the earlier version, and for suggestions to improve the quality of exposition. 2. Algebraic preliminaries In this section we shall obtain certain algebraic properties of the Fourier coefficients of the modular forms we are considering in Theorem 1. This is the case (c1 ) in [Ser77]. In this set-up, Serre has determined the image ρ(GQ ) in the following way. ρ(GQ ) consists of all elements s ∈ GL(2, C) whose image s˜ ∈ PGL(2, C) lies in ρ˜(GQ ) and satisfies the condition det(s) = sgn(˜ s) (3) where sgn : S4 −→ {±1} is the map that assigns +1 to even permutations and −1 to odd permutations. The above condition comes from the observation that det(ρ) and sgn(˜ ρ) are both quadratic characters of GQ unramified outside p and therefore must be the same. This forces ρ(GQ ) to be isomorphic to GL(2, F3 ). This is shown in the next proposition and the following lemma. Proposition 2. ρ(GQ ) is a central extension of S4 by Z/2Z with the property that the transpositions lift to involutions. Proof. Let us denote the projection map from GL(2, C) to PGL(2, C) by π and ρ(GQ ) by G. We restrict π to G. Then we claim the ker that 1 0 nel of π|G must be {±I}, where I is the identity matrix . This 0 1 λ 0 is because, if s is in the kernel, then s must be diagonal, s = 0 λ 2 and λ = det s = sgn(˜ s) = 1(since s˜ = trivial permutation) ⇒ λ = ±1. Now suppose, s ∈ G maps to s˜, a transposition. We shall show that s2 = I. By (3), we have det s = −1. Also, s˜2 = 1. Therefore, u 0 π(s2 ) is trivial in PGL(2, C). Hence s2 is diagonal, say . 0 u So u2 = dets2 = (det s)2 = 1 by (3). So u = ±1. Now, suppose a b s= . We have the conditions that c d 2 a + bc b(a + d) 1 0 2 s = = ±I = ± , c(a + d) d2 + bc 0 1 4

det s = ad − bc = −1, and s˜ is not-trivial in PGL(2, C). We want to show that s2 = I and not −I. Suppose s2 = −I. Note that this will mean, either a + d = 0 or a − d = 0. If a − d = 0, then a + d 6= 0, because then we would get det s = −bc = −(−1) = 1, a contradiction. So, if a − d = 0, then b = c = 0, which would mean s is trivial projectively, and that is not allowed. Hence, we must have a + d = 0. But then we must have a2 + bc = −1 = − det s, a contradiction. So we have proved that G is a central Z/2Z extension of S4 which has the property that the transpositions lift to involutions. Lemma 3. A central Z/2Z extension of S4 which has the property that the transpositions lift to involutions must be isomorphic to GL(2,F3 ). Proof. This proof was explained to me by R. Curtis. Let us call the group in question G. Then G has some involution x which projects to a transposition, say x˜, in S4 . G also has an element of order 3, say y, by one of the Sylow theorems and let y˜ be its image in S4 . Then y˜ must be a three cycle. It is an easy calculation to check that x˜y˜ is either a transposition or a four cycle. In any case, (˜ xy˜)4 is the trivial element 4 in S4 and hence (xy) is in the kernel of the projection map from G to S4 and in particular, in the centre of G. So, we have a subgroup of G having the presentation hx, y|x2 = y 3 = (xy)8 = [x, (xy)4 ] = 1i, where by [a, b], we denote aba−1 b−1 . But this is the presentation of GL(2, F3 ) which has 48 elements and therefore must be G itself. An explicit choice of x and y inside GL(2, F3 ) is given by 0 1 1 1 x= , y= . 1 0 0 1 Once we know that the image of the Galois representation associated to our modular form is isomorphic to GL(2,F3 ), we can infer that √ the traces of the matrices in the image of ρ lie in the number field Q( −2) by reading off from the character table of GL(2,F3 ). Now the DeligneSerre theorem (1) gives us the following proposition. Proposition 4. Let f ∈ S1 (q, χ) be of octahedral type. Moreover, let q be a prime, q ≡ 3(mod 4). Then the number √ field generated by the Fourier coefficients af (p) for primes p 6= q is Q( −2). 5

We can say more about the Fourier coefficients. The following proposition says that the Fourier coefficients of these modular forms can take a very restricted set of values. It is stated without any proof in §9 of [Ser77]. Proposition 5. For a modular form f ∈ S1 (q, χ) of octahedral type, where q ≡ 3(mod 4) is a prime, we have χ(p)af (p)2 = 0, 1, 2, or 4.

