Fourier Coefficients of Modular Forms and the Petersson Formula

Don Larson UCLA March 15, 2005 I. Introduction

During the Winter 2005 term at UCLA, Professor Charles Li and I conducted an informal reading course on the theory of modular forms beyond what is covered in the introductory graduate number theory courses. Our main goals were to derive the standard bounds on the size of Fourier coefficients of cusp forms (leading up to the Ramanujan conjectures) and to prove the Petersson formula. Sarnak’s book covered much of this theory but omitted many details from the calculations; I attempted to fill in these details through a series of weekly presentations at the blackboard. What follows is a written summary of our discussions throughout the term.

II. Preliminaries

Let H denote the upper half complex plane. The group Γ(1) = SL(2, Z) of 2 × 2 integral matrices with determinant 1 is called the full modular group. Often in the literature Γ(1) is instead defined to be PSL(2, Z) = SL(2, Z)/ ± I which merely changes some formulas by a factor of 2. In either case Γ(1) acts on H by linear fractional transformations

(\begin{matrix}a & b \\ c & d \end{matrix}) · z = \frac{az + b}{cz + d}.

This is a bona fide group action since

(\begin{matrix}a & b \\ c & d \end{matrix}) · ((\begin{matrix}α & β \\ γ & δ \end{matrix}) · z) = (\begin{matrix}a & b \\ c & d \end{matrix}) · (\frac{αz + β}{γz + δ} )

= \frac{a (\frac{αz+β}{γz+δ} ) + b}{c (\frac{αz+β}{γz+δ} ) + d}

= \frac{(aα + bγ)z + aβ + bδ}{(cα + dγ)z + cβ + dγ} = ((\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}α & β \\ γ & δ \end{matrix})) · z

and it is well-defined on H as

Im ((\begin{matrix}a & b \\ c & d \end{matrix}) · z) = \frac{\frac{az+b}{cz+d} − \frac{az̄+b}{cz̄+d}}{2i}
= \frac{(ad − bc)z − (ad − bc)z̄}{2i(cz + d)(cz̄ + d)}
= \frac{z − z̄}{2i|cz + d|^{2}} = \frac{Im z}{|cz + d|^{2}} > 0

for all z ∈ H. Furthermore, the orbits of this action define an equivalence relation ∼ on H where z ∼ w iff there exists γ ∈ Γ(1) such that γ · z = w. Denote the set of equivalence classes under ∼ by Γ(1)\H. The region \{−\frac{1}{2} < Re z < \frac{1}{2}\} ∩ \{|z| > 1\}, along with half of its boundary (to the left or to the right of the imaginary axis), constitutes a complete set of representatives for Γ(1)\H in the complex plane with no two points lying in the same equivalence class. This is called a fundamental domain for Γ(1) and is denoted by F. Note that this definition of F is standard but not unique. Adding the point ∞ makes F (and therefore Γ(1)\H) a compact Riemann surface of genus zero - the coordinate charts are straightforward except in the neighborhoods of i and ρ = \frac{−1+i\sqrt{3}}{2}.

Recall that the Cayley transform z ↦ \frac{z−i}{z+i} maps H onto the open unit disk and maps the real line into the unit circle S^{1}. It also sends the point ∞ to 1. A point of the fundamental domain F whose image under the Cayley transform has unit modulus is called a cusp of Γ(1)\H. In other words, cusps are points of F that lie in the boundary R ∪ ∞ of H. Hence Γ(1)\H as defined above has only one cusp, namely ∞ (sometimes written i∞). Note that the action of certain proper subgroups of Γ(1) can give rise to fundamental domains with multiple cusps; this generalization will be important later.

The first critical definition is that of a modular form for the full modular group. A function f on H is a modular form of weight k for Γ(1) if it has the following three properties:

1. f is holomorphic on H

2. f(γz) = (cz + d)^{k}f(z) for all γ ∈ Γ(1)

3. f is holomorphic at the cusp at ∞

Only the last property requires clarification. Note that both the exponential map e : z ↦ e^{2πiz} and f are perodic with period 1 since e(z + 1) = e^{2πi(z+1)} = e^{2πi}e^{2πiz} = e(z) and

f(z + 1) = f ((\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) · z) = (0z + 1)^{k}f(z) = f(z).

Observe also that when restricted to the upper half plane, the image of e is the open unit disk with the origin removed. Making the change of variables q = e^{2πiz} then gives f(z) = g(e(z)) = g(q) for some function g which is holomorphic on the punctured unit disk. The function f is said to be holomorphic at ∞ if the induced function g is holomorphic at 0.

If f is a modular form for Γ(1) then the corresponding function g(q) has a Laurent series expansion at 0 by holomorphicity, say

g(q) = ∑_{−∞}^{∞} a_{n}q^{n}

for some a_{i} ∈ C. That g is holomorphic at zero also implies a_{n} = 0 for all n < 0 since otherwise g would have poles. Hence f has a Fourier expansion at ∞ given by

f(z) = ∑_{n=0}^{∞} a_{n}e(nz).

The complex number a_{n} is the nth Fourier coefficient of f at ∞. A modular form f is called a cusp form if a_{0} = 0. Cusp forms are equivalently characterized as those modular forms that vanish at ∞ since

lim_{z→∞} e(nz) = lim_{y→∞} e^{2πin(x+iy)} = lim_{y→∞} e^{2πinx}e^{−2πny} = 0.

