Large sieves and cusp forms of weight one Satadal Ganguly∗ 10 October 2007 School of Mathematics, Tata Institute of Fundamental Research Homi Bhabha Road, Colaba Mumbai - 400 005, India

1

Introduction

The problem of estimating the dimension of the space of (holomorphic) cusp forms of weight one and a given level was first considered by Serre [Ser]. The problem is markedly different from the case when the weight k is two or more, in which case the dimension is O(kq), q being the level [Shim]. In the case of weight one, it is believed (see [Duke], [MV]) that the dimension should be 1 O(q 2 +ε ) where the implied constant is absolute. Recall that by the DeligneSerre theorem (Theorem 4), normalized newforms of weight one correspond to isomorphism classes of complex, two-dimensional, irreducible, continuous Galois representations. Here, the absolute Galois group is given the profinite topology, which means that continuity of such a representation is equivalent to its image in GL(2, C) being finite. So the image of the projectivization of one such representation is a finite subgroup in PGL(2, C) and therefore, by a classical result of Klein [Klein], must be either a dihedral group or one of A4 , S4 , and A5 . Depending on the images of the associated Galois representations, the weight one forms are referred to as of the dihedral, tetrahedral, octahedral, or icosahedral type. It is not difficult to prove that the dimension of the subspace of forms 1 of dihedral type is O(q 2 +ε ) (see [Hec], [Duke]). And it is expected that the dimension of the space spanned by those new forms (of a fixed nebentypus) that correspond to a non-dihedral Galois representation should only be O(q ε ) (see [Duke], [MV]). Serre himself speculated that the dimension of the space of forms of non-dihedral types should be O(q α ) for some real number α < 12 . However, proving even this is probably going to be quite difficult. To understand where the difficulty lies, one needs to consider the entire space of Maass cusp forms of which the holomorphic ones is just one eigenspace of an operator called the Laplace-Beltrami operator, namely the eigenspace with the eigenvalue 14 . The problem is that the point 41 is not isolated in the continous spectrum (which is ∗ E-mail:

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spanned by the Eisenstein series, see Section 3) and therefore it is difficult to use standard tools like the Selberg trace formula to estimate the dimension of the eigenspace of cusp forms with eigenvalue 14 . Using the special arithmetic properties of these weight-one forms, however, Duke [Duke] used techniques from analytic number theory, most importantly large sieves, to obtain a non-trivial improvement over the bound which was known before, which was essentially 11 O(q) (up to a factor of a power of log q). Duke showed that it is O(q 12 +ε ) when the level q is prime and the nebentypus is real. Many different authors worked on this problem after Duke (see [MV], [Wong],[Gan], [Klu]), and the best known result today, which is due to P. Michel and A. Venkatesh [MV], gives the bound 6 O(q 7 +ε ). In this talk, we shall show (Theorem 5) that if we assume a conjectural large sieve inequality for the Fourier coefficients of Maass cusp forms then the bound 1 for the space in question is indeed O(q 2 +ε ).

Acknowledgement This talk grew out of a part of the author’s PhD thesis written under the guidance of Prof. H. Iwaniec at Rutgers University, New Brunswick, New Jersey. The author would like to thank him for his support and generosity. He also thanks the Mathematics department of Rutgers University, the Institute of Mathematical Sciences, Chennai and the Tata Institute of Fundamental Research, Mumbai for providing excellent environment for working. And above all, he thanks Harish-Chandra Research Institute, Allahabad and the organizers for giving him the opportunity to speak at the memorable conference.

2

Maass forms

We give a brief overview of the theory. For details, the reader may consult [Iw], [DFI] or the papers by Maass and Selberg. We consider the Poincar´e upper half plane H = {z ∈ C : y = Im z > 0}. This becomes a complete Riemannian manifold when equipped with the metric ds2 = y −2 (dx2 + dy 2 ). Associated to the above Riemannian metric is a natural invariant measure dµ(z) =

dxdy y2

where z = x + iy. As a Riemannian manifold, it has dimension 2 and constant negative curvature −1. The action of the group SL(2, R) on H by linear fractional transformation az + b γz = cz + d

2

 for γ =

a c

b d

 ∈ SL(2, R) is an isometry. 

Definition 1. For any γ =

a c

b d

 ∈ SL(2, R), we define a function

jγ : H −→ C by jγ (z) =

cz + d = eiarg(cz+d) . |cz + d|

(1)

For an integer k, we define the Laplace-Beltrami operator of weight k to be

∆k = −y

2



∂2 ∂2 + ∂x2 ∂y 2



∂ k ∂2 + iky = (z − z)2 + (z − z) ∂x ∂z∂z 2



∂ ∂ − ∂z ∂z

 (2)

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. Definition 2. The congruence subgroup of level q is defined to be the group    a b Γ0 (q) = ∈ SL(2, Z) : c ≡ 0(mod q) . c d Let χ(mod q) be a primitive Dirichlet character such that χ(−1) = (−1)k .

