ON THETA CONSTANT IDENTITIES AND THE EVALUATION OF TRIGONOMETRIC SUMS

HERSHEL M. FARKAS AND IRWIN KRA

1. Introduction and statements of results

Theta functions and theta constants are a powerful tool for the study of function theory. This theory also yields deep applications to number theory and combinatorics. In a sense this has been known for a long time (see, for example, [4], a reprint of an earlier version, [1] and [2]) and is the main theme of [3], whose notation and results we use throughout this paper^{1}. The main purpose of this note is to indicate some of the by products of our investigations and to show that many question are suggested by [3]; where similar questions are answered in our book, we only outline the arguments needed in the proofs. We start with evaluations of trigonometric sums – a result established in slightly different form in [3].

Theorem 1. (a) For all integers k ≥ 2,

∑_{m=1}^{k−1} tan^{2} \frac{πm}{2k} = \frac{(k − 1)(2k − 1)}{3}.

(b) For each odd integer k ≥ 3,

∑_{l=0,1,..., \frac{k−3}{2} , (1+2l,k)=1} tan^{2} \frac{π(1 + 2l)}{2k} = \frac{1}{k} ∑_{l=0}^{\frac{k−3}{2}} \frac{(1 + 2l)^{2}}{(k, 1 + 2l)}φ((k, 1 + 2l))

and^{2}

∑_{l=0}^{\frac{k−3}{2}} tan^{2} \frac{π(1 + 2l)}{2k} = \frac{(k − 2)(k − 1)}{6}.

circulation.

1We repeat only the most important definitions of [3] that we need. The reader should consult that book for any nonstandard terms encountered.

2Doron Zeilberger, after a lecture by HF where the next equality was demonstrated, posted on his web journal on October 5, 1999 a note entitled The revenge of the plain mathematician that contains a proof of the next equality using properties of Chebychev polynomials. L. Ehrenpreis has informed us that a more elementary argument than Zeilberger’s for establishing the next equality is also apparently in

(c) For even integers k ≥ 2,

∑_{l=0,1,..., \frac{k−2}{2} , (1+2l,k)=1} tan^{2} \frac{π(1 + 2l)}{2k} = \frac{2}{k} ∑_{l=1}^{k−1} \frac{l^{2}}{(k, l)}φ((k, l)).

(d) For odd integers k ≥ 3,

∑_{m=1,2,..., \frac{k−1}{2} , (m,k)=1} tan^{2} \frac{πm}{k} = \frac{1}{k} ∑_{l=1}^{k−1}(2, l)\frac{l^{2}}{(k, l)}φ((k, l)).

The first identity in the theorem states, in particular, that for all even integers k ≥ 4,

∑_{m=1}^{\frac{k}{2} −1} tan^{2} \frac{πm}{k} = \frac{(k − 1)(k − 2)}{6},

which is a positive integer provided k ≢ 0 mod 3.

In the statement of the theorem, we have used standard number theoretic notation. Thus (a, b) is the greatest common divisor of the integers a and b, and φ is the Euler φ-function (for n ∈ ℤ^{+}, φ(n) is the number of positive integers ≤ n that are relatively prime to n). Out of respect for the formulae established, as we have done for other identities we derived, a number of cases, with relatively small k, were numerically verified using MATHEMATICA.

The above theorem is a consequence of an identity involving powers of ratios of θ-constants \frac{θ′[χ]}{θ′[χ]}, Theorem 2. Identities among these ratios are plentiful. See Corollary 1 and the remarks that follow. We are interested in identities among k-th powers of θ-constants generalizing the Jacobi quartic (7) (k = 4 and involving characteristics in X_{o}(2)). Among the formulae obtained is (6) (k = 8 and involving characteristics in X_{o}(4) and X_{o}(2)). We have not obtained identities among 4-th powers of θ-constant derivatives; the simplest relation obtained among such k-th powers is (9) (k = 8 and involving characteristics in X_{o}(4) and X_{o}(1)).

2. Preliminaries

The main tool in our investigations is the genus one θ-function with

characteristic [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] ∈ ℝ^{2}, variable ζ ∈ ℂ, and parameter τ ∈ ℍ^{2}, the

upper half plane, defined by

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (ζ, τ) = ∑_{n∈ℤ}exp 2πı [\frac{1}{2} (n + \frac{ϵ}{2})^{2} τ + (n + \frac{ϵ}{2}) (ζ + \frac{ϵ′}{2} )] .

The theta constants are defined by setting the variable ζ = 0 (they are holomorphic and generically nonvanishing functions on ℍ^{2})

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] = θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (0, τ).

The theta constant derivatives are

θ′ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] = θ′ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (0, τ),

where

θ′ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (ζ, τ) = \frac{∂}{∂ζ} θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (ζ, τ).

For fixed value of the parameter τ, θ-functions are used to study function theory on tori: For m and n ∈ ℤ,

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (ζ + n + mτ, τ)  (1)

= exp 2πı \{\frac{nϵ − mϵ′}{2} − mζ − \frac{m^{2}}{2} τ\} θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (ζ, τ).

The function θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (·, τ) has precisely one simple zero in a funda-

mental parallelogram for the torus with periods 1 and τ; at

ζ = \frac{1 − ϵ′}{2} + \frac{1 − ϵ}{2}τ.

The ℤ^{2}-quasi invariance of the θ-functions is not too relevant for most of our development; however, the invariance with respect to a second

group is a key ingredient in everything that follows. For γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ]

in SL(2, ℤ), we have

\frac{exp πı \{\frac{−cζ^{2}}{cτ+d}\} θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (\frac{ζ}{cτ+d}, \frac{aτ+b}{cτ+d})}{θ [ \begin{matrix}aϵ + cϵ′ − ac \\ bϵ + dϵ′ + bd \end{matrix} ] (ζ, τ)} = κ ([ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] , γ) (cτ + d)^{\frac{1}{2}} , (2)

where κ is a constant of absolute value 1. The ℤ^{2}-quasi invariance of the θ-functions (1) is one of the three key ingredients in establishing (2).

We view ℝ^{2} as the set of characteristics. We have a map

ℝ^{2} × SL(2, ℤ) → ℝ^{2}

defined by sending the pair (χ, γ) with χ = [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] ∈ ℝ^{2} and γ =

[ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ SL(2, ℤ) to

χγ = [ \begin{matrix}aϵ + cϵ′ − ac \\ bϵ + dϵ′ + bd \end{matrix} ] ∈ ℝ^{2}.