(4)

for primes p 6= q. 2

(ρ) Proof. Note that σ = Sym depends only on the projectivization of det(ρ) ρ and it is thus an irreducible three-dimensional representation of S4 . From the character table of S4 , we see that there are only two threedimensional irreducible representations of S4 (the standard representation and its twist by the sign character) and either of their characters take values from {3, −1, 0, 1}. Since

χ(p)af (p)2 = T r(ρ)2 /det(ρ) = 1 + T r(σ), the proposition follows. We shall use the above two propositions to find a linear relation between Fourier coefficients at prime powers. First note that by the Hecke’s recurrence formula mn X af (m)af (n) = χ(d)dk−1 af , (5) d2 d|(m,n)

we obtain the relation af (p2 ) = af (p)2 − χ(p)

(6)

so that χ(p)af (p2 ) = −1, 0, 1, or 3 (7) for primes p 6= q. One gets, as in [Du95], a linear relation between Fourier coefficients at various powers of prime p by applying (5) repeatedly. af (p8 ) − af (p4 ) − χ(p)af (p2 ) = 1 (8) This kind of relations plays a crucial role in Duke’s work. We establish below a similar linear relation, but the highest exponent of p that appears will be 6 instead of 8 and this is the source of the improvement. This lowering of the exponent comes from partitioning the set of primes into quadratic residues and non-residues modulo q and considering them separately. 6

Proposition 6. Let f ∈ S1 (q, χ) be a modular form of octahedral type. Suppose the level q is a prime and q ≡ 3(mod 4). Then for primes p 6= q, we have af (p6 ) − af (p2 ) − χ(p)af (p2 ) = 1.

(9)

Proof. Suppose first that the prime p is such that χ(p) = −1. Then from (4), we have √ √ √ af (p) ∈ {0, ± −1, ± −2, ±2 −1}. However, √ according to proposition (4), the numbers af (p) lie in the field Q( −2). So, we have even more restricted choice, namely, √ af (p) ∈ {0, ± −2}. So, applying Hecke’s recurrence formula (5) again, we get af (p2 ) = af (p)2 − χ(p), we have, af (p2 ) = af (p)2 + 1 = 1 or − 1. Now, applying (5), we can write X af (p2 )2 = χ(d)af (p4 d−2 ) = af (p4 ) + χ(p)af (p2 ) + 1.

(10)

(11)

d|p2

Since af (p2 )2 = 1 by (10), the above two equations and the fact that χ(p) = −1 implies that af (p4 ) = af (p2 ).

(12)

Applying (5) again, we can write af (p4 )af (p2 ) =

X

χ(d)af (p6 d−2 )

d|p2 6

= af (p ) + χ(p)af (p4 ) + af (p2 ) = af (p6 ) − af (p4 ) + af (p2 ) = af (p6 ).

(13)

Since the left hand side is 1 by (10) and (12), we have, for χ(p) = −1, af (p6 ) = 1.

(14)

Now, we shall consider primes p, so that χ(p) = 1. From (4), we must have af (p)2 = 0, 1, or 4 (15) √ since 2 is not in the field generated by the Fourier coefficients af (p) (see Proposition 4). So, af (p2 ) = af (p)2 − χ(p) = af (p)2 − 1 = −1, 0, or 3 respectively. (16) 7

Hence, from (12), af (p4 ) = 1, −1, or 5 respectively,

(17)

and (13) now implies, af (p6 ) = −1, 1, or 7 respectively.