(3)

This gives rise to a character of Γ = Γ0 (q) by setting χ(γ) = χ(d) for  γ=

a c

b d

 ∈Γ

A function f : H −→ C which satisfies the condition f (γz) = χ(γ)jγ (z)k f (z)

(4)

for all γ ∈ Γ is called is called automorphic of weight k and character χ. We shall denote the space of all such automorphic function by Ak (Γ, χ). Definition 3. 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. 3

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 integrable automorphic functions of weight k with respect to the inner product Z (5) hf, gi = f (z)g(z)dµz Γ\H

where dµz = y −2 dxdy is the hyperbolic measure. Note that this inner product is the same as the one usually defined for holomorphic forms (Petersson inner product), except 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 symmetric and bounded below by λ k2 and Bk (Γ, χ) is a dense subspace of the whole L2 space Lk (Γ, χ). Therefore, by results in functional analysis, the space Lk (Γ, χ) has a complete spectral resolution with respect to ∆k . This can be concretely realised in terms of the Eisenstein series and cusp forms. Definition 4. A cusp a of the congruence group 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 5. 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 , (6) 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 Γ∞

  1 = ± 0

b 1



 :b∈Z

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 4

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, [DFI]) with constant residue. Note that the Eisenstein series need not be square-integrable! So in order to obtain the spectral resolution of the operator ∆k , one defines a family of functions called the incomplete Eisenstein series which are automatically in Lk (Γ, χ). Definition 6. For a smooth, compactly supported function ψ : R+ −→ C, we define its associated incomplete Eisenstein series to be X χ(γ)jσa−1 γ (z)−k ψ(Im σa−1 γz) Ea (z|ψ) = (7) γ∈Γa \Γ

We shall denote the linear space of all incomplete Eisenstein series by E(Γ, χ). ˜ χ) be its closure in Lk (Γ, χ) (in the L2 -norm topology). Let E(Γ, ˜ χ) and the Then the operator ∆k has a purely continuous spectrum in E(Γ, ˜ χ) is “spanned” (in the sense of integrating with respect to the t space E(Γ, 1 parameter)  1 by
Eisenstein series with Re s = 2 ; in1 particular, the spectrum covers 4 , ∞ (see Proposition 4.1 in [DFI]). So 4 is at the bottom of the continuous spectrum. But
note that when the weight k is one, the operator ∆1 is bounded below by λ 12 = 14 ! We shall see below that the holomorphic cusp forms lie in the 14 -eigen space, but in the discrete spectrum, which is the topic of the next paragraph. 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). This space will play a very important role later on. Let us denote its closure in Lk (Γ, χ) by C˜k (Γ, χ). Then we have ˜ χ) = Lk (Γ, χ) C˜k (Γ, χ) ⊕ E(Γ,

(8)

It can be shown that he space Ck (Γ, χ) consists of functions whose constant term in the Fourier expansion at every singular cusp is zero. Definition 7. The functions in Ck (Γ, χ) which are also eigenfunction of the Laplace operator ∆k are called Maass cusp forms. Proposition 1. 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 Theorem 2. Any function f ∈ Lk (Γ, χ) can be expanded as  ∞ X X 1 Z ∞ 1 1 f (z) = hf, uj iuj (z) + f, Ea (∗, + it) Ea (z, + it)dt. 4π 2 2 −∞ a j=1 5

(9)

Here {uj }∞ j=1 is a complete orthonormal system of Maass cusp forms of weight k, the group Γ and character χ and this formula is to be understood in the L2 sense.

3

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. To make matters explicit, we shall define the holomorphic modular forms. The proofs can be found in [DFI]. Definition 8. A holomorphic function f : H −→C is called  a modular form of a b weight k, level q, and character χ, if for all γ = ∈ Γ0 (q), f satisfies c d the relation (cz + d)−k f (γz) = χ(γ)f (z) (10) and f is holomorphic at all the cusps of Γ0 (q). The following proposition makes the relation explicit. Proposition 3. Let f be
a Maass form
of weight k, level q, character χ, and the lowest eigenvalue λ k2 = k2 1 − k2 . Then k

F (z) = y − 2 f (z) is a holomorphic modular form of weight k, level q, and character χ and all such modular forms must arise this way. Note that when k > 1, the lowest eigen value is isolated from the continuous spectrum. This makes the problem easier when the weight is greater than one than when the weight is one. 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 2|n|

(4πny)e(nx)

(11)

n6=0

where the eigenvalue sj is given by λj = sj (1 − sj ) =

1 + t2j . 4

Recall that sj = 12 + itj , tj ∈ C. Here Wk,s (z) is a function called the Whittaker function (see [DFI]). Assume now that f is a Maass cusp form of weight one. Then the above Fourier expansion simplifies to 6

1

f (z) = y 2

X

1

(4πn) 2 ρf (n)e(nz)

n≥1

so that F (z) = y

− 21

f (z) s Fourier expansion X F (z) = aF (n)e(nz) n≥1

where

1

aF (n) = (4πn) 2 ρf (n).