This does not define a SL(2, ℤ) action on ℝ^{2}. However, we obtain a group action of Γ = PSL(2, ℤ) on equivalence classes, ℝ^{2}/ ≡, of characteristics if we define

χ_{1} ≡ χ_{2} iff χ_{1} ± χ_{2} ∈ 2ℤ^{2}.

As a bonus we obtain that theta functions for equivalent characteristics are almost the same in the sense that they differ by a well understood multiplicative nonzero constant and possible replacement of the variable by its negative. From now on, we identify a characteristic with its equivalence class.

If G is a subgroup of Γ, we let X(G) denote the set of equivalence classes of characteristics that are fixed pointwise by the elements of G. For^{3} k ∈ ℤ^{+}, we let Γ(k) denote the level k principal congruence subgroup of the modular group Γ (it consists of elements of Γ represented

by matrices [ \begin{matrix}a & b \\ c & d \end{matrix} ] with a ≡ 1 ≡ d mod k and b ≡ 0 ≡ c mod k)

and X(k) the set of equivalence classes of the characteristics

\{[ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] ∈ ℝ^{2}; ϵ = \frac{m}{k}, ϵ′ = \frac{m′}{k} \} ,

with m and m′ ∈ ℤ of the same parity as k.

It is easily shown that

X(k) = X(Γ(k)).

We define

X_{o}(k) = {χ ∈ X(k); χ ∉ X(k′) for all 1 ≤ k′ < k, k′|k} .

The group Γ(k)\Γ acts by permutations on the two finite sets X(k) and X_{o}(k).

Let χ = [ \frac{m}{k} ; \frac{m′}{k} ] ∈ X(k), m and m′ ∈ \{0\} ∪ ℤ^{+} (of the same par-

ity as k), d = (m, m′, k). Then χ ∈ X_{o}(k) if and only if (for k ≡ 1 mod 2) d = 1, (for k ≡ 2 mod 4) d = 2 and at least one of \frac{m}{2} and \frac{m′}{2} is even, (for k ≡ 0 mod 4) d = 2 (it is automatic that not both \frac{m}{2} and \frac{m′}{2} are even).

3Throughout this paper k denotes a positive integer.

The punctures on ℍ^{2}/Γ(k) are in one-to-one almost canonical correspondence with the points in X_{o}(k). The action of Γ(k)\Γ on X_{o}(k) is transitive and minimal.

Theta constants and theta constant derivatives satisfy, as a result of (2), important functional equations. For all τ ∈ ℍ^{2}, all χ ∈ ℝ^{2} and all γ ∈ SL(2, ℤ),

\frac{θ[χ](0, γ(τ))}{θ[χγ](0, τ)} = κ(χ, γ)γ′(τ)^{− \frac{1}{4}}  (3)

and

\frac{θ′[χ](0, γ(τ))}{θ′[χγ](0, τ)} = κ(χ, γ)γ′(τ)^{− \frac{3}{4}} .

As a consequence of this invariance, for χ ∈ X(k), θ[χ] is a holomorphic

\frac{1}{4}-form for Γ(k); it is identically zero if χ = [ \begin{matrix}1 \\ 1 \end{matrix} ], it is free of zeros

on ℍ^{2} otherwise, it is a cusp form (vanishes at the cusps) if k is odd, and in general a modular form (it is holomorphic and may be nonzero at the cusps) if k is even. The derivative θ′[χ] is a holomorphic \frac{3}{4}-form

for Γ(k); it is identically zero if χ = [ \begin{matrix}0 \\ 0 \end{matrix} ], [ \begin{matrix}0 \\ 1 \end{matrix} ] or [ \begin{matrix}1 \\ 0 \end{matrix} ]; it is free

of zeros on ℍ^{2} if χ = [ \begin{matrix}1 \\ 1 \end{matrix} ]; otherwise, it is a cusp form. Its study is

complicated because, in general, it may have zeros on ℍ^{2}. If in addi-

tion χ ≠ [ \begin{matrix}1 \\ 1 \end{matrix} ], then (\frac{θ′[χ]}{θ[χ]})^{2} is a modular 1-form, in fact, belonging to

the canonical factor of automorphy, for Γ(k). It is identically zero for

χ = [ \begin{matrix}0 \\ 0 \end{matrix} ], [ \begin{matrix}0 \\ 1 \end{matrix} ] or [ \begin{matrix}1 \\ 0 \end{matrix} ]. For odd k, it does not vanish at any of the

cusps of Γ(k).

The Fourier series expansion in terms of the local coordinate ζ =

e^{\frac{2πıτ}{k}} of θ [ \frac{m}{k} ; \frac{m′}{k} ] and θ′ [ \frac{m}{k} ; \frac{m′}{k} ] with k a positive integer and m, m′

integers of the same parity as k which without loss of generality we can assume to also satisfy 0 ≤ m ≤ k, 0 ≤ m′ < 2k,^{4} can easily be computed. From these calculations, we obtain more precise information for the order of vanishing at ∞ of the forms under study.^{5} We summarize the relevant calculations. All orders of automorphic forms are computed with respect to the local coordinate ζ = e^{\frac{2πıτ}{k}} at the puncture determined by the cusp ∞ on the surface ℍ^{2}/Γ(k). For

[ \frac{m}{k} ; \frac{m′}{k} ] ∈ X(k) − X(1),

ord_{∞}θ[χ] = \frac{m^{2}}{8k} and θ[χ](0, ∞) = \{ \begin{matrix}0 if m ≠ 0 \\ 1 if m = 0 \end{matrix} ,

for χ ∈ X(k) − X_{o}(2),

ord_{∞}θ′[χ] = \{ \begin{matrix}\frac{m^{2}}{8k} & if m ≠ 0 \\ \frac{k}{2} & if m = 0 \end{matrix} and hence θ′[χ](0, ∞) = 0

and for χ ∈ X(k) − X(2),

ord_{∞} \frac{θ′[χ]}{θ[χ]} = \{ \begin{matrix}0 & if m = k and 0 ≤ m′ < k \\ 0 & if 0 < m < k and 0 ≤ m′ < 2k \\ \frac{k}{2} & if m = 0 and 0 ≤ m′ < k \end{matrix} .