Let M_{k}(Γ(1)) be the set of modular forms of weight k for Γ(1) and S_{k}(Γ(1)) the subset of weight k cusp forms. If f_{1}, f_{2} ∈ M_{k}(Γ(1)) then their sum is also contained in M_{k}(Γ(1)) since

(f_{1} + f_{2})(γz) = (cz + d)^{k}f_{1}(z) + (cz + d)^{k}f_{2}(z) = (cz + d)^{k}(f_{1} + f_{2})(z),

and since the sum of two holomorphic functions is holomorphic. It is equally evident that the sum of two weight k cusp forms yields a weight k cusp form. In addition, if f is a modular form (resp. a cusp form) then so is cf for any complex scalar c. Thus M_{k}(Γ(1)) is a complex vector space with S_{k}(Γ(1)) a vector subspace. Moreover the following theorem from Riemann surface theory implies that M_{k}(Γ(1)) is finite dimensional over C. Note that what is important here is the finiteness and not the actual dimension itself.

Theorem 1 Let X be a compact Riemann surface and p_{1}, …, p_{n} ∈ X. Let r_{1}, …, r_{n} be integers. Denote by V the space of meromorphic functions f on X such that f is holomorphic except possibly at the points p_{i}, and such that at each p_{i}, either f is holomorphic or f has a pole of order r_{i}. Then dim V ≤ r_{1} + ⋯ + r_{n} + 1.

Corollary 2 M_{k}(Γ(1)) is finite dimensional.

To prove the corollary, consider 0 ≢ f_{0} ∈ M_{k}(Γ(1)) which is defined on the Riemann surface X = Γ(1)\H (compactified with a point at ∞). Now f_{0} only has finitely many zeros in X by compactness, for otherwise f_{0} would vanish identically. Label these zeros p_{1}, …, p_{n} and let r_{1}, …, r_{n} be the respective orders of vanishing. Let V be defined as in the theorem. If f ∈ M_{k}(Γ(1)) then the quotient \frac{f}{f_{0}} is meromorphic on H and ∞. This quotient is also invariant under Γ(1) since

\frac{f(γz)}{f_{0}(γz)} = \frac{(cz + d)^{k}f(z)}{(cz + d)^{k}f_{0}(z)} = \frac{f(z)}{f_{0}(z)}

for γ ∈ Γ(1), making it a well-defined function on Γ(1)\H. Therefore the map f ↦ \frac{f}{f_{0}} is an isomorphism from M_{k}(Γ(1)) to V. The result follows from Theorem 1.

Let Γ(N) = {γ ∈ Γ(1) | γ ≡ I mod N}, i.e., the subgroup of matrices that are entrywise congruent to the identity matrix modulo N for some N ∈ Z^{+}. A group Γ such that Γ(N) ≤ Γ ≤ Γ(1) is called a congruence subgroup of the full modular group. An important example of a such a subgroup is

Γ_{0}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ(1) | N divides c\} .

Each congruence subgroup inherits the action of Γ(1) on H, and thus associated to each congruence subgroup is a (not necessarily unique) fundamental domain F_{Γ} with a definition analogous to that of F for Γ(1). In fact if Γ is a congruence subgroup with [Γ(1) : Γ] = m then one possibility is

F_{Γ} = ⋃_{j=1}^{m} γ_{j}F

where γ_{1}, …, γ_{m} are coset representatives of Γ\Γ(1) and F is a fundamental domain for Γ(1).

With more than just the full modular group in hand, it is possible to define a more general notion of modular forms. Let Γ be a congruence subgroup. A modular form of weight k for Γ is a function f on H with the following three properties:

1. f is holomorphic on H

2. f(γz) = (cz + d)^{k}f(z) for all γ ∈ Γ

3. f is holomorphic at each cusp of Γ\H

Here again, the last property requires explanation. Consider the action of Γ(1) on Q ∪ ∞ ⊂ ∂H. This action has only one orbit since any rational number q = \frac{a}{b} with (a, b) = 1 is sent to ∞ by some γ ∈ Γ(1). To see this, let

γ_{q} = (\begin{matrix}x & y \\ −b & a \end{matrix})

where x and y are integers such that xa + yb = 1 given by the Euclidean algorithm. Then γ_{q} ∈ Γ(1) and

γ_{q} · q = (\begin{matrix}x & y \\ −b & a \end{matrix}) · \frac{a}{b} = \frac{x (\frac{a}{b}) + y}{−b (\frac{a}{b}) + a} = \frac{b^{−1}}{0} = ∞

since b ≠ 0. This is another way to see why a fundamental domain for the full modular group has only the one cusp at ∞. However for the action of a congruence subgroup Γ on Q ∪ ∞ the above construction is not always possible due to the congruence condition imposed on the matrix entries. Thus Q∪∞ can be split into multiple equivalence classes under the action of a proper congruence subgroup Γ, say with representatives q_{1}, …, q_{r} ∈ Q ∪ ∞, resulting in multiple cusps for the fundamental domain F_{Γ}. There will only be a finite number r of equivalence classes since any Γ is of finite index in Γ(1) and since SL(2, Z) acts transitively on Q ∪ ∞. It is customary to set q_{1} = ∞ as ∞ is always a cusp. For an example, consider the fundamental domain for the congruence subgroup Γ(2) which has the cusp at ∞ plus two additional cusps at 1 and 0.

It remains to define holomorphicity of a modular form at an arbitrary cusp. Let f be a modular form of weight k for Γ. For a cusp q_{j} of Γ\H, the argument above shows that there exists γ_{j} ∈ Γ(1) such that γ_{j} · q_{j} = ∞. Consider the function F : z ↦ f(γ_{j}^{−1}z). This function is a modular form for Γ_{j} = γ_{j}Γγ_{j}^{−1} since

F (γ_{j}γγ_{j}^{−1}z) = f(γ_{j}^{−1}γ_{j}γγ_{j}^{−1}z) = (cz + d)^{k}f(γ_{j}^{−1}z) = (cz + d)^{k}F (z)

where γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ. The function f is said to be holomorphic at q_{j} if F is holomorphic at ∞.