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Later we shall have to work with new forms. So we just remark that the Hecke operators can be defined in the case of Maass forms in the similar way as in the case of holomorphic modular forms. However, due to our different choice of normalization, we now define the n-th Hecke operator Tn acting on functions f : H −→ C by   X 1 X az + b (Tn f )z = √ χ(a) f . d n ad=n

b(mod d)

Note that this definition is independent of the weight k. If F ∈ M1 (q, χ) is an eigenfunction of the n-th Hecke operator (as defined in the first chapter) with √ eigenvalue λF (n), then f (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 n-th Hecke eigenvalue of f is λf (n), then λf (n) = λF (n).

4

A large sieve inequality

Given any natural harmonics (say, values of a character χ(mod q)) and given a fixed vector c = (c1 , c2 , c3 , · · · ) the philosophy of large sieve says that as one varies χ over a set of primitive characters modulo some integer q, the vectors (χ(1), χ(2), · · · ) become approximately orthogonal to c. This means that the l2 -norm of the linear form X cn χ(n) n≤N

becomes small compared the l2 -norm, kck2 =

X

|cn |2

n≤N

of c for most of the χ’s. In other words, one gets a non-trivial bound for sums like 2 X X cn χ(n) χ(mod q)primitive n≤N 7

due to cancellation among the off-diagonal terms. Many large sieve inequalities have been established for Dirchlet characters, Fourier coefficients of modular forms and many other harmonics by many differentauthors. For an introduction to this beautiful subject, the reader may wish to consult chaper VII in the book by Iwaniec and Kowalski [IK]. We conjecture below a general large sieve inequality for symmetric powers of Hecke-Maass cusp forms of weight one. Let χ(mod q) be primitive. Let H1 (q, χ) = {uj } be an orthonormal basis of C1 (Γ0 (q), χ) consisting of eigenfunctions of all the Hecke operators. Such a basis exists by the primitivity of the character. Let {λj (n)} be the corresponding Hecke eigenvalues.

Conjecture For r ≥ 1, q ≥ 1, T ≥ 1, N ≥ 1, and any sequence {cn }N n=1 of complex numbers, we have 2 X X ε 2 r |cn |2 . c λ (n ) n j  (qN ) (qT + N ) n≤N uj ∈C1 (Γ0 (q),χ),|tj |≤T n≤N X

(13)

uj non-dihedral

Note that by Weyl’s law, the number of Maass cusp forms uj with |tj | ≤ T is about qT 2 . This conjecture is quite natural from the point of view of the philosophy of large sieve (see chapter VII of [IK]).

5

The Main Result

We are going to show that the dimensionof the space of non-dihedral cusp forms for prime level q and real character q· is Oε (q ε ). Note that this large sieve inequality involves a sum over a much larger set than what we are actually interested in, which is just one eigenspace corresponding to the eigenvalue 1/4. Nevertheless we are able to “amplify” the contribution from the forms we are interested in by taking advantage of the arithmetic nature of these forms. The arithmetic comes from the Deligne-Serre theorem stated below (see [DS] or [Ser]). Theorem 4. Let f be a normalized new form of weight 1, level q and nebentypus χ(mod q). Then there exists an odd, irreducible, two-dimensional, continuous Galois representation ρ with Artin conductor q and determinant character χ such that L(s, f ) = L(s, ρ). The modular form f determines ρ uniquely up to isomorphism.

8

In particular, for all primes p not dividing the level q, we have af (p) = Tr(ρ(Frp ))

(14)

det(ρ(Frp )) = χ(p)

(15)

and where af (n) is the n-th Fourier coefficient of f Since we are dealing with non-dihedral forms, the associated Galois representations have projective image S4 or A5 . The case A4 does not occur here because of our assumptions (see [Ser]). If we take a normalized new form f of weight one and level q, then its Fourier coefficients af (p) at primes p not dividing q are given by the trace of the image under the associated Galois representation of the Frobenius element. So, the possible values of af (p) become limited. Serre has explicitly computed all the possible values in these cases in [Ser] and using that data, Duke in has constructed ([Duke]) some linear relations satisfied by the Fourier coefficients of these type of forms at prime powers. For example if f is a form of icosahedral type,the level q is prime, and the nebentypus χ is real, then Duke shows that for primes p different from q, af (p12 ) − af (p8 ) − χ(p)af (p2 ) = 1

(16)