Averaging Γ(k)-forms over Γ(k)\Γ produces Γ-forms; in particular, for

all k > 1, ∑_{χ∈X_{o}(k)} (\frac{θ′[χ]}{θ[χ]})^{2} and ∑_{χ∈X(k)−X(1)} (\frac{θ′[χ]}{θ[χ]})^{2} are modular 1forms (with canonical factor of automorphy) for Γ (their projections to the sphere ℍ^{2}/Γ are abelian differentials of the third kind whose only singularities are possible simple poles at the puncture P_{∞}). The sum of the residues must hence be zero.

But, the residue of (\frac{θ′[χ]}{θ[χ]})^{2} at ∞ is, up to an easily computed universal constant, the constant term in its Fourier series expansion. We

compute for k ≥ 3 and χ = [ \frac{m}{k} ; \frac{m′}{k} ] ∈ X(k) with m and m′ ∈ ℤ,

0 ≤ m ≤ k, 0 ≤ m′ < 2k (except when m = 0 or k where we assume that 0 ≤ m′ ≤ k), excluding the case k = m = m′,

\frac{θ′[χ]}{θ[χ]}(0, ∞) = \{ \begin{matrix}−π tan (\frac{πm′}{2k} ) & if m = k and 0 ≤ m′ < k \\ \frac{πım}{k} & if 0 < m < k and 0 ≤ m′ < 2k \\ 0 & if m = 0 and 0 < m′ ≤ k \end{matrix} .

4For m = 0 or k, we can and do assume that 0 ≤ m′ ≤ k.

5The functional equations permit us to calculate these invariants at all the cusps.

In general, let G be a subgroup of the modular group Γ. Assume

that the stabilizer of ∞ in G is generated by the translation [ \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ]

for some b ∈ ℤ^{+}. An arbitrary modular 1-form with canonical factor of automorphy, φ, for G on ℍ^{2} has a Fourier series expansion

∑_{i=0}^{∞} a_{i}e^{\frac{2πıτ}{b}} , a_{i} ∈ ℂ.

The sum of the residues of φ vanishes. Here

Res_{∞}φ = width(∞)φ(∞) = ba_{0}.

3. Proof of Theorem 1

For k > 2, the sums ∑_{χ∈X_{o}(k)} (\frac{θ′[χ]}{θ[χ]})^{2} and ∑_{χ∈X(k)−X(1)} (\frac{θ′[χ]}{θ[χ]})^{2}

project to abelian differentials of the third kind on the surface \overline{ℍ^{2}/Γ}, the compactification of ℍ^{2}/Γ. They are holomorphic except at the one puncture P_{∞} of ℍ^{2}/Γ where they have, at worse, simple poles. It is quite obvious that these differentials have no singularities at the projections of the elliptic fixed points of Γ. The residues of these differentials must be zero. This is the content of the identities of the theorem. For more details, we refer the reader to [3, Ch. 3]. We can conclude more. We have produced abelian differentials on the sphere with at most one simple pole. Such differentials vanish identically. Hence

Theorem 2. We have for each integer k ≥ 3, the two theta constant identities

∑_{χ∈X_{o}(k)} (\frac{θ′[χ]}{θ[χ]})^{2} = 0 and ∑_{χ∈X(k)−X(1)} (\frac{θ′[χ]}{θ[χ]})^{2} = 0.

The Fourier series coefficients of the two functions defined by the left hand sides of the above equalities must all vanish. The higher order relations obtained are more difficult to describe.

4. Γ q-forms, for some big q

4.1. Using only θ constants.

Proposition 1. Let k ∈ ℤ^{+}. Assume that k = 4 or that 8|k. Then for some χ ∈ Q^{2} and some m ∈ ℤ^{+}, θ^{k}[χ] is a modular \frac{k}{4}-form for Γ(m).

(a) If k ≠ 4, \frac{k}{8} is odd, and χ ∈ X (\frac{k}{8}), then θ^{k}[χ] is a cusp \frac{k}{4}-form for Γ (\frac{k}{8}). Hence also for every group Γ(m) for which \frac{k}{8} |m. For k = 8, this produces the 0 form.

(b) If χ ∈ X (\frac{k}{4}), then θ^{k}[χ] is a modular (cusp, if χ is of odd level)

\frac{k}{4}-form for Γ (\frac{k}{4}). Hence also for every group Γ(m) for which \frac{k}{4}|m.

(c) If χ ∈ X (\frac{k}{2}), then θ^{k}[χ] is a modular (cusp, if χ is of odd level) \frac{k}{4}-form for Γ (\frac{k}{2}). Hence also for every group Γ(m) for which \frac{k}{2}|m.

Proof. The proposition is a restatement of [3, Ch. 3, Th. 2.8].

We need some preliminaries to obtain identities among fixed powers of theta constants. Let δ ∈ ℤ^{+} be fixed. The primitive δ-characteristic is defined by

χ_{o} = χ_{o}(δ) = [ 1_{\frac{α}{δ}} ] = \{ \begin{matrix}[ 1_{\frac{1}{δ}} ] & for δ ≡ 1 mod 2 \\ [ \begin{matrix}1 \\ 0 \end{matrix} ] & for δ = 2 \\ [ 1_{\frac{4}{δ}} ] & for δ ≡ 2 mod 4, δ ≥ 6 \\ [ 1_{\frac{2}{δ}} ] & for δ ≡ 0 mod 4 \end{matrix} .

If χ ∈ X_{o}(δ) is arbitrary, then there exist δ left Γ(δ)-cosets Γ(δ)γ such that χ ≡ χ_{o}γ. We define for k ∈ ℤ^{+} with δ|k (here χ_{o} = χ_{o}(δ)),

ϵ_{δ,χ,k} = \frac{1}{δ} ∑_{γ∈Γ(δ)\Γ;χ≡χ_{o}γ} κ^{k}(χ_{o}, γ).