The notion of a cusp form generalizes as well. Note first that the subgroup

Γ_{∞} = \{(\begin{matrix}1 & n \\ 0 & 1 \end{matrix}) | n ∈ Z\}

of Γ(1) is precisely the stabilizer of the point ∞ under the action on H by linear fractional transformations. This subgroup is free abelian of rank 1 with generator ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Thus given any congruence subgroup Γ, the subgroup Γ ∩ Γ_{∞} is of finite index in Γ_{∞} and is generated by ( \begin{matrix}1 & m \\ 0 & 1 \end{matrix} ) for some m ∈ Z. Now let f be a modular form of weight k for Γ a congruence subgroup, and define Γ_{j}, F (z) as before so that F is a modular form for Γ_{j} and holomorphic at ∞. Then Γ_{j} ∩ Γ_{∞} is generated by ( \begin{matrix}1 & M \\ 0 & 1 \end{matrix} ) for some M ∈ Z so that

F (z + M) = F ((\begin{matrix}1 & M \\ 0 & 1 \end{matrix}) · z) = F (z).

Since the function e(z/M) is also periodic with period M, the change of variables q = e(z/M) gives F (z) = F^{∗}(q) for some function F^{∗} defined on the punctured unit disk. Thus F has a Fourier development at ∞ given by

(1)  F (z) = ∑_{n=0}^{∞} a_{n}e(nz/M)

and the Fourier coefficents of f at q_{j} are exactly the {a_{i}}_{i=0}^{∞} given in (1). The function f is a cusp form if a_{0} = 0 at each cusp q_{j}. Denote by M_{k}(Γ) the set of modular forms of weight k for Γ, and let S_{k}(Γ) be the subset of weight k cusp forms.

III. The trivial bound

Theorem 3 Let f ∈ S_{k}(Γ) with Fourier coefficients {a_{i}}_{i=1}^{∞}. Then

a_{n} = O(n^{k/2}),

i.e., a_{n} ≤ Cn^{k/2} for some constant C independent of n.

Proof: Define Φ(z) = |f(z)|(Im z)^{k/2}. The critical property of Φ is that it vanishes at each cusp since f is itself a cusp form. For example, at ∞, with z = x + iy and q = e(z),

Φ(z) = |∑_{n=1}^{∞} a_{n}q^{n}| y^{k/2} ≤ e^{−2πy}y^{k/2} ∑_{n=1}^{∞} |a_{n}|e^{−2π(n−1)y}

where the sum on the right-hand side vanishes term by term as y → ∞ since the exponential decay of e^{−2π(n−1)y} dominates the growth of y^{k/2} for all n ≥ 1. Moreover Φ is Γ-invariant since

Φ(γz) = |f(γz)|(Im γz)^{k/2} = |cz + d|^{k}|f(z)|\frac{(Im z)^{k/2}}{|cz + d|^{k}} = Φ(z)

for any γ ∈ Γ. Thus Φ is a well-defined function on Γ\H which is bounded on all of Γ\H and therefore bounded on H. That is, there exists M such that |F (z)| ≤ M for all z ∈ H. In other words |f(z)| ≤ My^{−k/2}. Now by orthogonality

a_{n} = ∫_{0}^{1} f(z)e(−nz)dz,

so for a fixed value of y

|a_{n}| = |∫_{0}^{1} f(z)e(−nz)dx|
= |∫_{0}^{1} f(z)e^{−2πinx}e^{2πny}dx|
≤ e^{2πny} ∫_{0}^{1} |f(z)|dx
≤ e^{2πny}My^{−k/2}.

Choosing y = 1/n gives |a_{n}| ≤ e^{2π}Mn^{k/2} = Cn^{k/2} for C independent of n. Here the independence of the constant is thanks to the optimal choice for y. □

IV. Eisenstein series

Up to now there has been no evidence that modular forms, let alone cusp forms, exist in nature. Let k > 2 even and define the Eisenstein series of weight k to be

E_{k}(z) = \frac{1}{2} ∑_{\begin{matrix}m,n∈Z \\ (m,n)≠(0,0) \end{matrix}}(mz + n)^{−k} = \frac{1}{2}∑_{m,n}′(mz + n)^{−k}.

As a function on H the series E_{k}(z) is absolutely convergent. To see this, fix z ∈ H, let L = {mz + n | m, n ∈ Z} be the integer lattice in the complex plane generated by 1 and z, and let P_{r} be the parallelogram with vertices ±r, ±rz. Notice that the subsets P_{r} form “concentric parallelograms” in the complex plane as r runs over the positive integers, and that

⋃_{r∈Z^{+}} P_{r} ∩ L = L \ {0}.