Using such relation and the conjecture above, we shall prove the following theorem. Theorem 5. Suppose q is a prime and χ is the character given by the Legendre symbol, χ(n) = ( nq ). We consider the set S1 (q, χ) of all cusp forms of weight one, level q, and character χ. Let N1 (q, χ) be the set of normalized new forms and let N1oct (q, χ) and N1ico (q, χ) be the subsets consisting of forms of octahedral and icosahedral type respectively. If the above conjecture holds then oct N1 (q, χ) + N1ico (q, χ) ε q ε . Proof. We prove it only for the icosahedral forms. The other case of the octahedral forms will be identical except we shall have a different relation than 16 We have the relation (16) af (p12 ) − af (p8 ) − χ(p)af (p2 ) = 1 Here p is any prime other than q. We choose the sequence {cn } to be  1 if n = p12 ≤ N for some prime p, (p, q) = 1   2  −1 if n = p8 ≤ N 3 for some prime p, (p, q) = 1 cn = 1  −χ(p) if n = p2 ≤ N 6 for some prime p, (p, q) = 1   0 otherwise.

9

Then by the prime number theorem and the above relation, 1

X

cn af (n) ∼

n≤N

12N 12 . log N

(17)

On the other hand, by the conjectural large sieve inequality above, we can get a sharp estimate on 2 X X cn af (n) S= f ∈N ico (q,χ) n≤N 1

as follows. The icosahedral forms f we are considering form a subset of the Maass cusp forms in the eigenspace with tj = 0. And since we are dealing with normalized primitive forms, the Fourier coefficients af (n) and the eigenvalues λf (n) coincide. So, by positivity, we can write from (13) that 2 X X r ε c a (n ) |cn |2 n f  (qX) (q + X) n≤X f ∈N ico (q,χ) n≤X X

(18)

1

P after taking T to be 1. The inner sum n≤N cn af (n) is supported on those integers n which are of the form pm where p is a prime and m = 2, 8, or 12. We break this sum into three parts depending on n being a square, 8th, or 12th power of a prime p 6= q. This leads to the following inequality. S  S12 + S8 + S2 where

2 X r Sr = cpr af (p ) . 1 f ∈N1ico (q,χ) p≤N 12 X

Now, applying (18) to this sum, we have 2 X X 1 r Sr = cpr af (p )  (qN )ε (q + N 12 ) |cn |2 . 1 n≤N f ∈N1ico (q,χ) p≤N 12 X

Therefore, S  S12 + S8 + S2 X 1  (qN )ε (q + N 12 ) |cn |2 n≤N

by applying (18) to each of the three sums.

10

And again by the prime number theorem, we have 1

X

|cn |2 ∼

n≤N

12N 12 , log N

so we can write

1

1

S  (qN )ε (q + N 12 )

N 12 . log N

Also, (17) implies that 1

S∼

|N1ico (q, χ)|

12N 12 log N

!2 .

The above two show that |N1ico (q, χ)|  q ε after choosing N > q 12 . Corollary  16. Since the dimension of the subspace of dihedral forms is known to be Oε q 2 +ε , the above theorem implies that 1

dim S1 (q, χ) ε q 2 +ε .

References [Buh] Buhler, Joe P., Icosahedral Galois representations. Lecture Notes in Mathematics, Vol. 654. Springer-Verlag, Berlin-New York, 1978. ii+143 pp. ISBN: 3-540-08844-X. [DFI] W. Duke, J. Friedlander, and H. Iwaniec, The subconvexity problem for Artin L-functions. Invent. math. 149 (2002), 489-577. [DS]

P. Deligne and J.-P. Serre, Formes modulaires de poids 1. Ann. Sci. Ec. Norm. Super., IV. Ser. 7, 507-530 (1974).

[Duke] W. Duke, The dimension of the space of cusp forms of weight one. Internat. Math. Res. Notices 1995, no. 2, 99-109. [Frey] G. Frey, Construction and arithmetical applications of modular forms of low weight. in C.R.M. Proc. and Lecture Notes, V.4, Elliptic Curves and Related Topics, A.M.S. (1994), 1-21. [Hec] 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. ). 11

[IK]

H. Iwaniec and E. Kowalski, Analytic Number Theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004.

[Iw]

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. [Klu] Kl¨ uners, J¨ urgen(D-KSSL-MI), The number of S4 -fields with given discriminant, Acta Arith. 122 (2006), no. 2, 185–194. [Gan] Satadal Ganguly, Large sieve inequalities and application to counting modular forms of weight one, PhD Thesis, Rutgers University, May 2006. [MV] 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. 2002:38 (2002) 2021-2027. [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.

[Ser]

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.

[Shim] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, NJ, 1994. [Wong] S. Wong, Automorphic forms on GL(2) and the rank of class groups. J. Reine Angew. Math. 515 (1999), 125-153.

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