Now for γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ SL(2, ℂ),

κ^{4}(χ_{o}, γ) = e^{−πı\{(ab+ \frac{2bc(1+d)α}{δ})+ \frac{cdα^{2}}{δ^{2}} \}}κ^{4}(0, γ),

with κ(0, γ) an eighth root of unity. Hence for all k ∈ ℤ^{+} such that 8|k (throughout the rest of this section k represents a positive multiple of 8),

κ^{k}(χ_{o}, γ) = e^{− \frac{πık}{4} \{ \frac{2bc(1+d)α}{δ} + \frac{cdα^{2}}{δ^{2}} \}};

hence

κ^{k}(χ_{o}, γ) = 1 provided \{ \begin{matrix}δ = 2, \\ δ ≡ 1 mod 2 and 8δ^{2}|k, \\ δ ≥ 6, δ ≡ 2 mod 4 and 2δ^{2}|k, or \\ δ ≡ 0 mod 4 and 4δ^{2}|k. \end{matrix} (4)

We proceed to describe a simple method for calculating ϵ_{δ,χ,k}. Let

χ ∈ X_{o}(δ) be arbitrary. Choose a γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ Γ with χ_{o}γ ≡ χ. A

complete set of Γ(δ)\Γ-representatives that map χ_{o} to χ is

B^{l}γ, l = 0, 1, ..., δ − 1.

Once we recall that δ|k, we easily see that

ϵ_{δ,χ,k} = e^{− \frac{πık}{4} \{ \frac{2bc(1+d)α}{δ} + \frac{cdα^{2}}{δ^{2}} \}}

and that ϵ_{δ,χ,k} = 1 under the conditions on δ and k described in (4). Let 1 ≠ δ ∈ ℤ^{+}. We call such δ an admissible divisor of k provided (a) δ ≡ 0 mod 4 and 2δ|k,

(b) δ ≡ 2 mod 4 and 4δ|k, or

(c) δ ≡ 1 mod 2 and 8δ|k. Since 8|k, 2 is always admissible. We let D(k) be the number of admissible divisors of k and D_{o}(k), the number of its admissible odd divisors. It is routine to show that if

k = 2^{α}p_{1}^{α_{1}}p_{2}^{α_{2}}... p_{r}^{α_{r}}

is the prime factorization of k (as usual α ≥ 3, α ∈ ℤ, the p_{i} are distinct odd primes and α_{i} ∈ ℤ^{+}), then

D(k) = α ∏_{i=1}^{r}(α_{i} + 1) − 1 = d (\frac{k}{2}) − 1

and

D_{o}(k) = ∏_{i=1}^{r}(α_{i} + 1) − 1 = d_{o}(k) − 1,

where d(n) and d_{o}(n) are the usual number theoretic functions, the number of divisors and odd divisors respectively of n ∈ ℤ^{+}. Moreover, the admissible divisors of k are the divisors of \frac{k}{2} with 1 excluded.

If δ is an admissible divisor of k and χ ∈ X(δ), then θ^{k}[χ] is a modular \frac{k}{4}-form for Γ(δ), an element of 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ(δ)). The relative Poincaré series

F_{δ}(τ) = F_{k,δ}(τ) = \frac{1}{δ} ∑_{γ∈Γ(δ)\Γ} θ^{k}[χ](0, γ(τ))γ′(τ)^{\frac{k}{4}}

defines an element of 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ). We need to rewrite the relative Poincaré series in slightly different form. As a result of (3),

θ^{k}[χ](0, γ(τ))γ′(τ)^{\frac{k}{4}} = κ^{k}(χ, γ)θ^{k}[χγ](0, τ).  (5)

If we use χ = χ_{o} = χ_{o}(δ) (this will suffice for our purposes), we obtain that

F_{δ}(τ) = ∑_{χ∈X_{o}(δ)} ϵ_{δ,χ,k}θ^{k}[χ] ∈ 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ).

We have used two facts to obtain the last conclusion. First, that the set of left Γ(δ)-cosets of Γ acts transitively on X_{o}(δ). Second, that the stabilizer of an equivalence class of characteristics has constant order δ. The computations of ϵ_{δ,χ,k} are simplified by the fact that k is a multiple of 8 and hence κ^{k}(0, γ) = 1. In equation (5), the term θ^{k}[χγ] depends only on the equivalence class of the characteristic χ. For odd δ, we have the stronger conclusion F_{δ} ∈ 𝔸_{\frac{k}{4}} (ℍ^{2}, Γ). The form

F_{2} = ∑_{χ∈X_{o}(2)} θ^{k}[χ] ∈ 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ)

is particularly simple (and hence pleasing). If k is sufficiently big multiple of δ, then we have

F_{δ} = ∑_{χ∈X_{o}(δ)} θ^{k}[χ];

this occurs provided k and δ satisfy the divisibility conditions in (4).

It is also of interest to determine conditions on δ, χ = [ \begin{matrix}0 \\ ∗ \end{matrix} ] and k,

so that ϵ_{δ,χ,k} = 1. This situation only occurs for even δ and always for δ = 2. So assume that δ ≥ 4. The motion γ ∈ Γ with χ_{o}γ ≡ χ is of

the form [ \begin{matrix}∗ & ∗ \\ c\frac{δ}{2} & ∗ \end{matrix} ], with c ∈ ℤ. It now follows that ϵ_{δ,χ,k} = 1 except

for δ ≡ 0 mod 4, c and \frac{k}{8} odd, when ϵ_{δ,χ,k} = −1. Finally, the number of equivalence classes of characteristics in X_{o}(δ) in the tower over 0 is

2φ (\frac{k}{2}) if δ ≡ 2 mod 4 φ (\frac{k}{2}) if δ ≡ 0 mod 4 .

We now explore the consequences of the last proposition and the remarks that followed it. We start with the four simplest examples before considering the general case.

k = 4. This case is rather simple since the only admissible δ is δ = 2, and the averaging process described above leads us to the formally zero form. By this we mean that the coefficients ϵ_{2,χ,4} all vanish for χ ∈ X_{0}(2). In fact, the same phenomenon occurs for k = 4l with l an odd positive integer and δ = 2. This explains why divisors of k of the form 4l with l odd were not considered admissible.

It is however possible to be a bit more general and consider, in place

of a 1-form on Γ(2), a 2-form on Γ(2), say θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ], and average

this over Γ(2)\Γ. This averaging process leads to the 2-form ψ on Γ given by

2 (θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] + θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ] θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] − θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ]) .

We shall soon see that the formula for this 2-form is not surprising. We will also obtain an identity that will reduce to the Jacobi quartic.

Easy calculations show that if we average the product of the fourth powers of the three even theta constants, ∏_{χ∈X_{0}(2)} θ^{4} [χ], we once again obtain formally zero.

k = 8. Only δ = 2 and 4 are admissible, producing

F_{2} = ∑_{χ∈X_{o}(2)} θ^{8}[χ] and F_{4} = ∑_{χ∈X_{o}(4)} ϵ_{4,χ,8}θ^{8}[χ].