A simple computation shows that each P_{r} intersects the integer lattice L 8r times; the strategy will be to sum over the lattice points contained in each individual parallelogram separately. If h is the minimum modulus for a point in P_{1} then evidently |w| ≥ rh for w ∈ P_{r}. Thus

∑_{w∈P_{r}∩L} |w|^{−k} ≤ 8r(rh)^{−k}

which gives

E_{z}(k) = \frac{1}{2}∑_{m,n}′(mz + n)^{−k} = ∑_{r=1}^{∞} ∑_{w∈P_{r}∩L} |w|^{−k} ≤ 8h^{−k} ∑_{r=1}^{∞} r^{−k+1}

where the last sum clearly converges for k > 2. In particular E_{k}(z) is therefore holomorphic in H. Also note that E_{k}(z) is holomorphic at ∞ since all terms with m ≠ 0 vanish as z → ∞, leaving

lim_{z→∞} E_{k}(z) = \frac{1}{2} ∑_{n∈Z\\{0\}} n^{−k} = 2 · \frac{1}{2} ∑_{n=1}^{∞} n^{−k} < ∞

as k even, k > 2 by assumption.

Eisenstein series also translate like weight k modular forms for the full modular group. Let γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(1) and m, n ∈ Z. The condition that det γ = 1 forces gcd(a, b) = gcd(c, d) = 1, so by the Euclidean algorithm the map (m, n) ↦ (ma + nc, mb + nd) is a 1-1 correspondence that permutes the integer pairs in Z × Z \ (0, 0). Hence

E_{k} (\frac{az + b}{cz + d}) = \frac{1}{2}∑_{m,n}′ (m (\frac{az + b}{cz + d}) + n)^{−k}

= \frac{1}{2}∑_{m,n}′ (\frac{m(az + b) + n(cz + d)}{cz + d})^{−k}

= (cz + d)^{k} \frac{1}{2}∑_{m,n}′((ma + nc)z + mb + nd)^{−k}

= (cz + d)^{k} \frac{1}{2}∑_{m,n}′(mz + n)^{−k}

= (cz + d)^{k}E_{k}(z).

So E_{k}(z) is indeed a modular form of weight k for Γ(1), proving that such functions do exist.

Calculating the Fourier expansion of Eisenstein series reveals that they are not cusp forms, as follows. Consider the function π^{2}/ sin^{2} πz. This function is meromorphic with poles of order 2 at z = m for all integers m. A theorem from complex analysis due to Mittag-Leffler then gives the partial fraction decomposition

(2)  \frac{π^{2}}{sin^{2} πz} = ∑_{−∞}^{∞} \frac{1}{(z − m)^{2}} .

Observe that on the left-hand side is the derivative of −π cot πz and on the right are the derivatives of −1/(z − m). Because the series

∑_{m≠0} (\frac{1}{z − m} + \frac{1}{m})

converges uniformly on compact sets (compare with ∑ 1/m^{2}), term-by-term integration of (2) is possible. Thus

π cot πz = z^{−1} + ∑_{m≠0} (\frac{1}{z − m} + \frac{1}{m}) = z^{−1} + ∑_{m=1}^{∞}((z + m)^{−1} + (z − m)^{−1}).

On the other hand the identities

cos z = \frac{e^{iz} + e^{−iz}}{2} , sin z = \frac{e^{iz} − e^{−iz}}{2i}

give

π cot πz = π (\frac{cos πz}{sin πz} ) = πi (\frac{q + 1}{q − 1})

where q = e(z) as before. Moreover, summing the geometric series ∑_{n=1}^{∞} q^{n} = q/(1 − q) yields

\frac{q + 1}{q − 1} = 1 − 2 (1 + \frac{q}{1 − q} ) = 1 − 2 (1 + ∑_{n=1}^{∞} q^{n}) = 1 − 2 (∑_{n=0}^{∞} q^{n}) .

Therefore

(3)  ∑_{−∞}^{∞}(z + m)^{−1} = πi (1 − 2 ∑_{n=0}^{∞} q^{n}) .

The next step is to differentiate both sides of (3) with respect to z. Let LHS, RHS be the left-hand and right-hand sides of (3) respectively. Then

\frac{d}{dz} (LHS) = − ∑_{−∞}^{∞}(z + m)^{−2},

\frac{d^{2}}{dz^{2}} (LHS) = 2 ∑_{−∞}^{∞}(z + m)^{−3},

⋮
\frac{d^{k−1}}{dz^{k−1}} (LHS) = (−1)^{k−1}(k − 1)! ∑_{−∞}^{∞}(z + m)^{−k}

and

\frac{d}{dz} (RHS) = −(2πi)^{2} ∑_{n=1}^{∞} nq^{n},

\frac{d^{2}}{dz^{2}} (RHS) = −(2πi)^{3} ∑_{n=1}^{∞} n^{2}q^{n},

⋮
\frac{d^{k−1}}{dz^{k−1}} (RHS) = −(2πi)^{k} ∑_{n=1}^{∞} n^{k−1}q^{n}.

But since k −1 is odd, the negative signs on both sides cancel. Thus by equality of both (k − 1)st derivatives

(4)  ∑_{−∞}^{∞}(z + m)^{−k} = \frac{(2πi)^{k}}{(k − 1)!} ∑_{n=1}^{∞} n^{k−1}q^{n}.

Let σ_{k}(n) = ∑_{d|n} d^{k} for any n, k ∈ Z^{+} and consider a double sum of the form

∑_{m=1}^{∞} ∑_{n=1}^{∞} n^{k}q^{mn}.

Making the change of variables mn ↦ m makes the inner sum into a sum over the divisors of m, and collecting terms then gives

(5)  ∑_{m=1}^{∞} ∑_{n=1}^{∞} n^{k}q^{mn} = ∑_{n=1}^{∞} σ_{k}(n)q^{n}.