(18)

From the above three equations, we can easily check that for any p with χ(p) = 1, we must have, af (p6 ) − 2af (p2 ) = 1.

(19)

Therefore, from (19) and (14), for any prime p 6= q, we get (1 − χ(p))af (p6 ) + (1 + χ(p))(af (p6 ) − 2af (p2 )) = 2. This gives the proposition after rearranging terms. 3. Maass forms and Kuznetsov formula The first breakthrough on the problem Serre had posed was by Duke [Du95], who proved that the dimension of the space S1 (q, χ) was 11 Oε (q 12 +ε ). This improvement upon the trivial bound O(q) hinged on the simple idea of exploiting two conflicting properties of the Fourier coefficients. On the one hand, the sequence of Fourier coefficients are approximately orthogonal for different primitive forms. On the other, the coefficients are “rigid”, that is, these numbers can assume values from only a very small set, and therefore these satisfy some algebraic relation like (8). These two conflicting properties force the size of the set of forms of this type to be small. This approximate orthogonality is manifested in a large sieve inequality for the Fourier coefficients of cusp forms of weight one that Duke established. Michel and Venkatesh subsequently observed (independently) that if one considered the space of holomorphic forms of weight one as a subspace (albeit a very small one) of the whole spectrum of Maass forms and applied the Kuznetsov’s formula, which involves a sum over the whole spectrum, then one obtained a sharper large sieve inequality. We can isolate the contribution of a small sub-family of, say, octahedral forms inside the much larger family of all Maass forms by choosing a suitable sequence {cn } in the large sieve inequality. This is an example of the amplification method, invented by Iwaniec, which has been crucial in establishing subconvex bounds for various L-functions (see [DFI02] and the references therein). Below we shall give a very brief summary of the spectral theory of automorphic forms in order to state the Kuznetsov formula (30) which is crucial for our purpose. A detailed account can be found in the 8

chapters 3-7 of the monograph [Iw95] for the full modular group and in [DFI02] for the Hecke congruence subgroups. Definition 1. Let χ(mod q) be a primitive Dirichlet character such that χ(−1) = (−1)k . (20) This gives rise to a character of a b Γ = Γ0 (q) = ∈ SL(2, Z) : c ≡ 0(mod q) c d by setting χ(γ) = χ(d) for

a b γ= ∈ Γ. c d A function f : H −→ C which satisfies the condition f (γz) = χ(γ)jγ (z)k f (z),

(21)

with

cz + d = eiarg(cz+d) |cz + d| for all γ ∈ Γ is called is called an automorphic function of weight k and character χ. jγ (z) =

We shall denote the space of all such automorphic function by Ak (Γ, χ). Definition 2. We define a Maass form to be a smooth function f ∈ Ak (Γ, χ) which is also an eigenfunction of the Laplace-Beltrami operator ∆k , that is, (∆k − λ)f = 0 for some λ ∈ C. Such a Maass form f is said to have weight k and eigenvalue λ. Here the Laplace-Beltrami operator ∆k is defined by, 2 ∂2 ∂ k ∂ ∂ ∂2 2 ∂ ∆k = −y + +iky = (z−z) + (z−z) − . ∂x2 ∂y 2 ∂x ∂z∂z 2 ∂z ∂z (22) We shall write the eigenvalue λ in the form λ = λ(s) = s(1 − s) where s is a complex number. We shall further write s as s = 21 + it where t is again a complex number. We now assume that k is a positive integer. For harmonic analysis, we need to consider a larger space than the space of all Maass forms. Let Lk (Γ, χ) be the space of square 2