Since dim 𝔸_{2}^{+}(ℍ^{2}, Γ) = 1, we have the identities

θ^{8} [ 0_{\frac{1}{2}} ] + θ^{8} [ 0^{\frac{1}{2}} ] + ıθ^{8} [ \frac{1}{2} \frac{1}{2} ] − θ^{8} [ 1^{\frac{1}{2}} ] − ıθ^{8} [ \frac{1}{2} \frac{3}{2} ] − θ^{8} [ 1_{\frac{1}{2}} ]

= \frac{1}{2} (θ^{8} [ \begin{matrix}0 \\ 0 \end{matrix} ] + θ^{8} [ \begin{matrix}0 \\ 1 \end{matrix} ] + θ^{8} [ \begin{matrix}1 \\ 0 \end{matrix} ])  (6)

= θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] + θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ] − θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ] .

The verification^{6} of the computations of the various constants needed to evaluate F_{4} are facilitated by the entries in Table 1. The last equality involves the 2-form ψ constructed above in the discussion of the k = 4 case, and gives after some algebraic manipulation another proof of the Jacobi quartic identity

θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] − θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ] − θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] = 0.  (7)

As before, it is possible to average any of the 5 products of the form

θ^{8} [ 1_{\frac{1}{2}} ] θ^{8}[χ], with χ ∈ X_{o}(4), χ ≠ [ 1_{\frac{1}{2}} ], over the group Γ(4)\Γ

with 24 elements. The formulae obtained for the relative Poincaré series involve the weighted sums of 12 products and produce elements of 𝔸_{4}^{+}(ℍ^{2}, Γ). These can be used in addition to the forms discussed below Characteristic Motion γ

6Should the reader be inclined to do so.

χ  with χ_{o}γ ≡ χ ϵ_{4,χ,8}

[ 1_{\frac{1}{2}} ]  [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ]  1

[ 0^{\frac{1}{2}} ]  [ \begin{matrix}2 & 3 \\ 1 & 2 \end{matrix} ]  −1

[ 0_{\frac{1}{2}} ]  [ \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ]  −1

[ \frac{1}{2} \frac{1}{2} ]  [ \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ] −ı

[ 1^{\frac{1}{2}} ]  [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ]  1

[ \frac{1}{2} \frac{3}{2} ]  [ \begin{matrix}0 & −1 \\ 1 & 3 \end{matrix} ]  ı

Table 1. Computation of constants appearing in F_{8,4}.

for the case k = 16 to get other identities. Relative Poincaré series of products of 3 or more terms can also be used in the study of the more general cases.

k = 16. The admissible δ’s are 2, 4 and 8. They produce the nontrivial modular 4-forms

F_{2} = ∑_{χ∈X_{o}(2)} θ^{16}[χ], F_{4} = ∑_{χ∈X_{o}(4)} ϵ_{4,χ,16}θ^{16}[χ] and F_{8} = ∑_{χ∈X_{o}(8)} ϵ_{8,χ,16}θ^{16}[χ].

Since dim 𝔸_{4}^{+}(ℍ^{2}, Γ) = 1, any two are linearly dependent. Since it is easy to compute the values at ∞ of these modular forms (they are 2, 1 and 2, respectively), the linear relations are also easily described:

F_{2} = 2F_{4} and F_{2} = F_{8}.

k = 24. Here the values of δ are 2, 3, 4, 6 and 12, producing the nontrivial modular 6-forms

F_{2} = ∑_{χ∈X_{o}(2)} θ^{24}[χ], F_{3} = θ^{24} [ 1_{\frac{1}{3}} ]+ω^{2}θ^{24} [ \frac{1}{3} \frac{1}{3} ]+θ^{24} [ 1^{\frac{1}{3}} ]+ωθ^{24} [ \frac{1}{3} \frac{5}{3} ] ,

where ω = e^{\frac{2πı}{3}} . The calculations of the constants appearing in the last formula are based on the entries in Table 2. Also,

F_{4} = ∑_{χ∈X_{o}(4)} ϵ_{4,χ,24}θ^{24}[χ], F_{6} = ∑_{χ∈X_{o}(6)} ϵ_{6,χ,24}θ^{24}[χ],

and

F_{12} = ∑_{χ∈X_{o}(12)} ϵ_{12,χ,24}θ^{24}[χ].

They establish the identities of the form

c_{1}F_{2} + c_{2}F_{4} + c_{3}F_{3} = 0,
c_{4}F_{2} + c_{5}F_{6} + c_{6}F_{3} = 0,
c_{7}F_{2} + c_{8}F_{12} + c_{9}F_{3} = 0,

where the c_{i} are easily determined constants. The functions F_{2} and F_{3} form a basis for 𝔸_{6}^{+}(ℍ^{2}, Γ) and

F_{3} = 2^{2}3^{3}7η^{24},  (8)

where

η(τ) = e^{\frac{2πıτ}{24}} ∏_{n=1}^{∞}(1 − e^{2πıτn}),

an identity of the type to be discussed in the next subsection.

More examples can be constructed using the data in Table 3 with the added information that for k = 2^{α} ∏_{i=1}^{r} p_{i}^{α_{i}} (usual prime factorization),

dim 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ) = ⌊\frac{2^{α−3} ∏_{i=1}^{r} p_{i}^{α_{i}}}{3} ⌋ + 1.

The numerology encountered shows that

k >> 0 ⟹ D(k) << dim 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ);

14 HERSHEL M. FARKAS AND IRWIN KRA

Characteristic Motion γ

χ  with χ_{o}γ ≡ χ ϵ_{3,χ,24}
[ 1_{\frac{1}{3}} ]  [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ]  1
[ \frac{1}{3} \frac{1}{3} ]  [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ]  ω^{2}
[ 1^{\frac{1}{3}} ]  [ \begin{matrix}1 & 2 \\ 1 & 3 \end{matrix} ]  1
[ \frac{1}{3} \frac{5}{3} ]  [ \begin{matrix}1 & 1 \\ 1 & 2 \end{matrix} ]  ω

Table 2. Computation of constants appearing in F_{24,3}.

k  Admissible δ  D(k) dim 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ)
8  2, 4  2  1
16  2, 4, 8  3  1
24  2, 3, 4, 6, 12  5  2
32  2, 4, 8, 16  4  2
40  2, 4, 5, 10, 20  5  2
48  2, 3, 4, 6, 8, 12, 24  7  3
56  2, 4, 7, 14, 28  5  3
64  2, 4, 8, 16, 32  5  3
72 2, 3, 4, 6, 9, 12, 18, 36  8  4
80  2, 4, 5, 8, 10, 20, 40  7  4
88  2, 4, 11, 22, 44  5  4
96 2, 3, 4, 6, 8, 12, 16, 24, 48 9  5

Table 3. Computation of admissible divisors.

hence we can only produce identities by this method for relatively small values of the parameter k.