Thus by (4), (5), and the evenness of k,

E_{k}(z) = \frac{1}{2}∑_{m,n}′(mz + n)^{−k}

= \frac{1}{2} (2 ∑_{n=1}^{∞} n^{−k} + ∑_{m>0} ∑_{n=−∞}^{∞}(mz + n)^{−k} + ∑_{m<0} ∑_{n=−∞}^{∞}(mz + n)^{−k})

= ∑_{n>0} n^{−k} + \frac{1}{2} ( ∑_{m>0} ∑_{n=−∞}^{∞}(mz + n)^{−k} + ∑_{m>0} ∑_{n=−∞}^{∞} \frac{1}{(−1)^{k}} (mz − n)^{−k})

= ζ(k) + ∑_{m>0} ∑_{n=−∞}^{∞}(mz + n)^{−k}

= ζ(k) + \frac{(2πi)^{k}}{(k − 1)!} ∑_{m>0} ∑_{n>0} n^{k−1}q^{mn}

= ζ(k) + \frac{(2πi)^{k}}{(k − 1)!} ∑_{n>0} σ_{k−1}(n)q^{n}

where ζ(s) = ∑_{n>0} n^{−s} is the Riemann zeta function. It is well-known that the zeta function has no zeros to the right of 1 in C. The zeroth Fourier coefficient of E_{k}(z), which is equal to ζ(k) by the above calculation, is therefore non-zero since k > 2. This proves Eisenstein series are not cusp forms.

V. Poincaré series and the space of cusp forms

Fortunately, there are weight k cusp forms that are explicitly constructible. One such construction is of particular importance and is similar to the one used for building Eisenstein series. The notation j(γ, z) = cz + d for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) is standard and also convenient for what follows.

Recall that Γ_{∞} = \{( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) | n ∈ Z\} < Γ(1). For any integers m ≥ 0 and k > 2 even define the Poincaré series of weight k to be

P_{m}(z, k) = ∑_{γ∈Γ_{∞}\Γ(1)}(j(γ, z))^{−k}e(mγz).

Note first that this function is a sum over right coset representatives of the quotient group Γ_{∞}\Γ(1). If the element γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) is a representative of the right coset

γ̄ = Γ_{∞} (\begin{matrix}a & b \\ c & d \end{matrix}) = \{(\begin{matrix}a + rc & b + rd \\ c & d \end{matrix}) | r ∈ Z\} ∈ Γ_{∞}\Γ(1)

and if ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) ∈ γ̄, then a′ = a + rc and b′ = b + rd. In other words, the cosets of Γ_{∞}\Γ(1) are in 1-1 correspondence with pairs of integers (c, d) ≠ (0, 0). Thus for m = 0

P_{0}(z, k) = ∑_{(c,d)≠(0,0)}(cz + d)^{−k}e(0) = E_{k}(z)

is just the usual Eisenstein series, and for m > 0, E_{k}(z) dominates P_{m}(z, k) since the term e(mγ(x + iy)) decays exponentially as y → ∞. In particular the Poincaré series P_{m}(z, k) are holomorphic in H and vanish at ∞ for m > 0. They also satisfy the desired translation property P_{m}(γ_{0}z, k) = (c_{0}z + d_{0})^{k}P_{m}(z, k) for γ_{0} = ( \begin{matrix}a_{0} & b_{0} \\ c_{0} & d_{0} \end{matrix} ) ∈ Γ(1) since, in particular, right multiplication by γ_{0} simply permutes the coset representatives of Γ_{∞}\Γ(1) leaving the e(mγz) factor of each summand unchanged; the remaining details of this verification are similar to those from the analogous calculation for Eisenstein series. Thus for all m > 0, P_{m}(z, k) is a cusp form of weight k for Γ(1).

Poincaré series are crucial because they span the finite-dimensional vector space S_{k}(Γ(1)). To prove this, define an inner product ⟨ , ⟩ on S_{k}(Γ(1)) called the Petersson inner product

⟨f, g⟩ = ∫_{Γ(1)\H} y^{k}f(z)\overline{g(z)}\frac{dxdy}{y^{2}}

for f, g ∈ S_{k}(Γ(1)) and y = Im z. For f, g, h ∈ S_{k}(Γ(1)) and a, b ∈ C, this product satisfies ⟨af+bg, h⟩ = a⟨f, h⟩+b⟨g, h⟩ since integration is linear, ⟨g, f⟩ = \overline{⟨f, g⟩} since \overline{fḡ} = f̄g, and ⟨f, f⟩ > 0 since ff̄ = |f|^{2} > 0, hence it is a legitimite inner product. It is well-defined on Γ(1)\H since

(Im γz)^{k}f(γz)\overline{g(γz)} = \frac{(Im z)^{k}}{|cz + d|^{2k}} (cz + d)^{k}(cz̄ + d)^{k}f(z)\overline{g(z)} = y^{k}f(z)\overline{g(z)}

and since \frac{dxdy}{y^{2}} is Γ(1)-invariant as a measure. It is therefore independent of the choice of fundamental domain. The integral defining the inner product is convergent since f and g decay rapidly at the cusp at ∞. For P_{m}(z, k) a Poincaré series and f ∈ S_{k}(Γ(1)) any cusp form,

⟨P_{m}, f⟩ = ∫_{Γ(1)\H} P_{k}(m, z)\overline{f(z)}y^{k} \frac{dxdy}{y^{2}}

= ∫_{Γ(1)\H} ∑_{γ∈Γ_{∞}\Γ(1)}(j(γ, z))^{−k}e(mγz)\overline{f(z)}\frac{dxdy}{y^{2}}

= ∫_{Γ_{∞}\H}(cz + d)^{−k}e(mγz)(\overline{cz + d})^{−k}\overline{f(γz)}(Im γz)^{k}|cz + d|^{2k} \frac{dxdy}{y^{2}}