9

integrable automorphic functions of weight k with respect to the inner product Z hf, gi = f (z)g(z)dµz (23) Γ\H −2

where dµz = y dxdy is the hyperbolic measure. Note that this inner product is similar to the one usually defined for holomorphic forms (Petersson inner product), but the normalization is different. Let Bk (Γ, χ) be the linear space of smooth functions f ∈ Ak (Γ, χ) such that both f and ∆k f are bounded. One can show that the operator ∆k is symmetk ric and bounded below by λ 2 and Bk (Γ, χ) is a dense subspace of the whole L2 space Lk (Γ, χ). The space Lk (Γ, χ) has a complete spectral resolution with respect to ∆k and this can be concretely realised in terms of the Eisenstein series and cusp forms. Definition 3. A cusp a of the congruence group Γ is said to be singular with respect to the character χ if χ(γa ) = 1 or χ(−γa ) = 1, where one of γa and −γa is a generator of the stability group Γa of the cusp a. Definition 4. For every singular cusp a, the Eisenstein series Ea (z, s) is defined by setting X Ea (z, s) = χ(γ)jσa−1 γ (z)−k (Im σa−1 γz)s , (24) γ∈Γa \Γ

the series being absolutely convergent on Re s > 1. Here σa ∈ SL(2, R) is the matrix (unique up to translation on the right), such that σa ∞ = a and σa−1 Γa σa = Γ∞ where

1 b Γ∞ = ± :b∈Z 0 1 is the stability group for the cusp at infinity. Selberg [Sel] showed that the function Ea (z, s) has meromorphic continuation to the whole complex s-plane with no poles in the region Re s ≥ 21 if the character χ is non-trivial. If the character is trivial, then Ea (z, s) does have a pole at s = 1 (coming form the Riemann zeta function, see §4, [DFI02]). Since the Eisenstein series is not necessarily square-integrable, in order to obtain the spectral resolution of the 10

operator ∆k , one defines a family of functions called the incomplete Eisenstein series which are in Lk (Γ, χ). Definition 5. For a smooth, compactly supported function ψ : R+ −→ C, we define an associated incomplete Eisenstein series X Ea (z|ψ) = χ(γ)jσa−1 γ (z)−k ψ(Im σa−1 γz). (25) γ∈Γa \Γ

We shall denote the linear space of all incomplete Eisenstein series ˜ χ) be its closure in Lk (Γ, χ) (in the L2 -norm by E(Γ, χ). Let E(Γ, topology). Then the operator ∆k has a purely continuous spectrum in ˜ χ) and the space E(Γ, ˜ χ) is “spanned” (in the sense of integrating E(Γ, 1 with respect to the t parameter) by Eisenstein 1 series with Re s = 2 ; in particular, the spectrum contains 4 , ∞ (see Proposition 4.1 in [DFI02]). So 14 is at the bottom of the continuous spectrum. But note thatwhen the weight k is one, the operator ∆1 is bounded below by λ 21 = 14 . We shall see later that the holomorphic cusp forms lie in the 14 -eigen space, but in the discrete spectrum. Let us denote the orthogonal complement of Ek (Γ, χ) in Bk (Γ, χ) by Ck (Γ, χ). Since, ∆k acts on both Ek (Γ, χ) and Bk (Γ, χ), it also acts on Ck (Γ, χ). Let us denote its closure in Lk (Γ, χ) by C˜k (Γ, χ). Then we have (see [Iw95], page 64) ˜ χ) = Lk (Γ, χ) C˜k (Γ, χ) ⊕ E(Γ, (26) It can be shown that the space Ck (Γ, χ) consists of functions whose constant term in the Fourier expansion at every singular cusp is zero. Definition 6. The functions in Ck (Γ, χ) which are also eigenfunction of the Laplace operator ∆k are called Maass cusp forms. Proposition 7. The spectrum of ∆k on Ck (Γ, χ) is discrete and infinite, but of finite multiplicity. Finally we state the main theorem in the spectral theory of automorphic forms which gives the spectral decomposition of the space Lk (Γ, χ) with respect to ∆k . See [DFI02], §5. Theorem 8. Let {uj }∞ j=1 be a complete orthonormal system of Maass cusp forms of weight k, the group Γ and character χ. Then any function f ∈ Lk (Γ, χ) can be expanded as ∞ X X1 Z ∞ 1 1 f, Ea (∗, + it) Ea (z, + it)dt. f (z) = hf, uj iuj (z) + 4π −∞ 2 2 a j=1 (27) 11