Arbitrary k, a multiple of 8. For every admissible divisor δ of k, we have constructed a modular \frac{k}{4}-form for Γ. Of the D(k) forms constructed, D_{o}(k) are cusp forms. If we compare, the three integers D(k), D_{o}(k) and dim 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ), we see that we can produce identities among k-th powers of θ-constants for k = 2^{α}, α = 3, 4, 5, 6, 7 and a number of other k. For powers of 2, we do not obtain any relations among the cusp forms produced.

4.2. Using θ-constants and derivatives. The basic facts about the θ-constant derivatives θ′[χ] that we need are contained in

Proposition 2. Let k ∈ ℤ^{+}. Assume that k = 4 or that 8|k. Then for some χ ∈ Q^{2} and some m ∈ ℤ^{+}, (θ′)^{k}[χ] is \frac{3k}{4}-cusp form for Γ(m).

(a) If k ≠ 4, \frac{k}{8} is odd, and χ ∈ X (\frac{k}{8}), then we may take m to be any multiple of \frac{k}{8}.

(b) If χ ∈ X (\frac{k}{4}), then we may take m to be any multiple of \frac{k}{4}.

(c) If χ ∈ X (\frac{k}{2}), then we may take m to be any multiple of \frac{k}{2}.

(d) If χ ∈ X_{o}(2), then (θ′)^{k}[χ] = 0.

Let k ∈ ℤ^{+} be, as before, a multiple of 8. For δ = 1 or an admissible divisor of k, the relative Poincaré series

G_{δ}(τ) = G_{k,δ}(τ) = \frac{1}{δ} ∑_{γ∈Γ(δ)\Γ}(θ′)^{k}[χ](0, γ(τ))γ′(τ)^{\frac{3k}{4}} ∈ 𝔸_{\frac{3k}{4}} (ℍ^{2}, Γ).

We note that for δ = 1, we obtain a constant multiple of η^{\frac{3k}{4}} and for δ = 2, the zero form. Also as before,

G_{δ} = \frac{1}{δ} ∑_{χ∈X_{o}(δ)} ϵ_{δ,χ,k}(θ′)^{k}[χ].

Hence for every integer k that is a positive multiple of 24 and every admissible divisor δ of k, the modular form

F_{k,δ} ∈ 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ),

and for every admissible divisor ϵ of \frac{k}{3}, excluding 2, and for ϵ = 1, the cusp form

G_{\frac{k}{3} ,ϵ} ∈ 𝔸_{\frac{k}{4}} (ℍ^{2}, Γ).

We have constructed D(k) + D (\frac{k}{3}) modular \frac{k}{4}-forms for Γ; hence if D(k) + D (\frac{k}{3}) ≥ dim 𝔸+_{\frac{k}{4}}(ℍ^{2}, Γ), we have relations among modular

forms (sometimes among cusp forms). For k = 2^{α}3^{β} with α ≥ 3 and β ∈ ℤ^{+}, we obtain such relations as long as α(2β + 1) > 2^{α−3}3^{β−1} + 1; in particular for k = 24, 72 and 216 we have relations among modular forms and for k = 24 and 72, among cusp forms produced as \frac{k}{3} powers of θ-constant derivatives; most interesting perhaps is the relation

G_{8,3} = cG_{8,1}

between the two cusp forms. Its expanded version is

(θ′)^{8} [ 0_{\frac{1}{2}} ] + (θ′)^{8} [ 0^{\frac{1}{2}} ] + ı(θ′)^{8} [ \frac{1}{2} \frac{1}{2} ] − (θ′)^{8} [ 1^{\frac{1}{2}} ] − ı(θ′)^{8} [ \frac{1}{2} \frac{3}{2} ]

− (θ′)^{8} [ 1_{\frac{1}{2}} ] = −\frac{3 · 13}{2^{8}}(θ′)^{8} [ \begin{matrix}1 \\ 1 \end{matrix} ] .  (9)

The connection of the last identity to (8) is provided by the well known equality

θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, τ) = 2πıθ^{3} [ 1^{\frac{1}{3}} ] (0, 3τ) = −2πη^{3}(τ).

4.3. Using ratios. We recall a result used in establishing Theorem 1.

Proposition 3. For all k ∈ ℤ^{+}, k > 2 and all χ ∈ X_{o}(k), the ratio

(\frac{θ′[χ]}{θ[χ]})^{2} ∈ 𝔸_{1}^{+}(ℍ^{2}, Γ(k)).

For k = 1, the ratio is not defined, it is identically zero for k = 2

Corollary 1. For all k ∈ ℤ^{+} and all δ ∈ ℤ^{+}, δ > 2,

∑_{χ∈X_{o}(δ)} (\frac{θ′[χ]}{θ[χ]})^{2k} ∈ 𝔸_{k}^{+}(ℍ^{2}, Γ).

We have obviously produced infinitely many identities among sums of powers of ratios (the ratios are logarithmic derivatives in an appropriate sense). Theorem 2 tells us that for k = 1 we have produced the zero form. This suggests several problems that we proceed to describe. The cardinality of the set X_{o}(δ) is n(δ), the number of punctures on ℍ^{2}/Γ(δ). Choose a numbering, χ_{1}, ..., χ_{n(δ)}, of the characteristics in X_{o}(δ). Consider the map

H : ℂ^{n(δ)} → 𝔸_{1}^{+}(ℍ^{2}, Γ(k))

defined by

H(z_{1}, ..., z_{n(δ)}) = ∑_{i=1}^{n(δ)} z_{i} (\frac{θ′[χ_{i}]}{θ[χ_{i}]})^{2} .

(a) What is the rank of the operator H (we know that it is positive and less that n(δ))?

(b) Describe the range and kernel of the operator H.