= ∫_{0}^{∞} ∫_{0}^{1} e(mz)\overline{f(z)}y^{k} \frac{dxdy}{y^{2}}

= ∫_{0}^{∞} ∫_{0}^{1} e(mz) (∑_{n=1}^{∞} \overline{a_{n}e(nz)}) y^{k} \frac{dxdy}{y^{2}}

= ∫_{0}^{∞} \overline{a_{m}}|e(mz)|y^{k} \frac{dy}{y^{2}}

= \overline{a_{m}} ∫_{0}^{∞} e^{−4πmy}y^{k} \frac{dy}{y^{2}}

= \overline{a_{m}}(4πm)^{1−k}Γ(k − 1),

where f has Fourier expansion ∑ a_{n}e(nz), and

Γ(s) = ∫_{0}^{∞} y^{s}e^{−y} \frac{dy}{y} > 0

is the gamma function. So, if a cusp form f is orthogonal to P_{m} in S_{k}(Γ(1)) for all positive integers m, i.e., if ⟨P_{m}, f⟩ = 0 for all m > 0, then \overline{a_{m}} = 0 for all m > 0 by the above calculation, which implies f must be identically zero. Thus there are no nonzero weight k cusp forms which are simultaneously orthogonal to all weight k Poincaré series. This proves that {P_{m}}_{m=1}^{∞} spans S_{k}(Γ(1)).

The Fourier coefficients of Poincaré series are computable using standard orthogonality relations from Fourier analysis. Fix y ∈ R, y > 0, and note that although the integrals that appear below are taken from 0 to 1, the technically correct bounds are iy and iy + 1 since P_{m} is only defined on H. One can then take the limit as y → 0^{+}. If P_{m}(z, k) = ∑ a_{n}q^{n}, then

a_{n} = ∫_{0}^{1} P_{m}(z, k)e(−nx)dx

= ∫_{0}^{1} ∑_{γ∈Γ_{∞}\Γ(1)} j(γ, z)^{−k}e(mγz)e(−nx)dx

= 2 ∫_{0}^{1} e(mz)e(−nx)dx + ∫_{0}^{1} ∑_{\begin{matrix}γ∈Γ_{∞}\Γ(1) \\ c≠0 \end{matrix}} j(γ, z)^{−k}e(mγz)e(−nx)dx

Denote these two summands A and B. Then

A = 2 ∫_{0}^{1} e^{2πim(x+iy)}e^{−2πinx}dx
= 2e^{−2πmy} ∫_{0}^{1} e^{2πi(m−n)x}dx
= 2e^{−2πmy}δ_{m,n}

where δ_{m,n} is the Kronecker delta. In the limit as y → 0 this becomes simply A = 2δ_{m,n}. To calculate B there are three small facts that prove useful. First, note that since

(\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}1 & r \\ 0 & 1 \end{matrix}) = (\begin{matrix}a & ra + b \\ c & rc + d \end{matrix}) ,

the lower right entry d of a left coset representative ( \begin{matrix}a & b \\ c & d \end{matrix} ) of Γ(1)/Γ_{∞} is uniquely determined modulo its lower left entry c. Therefore a sum over the double cosets of Γ_{∞}\Γ(1)/Γ_{∞} is equivalent to a double sum over all c ≠ 0 and over residue classes d modulo c. Second is the easily proven equation

γz = \frac{az + b}{cz + d} = \frac{a}{c} − \frac{1}{c(cz + d)}.

Third, any exchanges of the order of summations and integrals are allowed since Poincaré series themselves are absolutely convergent.

Thus

B = ∑_{\begin{matrix}Γ_{∞}\Γ(1)/Γ_{∞} \\ c≠0 \end{matrix}} ∑_{r∈Z} ∫_{0}^{1} j (γ (\begin{matrix}1 & r \\ 0 & 1 \end{matrix}) , z)^{−k} e (mγ (\begin{matrix}1 & r \\ 0 & 1 \end{matrix}) z) e(−nx)dx

= ∑_{\begin{matrix}Γ_{∞}\Γ(1)/Γ_{∞} \\ c≠0 \end{matrix}} ∑_{r∈Z} ∫_{0}^{1} j ((\begin{matrix}a & ra + b \\ c & rc + d \end{matrix}) , z)^{−k} e (m (\begin{matrix}a & ra + b \\ c & rc + d \end{matrix}) z − nx) dx

= ∑_{\begin{matrix}Γ_{∞}\Γ(1)/Γ_{∞} \\ c≠0 \end{matrix}} ∑_{r∈Z} ∫_{0}^{1}(c(z + r) + d)^{−k}e (m (\frac{a(z + r) + b}{c(z + r) + d}) − nx) dx

= ∑_{\begin{matrix}Γ_{∞}\Γ(1)/Γ_{∞} \\ c≠0 \end{matrix}} ∑_{r∈Z} ∫_{−r}^{1−r}(cz + d)^{−k}e (m (\frac{az + b}{cz + d}) − nx − nr) dx

= ∑_{\begin{matrix}Γ_{∞}\Γ(1)/Γ_{∞} \\ c≠0 \end{matrix}} ∫_{−∞}^{∞}(cz + d)^{−k}e(mγz)e(−nx)dx

= 2 ∑_{c>0} ∑_{\begin{matrix}d(mod c) \\ (c,d)=1 \end{matrix}} ∫_{−∞}^{∞}(cz + d)^{−k}e (\frac{ma}{c} − \frac{m}{c(cz + d)} − nx) dx

= 2 ∑_{c>0} c^{−k} ∑_{\begin{matrix}d(mod c) \\ (c,d)=1 \end{matrix}} e (\frac{ma}{c}) ∫_{−∞}^{∞} (z + \frac{d}{c})^{−k} e (\frac{−m}{c^{2} (z + \frac{d}{c})} − nx) dx

= 2 ∑_{c>0} c^{−k} ∑_{\begin{matrix}d(mod c) \\ (c,d)=1 \end{matrix}} e (\frac{ma + nd}{c}) ∫_{−∞}^{∞} z^{−k}e (\frac{−m}{c^{2}z} − nx) dx.