3.1. Kuznetsov formula. Let {uj } be an orthonormal system of Maass cusp forms which forms a basis of the space Ck (Γ, χ). Each uj (z) has a Fourier expansion X uj (z) = ρj (n)W kn ,itj (4πny)e(nx), (28) 2|n|

n6=0

where the eigenvalue sj is given by 1 + t2j . 4 Recall that sj = 12 + itj , tj ∈ C. Here Wk,s (z) is a function called the Whittaker function (see [DFI02] for the definition and the basic propetries). Also, each Eisenstein series Ea (z, s) associated with a singular cusp a has a Fourier expansion of type X ρa (n, t)W kn ,it (4π|n|y)e(nx). (29) Ea (z, s) = δa y s + φa (s)y 1−s + λj = sj (1 − sj ) =

2n

n6=0

The following Kuznetsov type spectral summation formula is taken from [DFI02], §5. Proposition 9. For any two positive integers m and n, and a real number r, we have the following equality. X X 1 Z ∞ ρj (m)ρj (n) ρa (m, t)ρa (n, t) + dt chπ(r − t )chπ(r + t ) 4π chπ(r − t)chπ(r + t) j j −∞ a j √ Γ(1 − k − ir) 2 X 1 4π mn 2 √ δ(m, n) + . = Sχ (m, n; c)Ir 3 c c 4π mn c≡0(mod q)

(30) Here “ch” is the hyperbolic cosine function, 1 if m = n, δ(m, n) = ; 0 otherwise. and Ir is given by i

Z

(−iζ)k−1 K2ir (ζx)dζ,

Ir (x) = −2x

(31)

−i

Kν (z) being the K-Bessel function defined by Z ∞ Kν (z) = e−zch(t) ch(νt)dt 0

12

(32)

for Re z > 0 and Re ν > − 12 . Note that Ir (x) is not the I-Bessel function. And Sχ (m, n; c) is a twisted Kloosterman sum defined by ¯ X? dm + dn , (33) Sχ (m, n; c) = χ(d)e ¯ c d(mod c)

where the sum ranges over residue classes coprime to c. 3.2. Relation to holomorphic modular forms. The space of holomorphic weight one cusp forms is just one eigenspace in the discrete spectrum, corresponding to the lowest eigenvalue 14 . But strictly speaking, it is not a set-theoretic inclusion due to the choice of our normalization. For completeness and to make matters explicit, we shall define the holomorphic modular forms. Definition 7. A holomorphic function f : H −→ C is called a modular a b form of weight k, level q, and character χ, if for all γ = ∈ c d Γ0 (q), f satisfies the relation (cz + d)−k f (γz) = χ(γ)f (z)

(34)

and f is holomorphic at all the cusps of Γ0 (q). The following proposition makes the relation between modular forms and Maass forms (as defined here) explicit. A proof can be found in §4 in [DFI02]. Proposition 10. Let g be a Maass of weight k, level q, character form k k k χ, and the lowest eigenvalue λ 2 = 2 1 − 2 . Then k

f (z) = y − 2 g(z) is a holomorphic modular form of weight k, level q, and character χ and all such modular forms arise this way. Note that when k > 1, the lowest eigenvalue is isolated from the continuous spectrum unlike when k = 1. Let {uj } be an orthonormal system of Maass cusp forms which forms a basis of the space Ck (Γ, χ) and let g be a Maass cusp form of weight one. Then the Fourier expansion (28) becomes X 1 1 g(z) = y 2 (4πn) 2 ρg (n)e(nz), n≥1

so that f (z) = y

− 21

g(z) has Fourier expansion X f (z) = af (n)e(nz), n≥1

13

where 1

af (n) = (4πn) 2 ρg (n).