(c) Find necessary and sufficient conditions for H(z) = 0 for a given z ∈ ℂ^{n(δ)}.

5. Γ_{o}(k) and G(k)-forms

One of the main purposes of this section is to lay the foundation for an investigation of the map H introduced above that will be pursued elsewhere. For the odd positive integer k and all l ∈ ℤ, the function

(\frac{θ′[ 1_{\frac{1+2l}{k}} ]}{θ[ 1_{\frac{1+2l}{k}} ]} )^{2} is a modular 1-form for the group G(k) obtained by ad-

joining the translation [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] to Γ(k). For even k, we reach the same

conclusion if we replace 1+2l by 2l. We can also replace the 1 by 0. The

group G(k)\Γ_{o}(k), where Γ_{o}(k) = \{γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ Γ; c ≡ 0 mod k\},

acts by permutations on characteristics of the form [ 1_{\frac{ϵ+2l}{k}} ], l ∈ ℤ,

where ϵ = \{ \begin{matrix}1 for odd k \\ 0 for even k \end{matrix} . For even k it also acts on the character-

istics [ 0_{\frac{2l}{k}} ], l ∈ ℤ. Hence we have a number of different modular

1-forms for Γ_{o}(k) as summarized by the table in this section. Some of the entries in the table can be expressed in terms of the following functions defined for all k and n ∈ ℤ^{+}

S(k, n) = ∑_{l=1,2,...,k, (l,k)=1} l^{n},

S_{even}(k, n) = ∑_{l=1,2,...,⌊\frac{k}{2}⌋, (2l,k)=1}(2l)^{n} = 2^{n} ∑_{l=1,2,...,⌊\frac{k}{2}⌋, (2l,k)=1} l^{n},

and

S_{odd}(k, n) = ∑_{l=0,1,...,⌊\frac{k−1}{2}⌋, (l+2l,k)=1}(1 + 2l)^{n}.

We note that

S_{even}(2k, n) = 2^{n}S(k, n).

To analyze the modular forms in the table in this section, it is most convenient to use the η-function previously defined. For the remainder of this paper, we outline the solutions of some problems and suggest areas for possible future investigations.

18 HERSHEL M. FARKAS AND IRWIN KRA

Formula for form φ  k  Res_{∞}φ  Res_{0}φ

∑ \frac{k−3}{2} ; l=0 ^{(^{\frac{θ′[ 1_{\frac{1+2l}{k}} ]}{θ[ 1_{\frac{1+2l}{k}} ]}} )}^{2}  \begin{matrix}≥ 3 \\ ≡ 1 mod 2 \end{matrix}  π^{2} \frac{(k−2)(k−1)}{6}  −π^{2} \frac{(k−2)(k−1)}{6}

\begin{matrix}∑ & l=0,1,..., \frac{k−3}{2} ,(1+2l,k)=1 \\ ( \frac{θ′[ 1_{\frac{1+2l}{k}} ]}{θ[ 1_{\frac{1+2l}{k}} ]} ) & 2 \end{matrix}  \begin{matrix}≥ 3 \\ ≡ 1 mod 2 \end{matrix}  \begin{matrix}\frac{π2}{k} & ∑ & \frac{k−3}{2} ; l=0 \\ \frac{(1+2l)2}{(k,1+2l)} φ((k, 1 + 2l)) \end{matrix}  − \frac{π2}{k} S_{odd}(k, 2)

∑ \frac{k−2}{2} ; l=0 (^{\frac{θ′[ 1_{\frac{2l}{k}} ]}{θ[ 1_{\frac{2l}{k}} ]}} )^{2}  \begin{matrix}≥ 2 \\ ≡ 0 mod 2 \end{matrix}  π^{2} \frac{(k−2)(k−1)}{6}  −π^{2} \frac{(k−2)k}{6}

\begin{matrix}∑ & l=1,2,..., \frac{k−2}{4} ,(l, \frac{k}{2} )=1 \\ ( \frac{θ′[ 1_{\frac{4l}{k}} ]}{θ[ 1_{\frac{4l}{k}} ]} ) & 2 \end{matrix}  \begin{matrix}≥ 6 \\ ≡ 2 mod 4 \end{matrix}  \begin{matrix}\frac{2π2}{k} & ∑ & \frac{k−2}{2} ; l=1 \\ (2, l)_{\frac{l2}{( \frac{k}{2} ,l)}} φ (( \frac{k}{2} , l)) \end{matrix}  − \frac{16π2}{k} S ( \frac{k}{2} , 2)

\begin{matrix}∑ & l=1,3,..., \frac{k−2}{2} ,(l, \frac{k}{2} )=1 \\ ( \frac{θ′[ 1_{\frac{2l}{k}} ]}{θ[ 1_{\frac{2l}{k}} ]} ) & 2 \end{matrix}  \begin{matrix}≥ 4 \\ ≡ 0 mod 4 \end{matrix}  \begin{matrix}\frac{4π2}{k} & ∑ & \frac{k−2}{2} ; l=1 \\ _{\frac{l2}{( \frac{k}{2} ,l)}} φ (( \frac{k}{2} , l)) \end{matrix}  − \frac{4π2}{k} S ( \frac{k}{2} , 2)

∑ \frac{k−2}{2} ; l=0 (^{\frac{θ′[ 0_{\frac{2l}{k}} ]}{θ[ 0_{\frac{2l}{k}} ]}} )^{2}  \begin{matrix}≥ 2 \\ ≡ 0 mod 2 \end{matrix}  0  − \frac{π2}{6} (k − 2)(k − 1)

\begin{matrix}∑ & l=1,2,..., \frac{k−2}{2} ,(l, \frac{k}{2} )=1 \\ ( \frac{θ′[ 0_{\frac{2l}{k}} ]}{θ[ 0_{\frac{2l}{k}} ]} ) & 2 \end{matrix}  \begin{matrix}≥ 6 \\ ≡ 2 mod 4 \end{matrix}  0  − \frac{4π2}{k} S ( \frac{k}{2} , 2)

\begin{matrix}∑ & l=1,3,..., \frac{k−2}{2} ,(l, \frac{k}{2} )=1 \\ ( \frac{θ′[ 0_{\frac{2l}{k}} ]}{θ[ 0_{\frac{2l}{k}} ]} ) & 2 \end{matrix}  \begin{matrix}≥ 4 \\ ≡ 0 mod 4 \end{matrix}  0  − \frac{4π2}{k} S_{odd} ( \frac{k}{2} , 2)

Table 4. Modular forms φ for Γ_{o}(k). Some of the calculations of residues at ∞ use Theorem 1.