Summing up,

(6) a_{n} = 2δ_{m,n} + 2 ∑_{c>0} c^{−k} ∑_{\begin{matrix}d(mod c) \\ (c,d)=1 \end{matrix}} e (\frac{ma + nd}{c}) ∫_{−∞}^{∞} z^{−k}e (\frac{−m}{c^{2}z} − nx) dx

for P_{m}(z, k) = ∑ a_{n}q^{n}.

VI. The nontrivial bound

Because weight k Poincaré series span the finite-dimensional subspace of weight k cusp forms in M_{k}(Γ(1)), every weight k cusp form can be written as a finite complex-linear combination of the functions P_{m}(z, k). A bound on the size of the Fourier coefficients of Poincaré series therefore automatically gives a bound on the size of Fourier coefficients of an arbitraty cusp form up to a constant. The goal, then, is to bound the right-hand side of (6) in hopes of improving upon the trivial bound given in section III.

Fortunately there are two well-understood quantities hidden in equation (6) that will aid in calculating a bound, the first being a certain exponential sum known as a Kloosterman sum. For m, n, c ∈ Z a Kloosterman sum is of the form

K(m, n; c) = ∑_{\begin{matrix}d(mod c) \\ (c,d)=1 \end{matrix}} e (\frac{md̄+ nd}{c}) .

Since ad = 1 in (6), taking d̄= a gives

(7)  a_{n} = 2δ_{m,n} + 2 ∑_{c>0} c^{−k}K(m, n; c) ∫_{−∞}^{∞} z^{−k}e (\frac{−m}{c^{2}z} − nx) dx.

The second quantity is a function of the form

J_{n}(ζ) = \frac{1}{2πi} ∫_{R} \frac{e^{\frac{1}{2} ζ(t−t^{−1})}}{t^{n+1}} ,

called the nth Bessel function, where R is a region in the complex plane containing 0. To see how this function arises in (7), consider the integral

∫_{−∞}^{∞} x^{−k}e (\frac{−m}{c^{2}x} − nx) dx

which occurs as a factor of the right-hand side of (7) after letting y → 0. This integral vaguely resembles a Bessel function. In fact, with a clever change of variables it can be expressed as a constant times the Bessel function J_{k−1}(ζ) evaluated at (4π\sqrt{mn})/c. For illustrative purposes it is better to perform this change of variables in stages: x ↦ −x/n; x ↦ \frac{\sqrt{mn}}{c}x; x ↦ 2πix. This gives

∫_{−∞}^{∞} x^{−k}e (\frac{−m}{c^{2}x} − nx) dx = −n^{k−1} ∫_{−∞}^{∞} \frac{e (\frac{mn}{c^{2}x} + x)}{x^{k}}dx

= \frac{−n^{k−1}c^{k−1}}{(mn)^{(k−1)/2}} ∫_{−∞}^{∞} \frac{e (\frac{\sqrt{mn}}{c}(x + x^{−1}))}{x^{k}}dx

= (\frac{n}{m})^{\frac{k−1}{2}} c^{k−1}i^{−k−1} ∫_{−∞}^{∞} \frac{e (\frac{2π\sqrt{mn}}{c}(x − x^{−1}))}{x^{k}}dx

= (\frac{n}{m})^{\frac{k−1}{2}} c^{k−1}2πi^{−k}J_{k−1} (\frac{4π\sqrt{mn}}{c}) .

Plugging this back into (7) yields the following formula for a_{n}, now in terms of a Bessel function and a Kloosterman sum:

(8) a_{n} = 2 (\frac{n}{m})^{\frac{k−1}{2}} (δ_{m,n} + 2πi^{−k} ∑_{c>0} J_{k−1} (\frac{4π\sqrt{mn}}{c}) \frac{K(m, n; c)}{c}) .

The goal is now to bound a_{n} by bounding J_{k−1}(ζ) and K(m, n; c).

Proposition 4 For (c_{1}, c_{2}) = 1,

K(u, v, c_{1}c_{2}) = K(u, vc̄_{2}^{2}, c_{1})K(u, vc̄_{1}^{2}, c_{2}).

Proof: By the Chinese remainder theorem, if d and d′ run through invertible residues modulo c_{1} and c_{2} respectively, then elements of the form dc_{2} + d′c_{1} run through invertible residues modulo c_{1}c_{2}. Hence

K(u, vc̄_{2}^{2}, c_{1})K(u, vc̄_{1}^{2}, c_{2}) = ∑_{\begin{matrix}d(mod c_{1}) \\ (c_{1},d)=1 \end{matrix}} ∑_{\begin{matrix}d′(mod c_{2}) \\ (c_{2},d′)=1 \end{matrix}} e (\frac{ud + vc̄_{2}^{2}d̄}{c_{1}} + \frac{ud′ + vc̄_{1}^{2}d̄′}{c_{2}} )

= ∑_{d,d′} e (u (\frac{dc_{2} + d′c_{1}}{c_{1}c_{2}} ) + v (\frac{\overline{dc_{2} + d′c_{1}}}{c_{1}c_{2}} ))

= ∑_{\begin{matrix}d(mod c_{1}c_{2}) \\ (d,c_{1}c_{2})=1 \end{matrix}} e (\frac{ud̄+ vd}{c_{1}c_{2}})

= K(u, v, c_{1}c_{2})

as desired. □ Furthermore, for k ≥ 2 and p prime,

|K(u, v; p^{k})| ≤ 2p^{k/2},

and a highly nontrivial result of Weil gives

|K(u, v; p)| ≤ 2\sqrt{p}.