(35)

It is worthwhile at this point to remark that the Hecke operators can be defined in the case of Maass forms in a way similar to the case of holomorphic modular forms. However, due to the normalization we have chosen, the Hecke operators that we shall define will look a little different from the way these are usually defined for holomorphic modular forms. We define the n-th Hecke operator Tn acting on functions f : H −→ C by X 1 X az + b (Tn g)z = √ χ(a) . g d n ad=n b(mod d)

Note that this definition is independent of the weight k. If f (z) is an eigenfunction of the n-th Hecke operator (as usually defined for holo√ morphic forms) with eigenvalue λf (n), then g(z) = yf (z) is also an eigenfunction of Tn as defined above. Moreover, it can be checked that the corresponding Hecke eigenvalues are equal; i.e., if the eigenvalue of the n-th Hecke operator (as defined above) acting on g is λg (n), then λg (n) = λf (n).

4. Proof of the theorem We prove theorem 1 in this section. Proof. Let N1oct (q, χ) be the set of normalized primitive forms of octahedral type. Recalling proposition (10), we may note that for each f ∈ N1oct (q, χ), the function √ g(z) = yf (z) is a Maass cusp form with eigenvalue 1/4. From (35), the Fourier coefficients ρg (n) of g and af (n) are related by √ 4πnρg (n) = af (n). (36) Let us put h(t, r) =

4π 3 1 . . 1 2 |Γ(1 − 2 − ir)| chπ(r − t)chπ(r + t) 14

Then the Kuznetsov formula (30) can be rewritten as X X 1 Z ∞ √ √ h(t, r) mnρa (m, t)ρa (n, t)dt h(tj , r)ρj (m)ρj (n) mn + 4π −∞ a j X

= δ(m, n) +

c≡0(mod q)

Observe that the set (

1 Sχ (m, n; c)Ir c

√ 4π mn .(37) c

)

g hg, gi

1 2

:y

− 12

g ∈ N1oct (q, χ)

is a subset of the orthonormal basis {uj } of the whole space of Maass cusp forms C1 (Γ0 (q), χ) (corresponding to eigenvalue 1/4; i.e., forms with tj = 0). Thus, by the positivity of the function h(t, r) , for any N ≥ 1, and any sequence of complex numbers {cn }, we can bound the sum 2 X X h(0, r) S= c a (n) (38) n f hg, gi n≤N,(n,q)=1 −1 oct f =y

2 g∈N

1

(q,χ)

by the following sum which ranges over the complete spectrum; 2 2 Z ∞ ∞ X X X 1 X √ √ h(t, r) cn nρa (n, t) dt, cn nρj (n) + h(tj , r) 4π −∞ n≤N n≤N a j=1 (n,q)=1 (n,q)=1 where, as in (28), the ρj s are the Fourier coefficients of uj s. Applying the Kuznetsov formula, we obtain ∞ X

h(tj , r)

j=1

XX

√ cm cn mnρj (m)ρj (n)

m,n≤N (mn,q)=1

X X

X 1 Z ∞ + h(t, r) 4π −∞ a =

X n≤N (n,q)=1

2

|cn | +

XX m,n≤N (mn,q)=1

cm cn

m,n≤N (mn,q)=1

X c≡0(mod q)

15

√ cm cn mnρa (m, t)ρa (n, t)

1 Sχ (m, n; c)Ir c

√ 4π mn . c

Hence, using the Weil bound for Kloosterman sums 1√ |Sχ (m, n; c)| ≤ (m, n, c) 2 cτ (c), 1

1

and√ using the bound Ir (x) k x− 2 (|r| + 1) 2 for the terms with c ≤ 4π mn and the bound Ir (x) k x(1√+ | log x|) (see §12 of [DFI02] for these bounds) for those with c > 4π mn and taking r = 0, we finally obtain 2

S ε

X

|cn |2 +

n≤N (n,q)=1

1 2

X N (qN )ε |cn | . q n≤N

(39)

(n,q)=1

Now using the relation (9) af (p6 ) − af (p2 ) − χ(p)af (p2 ) = 1 we construct a specific sequence {coct n } such that X coct n af (n) n≤N, (n,q)=1

is comparatively large whenever f is a modular form of octahedral type but the individual terms coct n are small on average. We define 6 1 if n = p ≤ N for some prime p, (p, q) = 1 1 −1 − ( pq ) if n = p2 ≤ N 3 for some prime p, (p, q) = 1 coct n = 0 otherwise. The prime number theorem tells us that the number of primes p ≤ 1 6N 6 N is asymptotically log . Thus the above relation implies N 1 6