If N ∈ ℤ and |N| is sufficiently large, then (\frac{η(kτ}{η(τ)} )^{\frac{24N}{k−1}} is a holomorphic

Γ_{o}(k)-function that projects to a meromorphic function on \overline{ℍ^{2}/Γ_{o}(k)}, the compact surface obtained by filling in the punctures on ℍ^{2}/Γ_{o}(k), with poles only at the punctures^{7}. Hence Φ = \frac{d}{dτ} log (\frac{η(kτ)}{η(τ)}) projects

to an abelian differential of the third kind on \overline{ℍ^{2}/Γ_{o}(k)} whose only singularities are simple poles at the punctures of ℍ^{2}/Γ_{o}(k). Two of the residues of Φ are readily computed:^{8}

Res_{∞}Φ = \frac{πı(k − 1)}{12} = −Res_{0}Φ.

This accounts for all the residues for Φ provided k is prime.

Theorem 3. For all odd primes k,

\frac{d}{dτ} log (\frac{η(kτ)}{η(τ)}) + \frac{1}{2πı(k − 2)} ∑_{l=0}^{\frac{k−3}{2}} (\frac{θ′ [ 1_{\frac{1+2l}{k}} ]}{θ [ 1_{\frac{1+2l}{k}} ]})^{2} (0, τ) (10)

is a cusp 1-form (belonging to the canonical factor of automorphy) for the group Γ_{o}(k). This form is identically zero provided k ≤ 13, k ≠ 11.

The last statement of the theorem follows from the fact that

dim 𝔸_{1}(ℍ^{2}, Γ(k) = 0

for the specified values of k. We conjecture that the cusp form is identically zero for all primes k. We can prove this conjecture for primes k ≤ 19 by showing, using computer aided calculations, that this cusp form vanishes to sufficiently high order at ∞. Because of rounding errors, one rapidly (with increasing k) looses confidence in the calculations. For even k ≥ 4, (10) is a modular, but not a cusp, form. We do not know if the corresponding statement is true for odd composite k.

If we start with an automorphic form θ^{r}[χ], χ ∈ X(k), for Γ(k), then

\frac{1}{k} ∑_{l=0}^{k−1} θ^{r}[χ](0, τ + l)

defines a form for G(k). If we start with a form of the same type with χ ∈ X_{o}(k), we can simplify the sum considerably. Similar conclusions hold for functions of the form (θ′)^{r}[χ] and (\frac{θ′[χ]}{θ[χ]})^{2r}. We illustrate one case. Take χ ∈ X(k), χ ∉ X(2), then

7For primes k, there are only two punctures: P_{∞} and P_{0}.

8The widths of the cusps ∞ and 0 are 1 and k, respectively.

\frac{1}{k} ∑_{l=0}^{k−1} (\frac{θ′[χ]}{θ[χ]}(0, τ + l))^{2} = \frac{K}{k} ∑_{χ̂≡χ mod <B>} (\frac{θ′[χ̂]}{θ[χ̂]}(0, τ))^{2} ,

where K is the order of the subgroup of < B^{k} > \ < B > that

stabilizes χ. Each tower, characteristics of the form [ ∗^{\frac{m}{k}} ], breaks up

into a number of < B >-invariant segments. Each segment except the ones containing elements of X(2) produces an element of A_{1}^{+}(ℍ^{2}, G(k)). For k = 6 we get 6 segments (forms). We obtain a form ϕ_{1} by choosing

χ = [ 0_{\frac{1}{3}} ]; ϕ_{2} by choosing χ = [ 0^{\frac{1}{3}} ]; ϕ_{3} by choosing χ = [ \frac{1}{3} \frac{1}{3} ]; ϕ_{4}

by choosing χ = [ 0^{\frac{2}{3}} ]; ϕ_{5} by choosing χ = [ 1_{\frac{1}{3}} ] and ϕ_{6} by choosing

χ = [ 1_{\frac{2}{3}} ].

Since ℍ^{2}/G(k) is a four times punctured sphere, the dimension of the space of forms with simple poles at the four punctures is 3 and we expect 3 linear relations between the six forms produced. Two of these are

ϕ_{1} + ϕ_{2} + ϕ_{4} + ϕ_{6} = 0

and

ϕ_{3} + ϕ_{5} = 0.

A little linear algebra will help the reader find the third.

6. A more complicated trigonometric sum

We end with a result established in [3].

Theorem 4. Let k ≥ 3 be an odd integer and N an arbitrary positive integer. Define S_{N} as the set of \frac{k−1}{2}-tuples

(m_{1}, ..., m_{\frac{k−1}{2}} ) ∈ (ℤ^{+} ∪ \{0\})^{\frac{k−1}{2}}

satisfying

∑_{i=1}^{\frac{k−1}{2}} \frac{m_{i}(m_{i} + 1)}{2} = N.

Then

∑_{(m_{1},... ,m_{\frac{k−1}{2}} )∈S_{N}} ∏_{i=1}^{\frac{k−1}{2}} \begin{matrix}cos & (\frac{(2m_{i}+1)(2i−1)π}{2k}) \\ cos & (\frac{(2i−1)π}{2k}) \end{matrix} = A_{N},

with A_{N} an integer defined by

∏_{n=1}^{∞}(1 − x^{n})^{\frac{k−3}{2}} (1 − x^{kn}) = 1 + ∑_{N=1}^{∞} A_{N}x^{N}.

References

[1] H.M. Farkas and I. Kra, Automorphic forms for subgroups of the modular group, Israel J. Math. 82 (1993), 87–131.

[2] , Automorphic forms for subgroups of the modular group. II: Groups containing congruence subgroups, J. d’Analyse Math. 70 (1996), 91–156.

[3] , Theta Constants, Riemann Surfaces and the Modular Group, Graduate Studies in Mathematics, vol. 37, American Mathematical Society, 2001.

[4] M.I. Knopp, Modular Functions in Analytic Number Theory, Chelsea Publishing Company, 1993.

The Hebrew University of Jerusalem Jerusalem 91904, Israel and State University of New York at Stony Brook Stony Brook, NY 11794, USA