Now, by the fundamental theorem of arithmetic any c ∈ Z has a unique factorization into products of prime powers

c = ∏_{i=1}^{l} p_{i}^{k_{i}}.

Conbining these results gives

|K(m, n; c)| = |K (m, n; ∏_{i=1}^{l} p_{i}^{k_{i}})|

= |K(m, n_{1}; p_{1}^{k_{1}})||K(m, n_{2}; p_{2}^{k_{2}})| ⋯ |K(m, n_{l}; p_{l}^{k_{l}})|

≤ 2^{l} ∏_{i=1}^{l} p_{i}^{k_{i}/2}

= 2^{l}c^{1/2},

i.e.,

(9)  K(m, n; c) = O_{ε}(c^{1/2+ε}).

Thus by applying (9) to (8), the nth Fourier coefficient of P_{m} for a fixed m is

(10)  ≪_{k,ε} ∑_{c=1}^{∞} c^{ε− \frac{1}{2}} n^{\frac{k−1}{2}} |J_{k−1}(4π\sqrt{mn}/c)|.

There is a well-known bound for the Bessel function

(11)  J_{k−1}(ζ) ≪ min{ζ^{k−1}, ζ^{−1/2}}

where, clearly, the first choice is optimal for ζ small (ζ ≤ 1) and the second is optimal for ζ large (ζ > 1). The sum in (10) can be split around c = 4π\sqrt{mn} to take full advantage of (11), as follows:

∑_{c=1}^{∞} c^{ε− \frac{1}{2}} n^{\frac{k−1}{2}} |J_{k−1}(4π\sqrt{mn}/c)| ≪ ∑_{c≤4π\sqrt{mn}} c^{ε− \frac{1}{2}} n^{\frac{k−1}{2}} (\frac{4π\sqrt{m}}{c})^{−1/2} n^{−1/4}

+ ∑_{c>4π\sqrt{mn}} c^{ε− \frac{1}{2}} n^{\frac{k−1}{2}} (\frac{4π\sqrt{m}}{c})^{k−1} n^{\frac{k−1}{2}}

= ∑_{c≤4π\sqrt{mn}} c^{ε}n^{\frac{k−1}{2}} (4π\sqrt{m})^{−1/2}n^{−1/4}

+ ∑_{c>4π\sqrt{mn}} c^{ε−k+ \frac{1}{2}} n^{\frac{k−1}{2}} (4π\sqrt{m})^{k−1}n^{\frac{k−1}{2}} .

Call the first sum Σ_{finite} and the second Σ_{infinite}. Approximating each of them by integrals gives

Σ_{finite} ≤ n^{\frac{k}{2} − \frac{3}{4}} (4π\sqrt{m})^{−1/2} ∫_{1}^{4π\sqrt{mn}} c^{ε}

= n^{\frac{k}{2} − \frac{3}{4}} (4π\sqrt{m})^{−1/2} [\frac{c^{ε+1}}{ε + 1}] |_{1}^{4π\sqrt{mn}}

= (\frac{n^{\frac{k}{2} − \frac{3}{4}} (4π\sqrt{m})^{ε+1/2}}{ε + 1} ) (n^{ε+1/2} − 1) ≪ n^{\frac{k}{2} − \frac{1}{4} +ε},

Σ_{infinite} ≤ n^{k−1}(4π\sqrt{m})^{k−1} ∫_{4π\sqrt{mn}}^{∞} c^{ε−k+ \frac{1}{2}}

= n^{k−1}(4π\sqrt{m})^{k−1} lim_{t→∞} [\frac{c^{ε−k+ \frac{3}{2}}}{ε − k + \frac{3}{2}} ] |_{4π\sqrt{mn}}^{t}

= (\frac{n^{k−1}(4π\sqrt{m})^{k−1}}{ε − k + \frac{3}{2}} ) (−(4π\sqrt{mn})^{ε−k+ \frac{3}{2}} )

= (\frac{n^{k−1}(4π\sqrt{m})^{ε+1/2}}{ε − k + \frac{3}{2}} ) (−n^{ε− \frac{k}{2} + \frac{3}{4}} ) ≪ n^{\frac{k}{2} − \frac{1}{4} +ε}.

Thus a_{n} = O_{ε}(n^{\frac{k}{2} − \frac{1}{4} +ε}), a nontrivial step toward the Ramanujan’s conjecture that a_{n} = O_{ε}(n^{(k−1)/2+ε}).

References

[1] Elkies, N., notes from an analytic number theory course at Harvard,

www.math.harvard.edu/∼elkies/M259.02/index.html.

[2] Liu, J. and Ye, Y., Petersson and Kuznetsov trace formulas, to appear in Lie Groups and Automorphic Forms.

[3] Sarnak, P., Some Applications of Modular Forms, Cambridge Tracts in Math., Vol. 99 (1976).

[4] Selberg, A., On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, RI, 1965, pp. 1-15.

[5] Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, 4th edition, Cambridge Univ. Press, 1927.