1

X

coct n af (n)

n≤N,(n,q)=1

The sequence {cn } also satisfies X

N6 . log N

(40)

1

2 6 |coct n | N

n≤N, (n,q)=1

and 1

X

6 |coct n | N

n≤N, (n,q)=1

again by the prime number theorem. The above two bounds, together with (39) therefore imply, 5

N 6 (qN )ε S ε N + . q 1 6

16

(41)

Also, hg, gi satisfies Z Z dxdy 2 dxdy hg, gi = |g(z)| y |f (z)|2 2 = hf, f i = 2 y y Γ\H Γ\H and by Proposition 2 in [Du95], hg, gi q log3 q.

(42)

Hence, (38), (40), and (41) imply N1oct (q, χ)

5 6

ε

!

N (qN ) log N ε N + 1 q N6 1 1 ε N − 6 q + N 2 (qN )ε . 1 6

2

q log3 q.

3

Choosing N = q 2 , we get the required bound, with the implied constant depending only on ε. References [BG08]

M. Bhargava and E. Ghate, On the average number of octahedral newforms of prime level, preprint (2008). [Cu96] R. T. Curtis, Monomial modular representations and construction of the Held group, Journal of Algebra, 184, 1205-1227. [DS74] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ec. Norm. Super., IV. Ser. 7, 507-530 (1974). [Du95] W. Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices 1995, no. 2, 99-109. [DFI02] W. Duke, J. Friedlander, and H. Iwaniec, The subconvexity problem for Artin L-functions, Invent. math. 149 (2002), 489-577. [Ell03] Jordan S. Ellenberg, On the average number of octahedral modular forms, Math. Res. Lett. 10 (2003), no. 2-3, 269–273. [Gan08] S. Ganguly, Large sieves and cusp forms of weight one, Proceedings of the International Conference on Number Theory and Cryptography at HarishChandra Research Institute, Allahabad, February 2007, to appear. [Hecke] E. Hecke, Z¨ ur Theorie der elliptischen Modulfunktionen, (no. 23 in Mathematische Werke. (German) Mit einer Vorbemerkung von B. Schoenberg, einer Anmerkung von Carl Ludwig Siegel, und einer Todesanzeige von Jakob Nielsen. Zweite durchgesehene Auflage. Vandenhoeck Va & Ruprecht, G¨ ottingen, 1970. 956 pp. ). [Iw95] H. Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matem´atica Iberoamericana, Revista Matem´atica Iberoamericana, Madrid, 1995. xiv+247 pp. [Klein] Felix Klein, Vorlesungen u ¨ber das Ikosaeder und die Aufl¨ osung der Gleichungen vom f¨ unften Grade, Birkh¨auser Verlag, Basel, 1993. Reprint of the 1884 original, Edited, with an introduction and commentary by Peter Slodowy. 17

[Klu06] Kl¨ uners, J¨ urgen, The number of S4 -fields with given discriminant, Acta Arith. 122 (2006), no. 2, 185–194. [MV02] P. Michel and A. Venkatesh, On the dimension of the space of cusp forms associated to 2-dimensional complex Galois representations, International Math. Research Notices, no. 38 (2002), 2021-2027. ¨ [Sch] I. Schur, Uber die Darstellung der symmetrischen und der alterniereden Gruppe durch gebrochen lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250. [Sel] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. [Ser77] J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic Number Fields, ed A. Fr¨ohlich, Proc. Symp. Durham 1975, 193-268, Academic Press (London), 1977. [Wo99] S. Wong, Automorphic forms on GL(2) and the rank of class groups, J. Reine Angew. Math. 515 (1999), 125-153. School of Mathematics,Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Colaba, Mumbai - 400 005, India E-mail address, Satadal Ganguly: [email protected]

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