TORIC MODULAR FORMS OF HIGHER WEIGHT

LEV A. BORISOV AND PAUL E. GUNNELLS

Abstract. In the papers [1, 2] we used the geometry of complete polyhedral fans to construct a subring 𝒯 (l) of the modular forms on Γ_{1}(l), and showed that for weight two the cuspidal part of 𝒯 (l) coincides with the space of cusp forms of analytic rank zero. In this paper we show that in weights greater than two, the cuspidal part of 𝒯 (l) coincides with the space of all cusp forms.

1. Introduction

In [1, 2] we used the geometry of complete polyhedral fans to construct a subring 𝒯 (l) of the modular forms on Γ_{1}(l). If l ≥ 5 then 𝒯 (l) is generated in weight one by certain Eisenstein series, and in [1, Theorem 5.11] we showed that for weight two the cuspidal part of 𝒯 (l) coincides with the space of cusp forms of analytic rank zero. In this paper in Theorem 5.11 we show that in weights greater than two, the cuspidal part of 𝒯 (l) coincides with the space of all cusp forms.

The proof of Theorem 5.11 is formally very similar to the proof of [1, Theorem 5.11]. We define a certain endomorphism ρ of the space of cuspforms of weight k on Γ_{1}(l) and show that its image contains all newforms for k ≥ 3. We describe the map ρ in terms of Manin symbols, which allows us to write ρ(f) in terms of products of certain explicit toric Eisenstein series s̃_{a/l}^{(r)}. We are helped by the fact that there exist certain Manin symbols of weight k, denoted by R_{(m,n)}, m, n ∈ ℤ, that satisfy relations similar to weight two Manin symbols.

Here is an outline of the paper. Section 2 reviews relevant results about toric modular forms, and Section 3 reviews results about Manin symbols and introduces the symbols R_{(m,n)}. Section 4 describes a class of functions we call (mod l)-polynomials that are needed as a technical tool for the proof of Theorem 5.11. Finally, in Section 5 we define the map ρ and prove our main result and some corollaries.

2. Toric modular forms

2.1. We will briefly review the definition of toric forms; more details can be found in [1, 2]. Let k and l be positive integers, and suppose l ≥ 5. A modular form of weight k on the group Γ_{1}(l) is called toric if it can be expressed as a homogeneous polynomial of degree k in the functions s̃_{a/l}(q) given by

Date: May 25, 2000.

Key words and phrases. Modular forms, theta functions, Manin symbols.

s̃_{a/l}(q) := (\frac{1}{2} − \frac{a}{l}) + ∑_{n>0} q^{n} ∑_{d|n}(δ_{d}^{amod l} − δ_{d}^{−amod l}), a = 1, …, l − 1.

Here δ_{d}^{amod l} is 1 if a = d mod l and is 0 otherwise. The ring of toric forms 𝒯 (l) is thus generated in degree one. It is known to be stable under the action of the Hecke operators, and under Atkin-Lehner lifting.

Proposition 2.2. The ring of toric modular forms contains the modular forms

s̃_{a/l}^{(k)} := C + ∑_{n>0} q^{n} ∑_{d|n} d^{k−1}(δ_{d}^{amod l} + (−1)^{k}δ_{d}^{−amod l}), k > 0, a = 0, …, l,

except for k = 2, l = 0. Here C is a constant determined uniquely by the modular properties of s̃_{a/l}^{(k)}.

Proof. Indeed, it is easy to construct a given s̃_{a/l}^{(k)} as a linear combination of the modular forms

s_{a/l}^{(k)}(q) := (2πi)^{−k} (\frac{∂^{k}}{∂z^{k}} )_{z=0} log (\frac{zϑ(z + a/l, τ)ϑ′(0, τ)}{ϑ(z, τ)ϑ(a/l, τ)})

= C + ∑_{n>0} q^{n} ∑_{d|n} d^{k−1}(e^{2πika/l} + (−1)^{k}e^{−2πika/l} − 2δ_{k}^{0mod 2})

of [2], and the standard Eisenstein series

E_{k} := C + ∑_{n>0} q^{n} ∑_{d|n} d^{k−1}

for even k ≥ 4. Here ϑ(z, τ) is the Jacobi theta function (see [3]). The forms s_{a/l}^{(k)} and E_{k} for k > 2 are toric, see [2]. We use here and in what follows the convention denoting constant terms whose exact value is irrelevant by C.

We must exclude k = 2, a = 0 since E_{2} is not a modular form. However, it will be convenient to consider s̃_{0/l}^{(2)}, which amounts to working in the ring 𝒯 (l)[E_{2}] instead of 𝒯 (l). We will call the elements of 𝒯 (l)[E_{2}] toric quasimodular forms.

3. Manin symbols

This section closely follows [6], to which the reader is referred for more details.

3.1. Let l > 1 be an integer, and let E_{l} ⊂ (ℤ/lℤ)^{2} be the subset of pairs (u, v) such that ℤu + ℤv = ℤ/lℤ. The space of Manin symbols of weight k is the ℂ-vector space generated by the symbols x^{r}y^{s}(u, v), where r and s are non-negative integers that sum to k − 2 and (u, v) ∈ E_{l}, modulo the relations:

1. x^{r}y^{s}(u, v) + (−1)^{r}x^{s}y^{r}(v, −u) = 0.

2. x^{r}y^{s}(u, v) + (−1)^{r}y^{r}(x − y)^{s}(v, −u − v) + (−1)^{s}(x − y)^{r}x^{s}(−u − v, u) = 0.

Two subspaces of M will play an important role in what follows. Let ι: M → M be the involution that takes x^{r}y^{s}(u, v) ↦ (−1)^{r}x^{r}y^{s}(−u, v).

Definition 3.2. The space of plus symbols M_{+} ⊂ M is the subspace consisting of symbols w satisfying ι(w) = w. Similarly, the space of minus symbols M_{−} ⊂ M is the subspace consisting of symbols w satisfying ι(w) = −w.

We have symmetrization maps ( , )_{±}: M → M_{±} given by x^{r}y^{s}(u, v)_{±} := (x^{r}y^{s}(u, v)± (−1)^{r}x^{r}y^{s}(−u, v))/2.

3.3. Let M^{∗} = Hom_{ℂ}(M, ℂ) be the dual space to the space of Manin symbols. For any φ ∈ M^{∗}, we define φ on “degenerate” symbols x^{r}y^{s}(u, v) with ℤu + ℤv ≠ ℤ/lℤ by setting φ(x^{r}y^{s}(u, v)) = 0. This convention is somewhat artificial but turns out to be quite useful.

3.4. There exists a natural pairing between the spaces of Manin symbols and the spaces of cusp forms, see [6]. Let ℳ (l) be the ℂ-vector space of weight k holomorphic modular forms on Γ_{1}(l), and let 𝒮 (l) ⊂ ℳ (l) be the subspace of cusp forms. For

x^{r}y^{s}(u, v) ∈ M and a cusp form f(τ) find an element g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) of Γ(1) with

(c, d) = (u, v) mod l. Then the integral

∫_{0}^{i∞}(cτ + d)^{−k}f(\frac{aτ + b}{cτ + d})τ^{r} dτ

does not depend on the choice of g. Moreover, it is compatible with the relations on modular symbols, and we obtain a pairing

M × 𝒮 (l) ⟶ ℂ,
(x^{r}y^{s}(u, v), f) ⟼ ⟨f, x^{r}y^{s}(u, v)⟩.

In general, this pairing is degenerate, but one can identify a subspace of cuspidal Manin symbols S such that the pairing is non-degenerate on S_{±} × 𝒮 (l). We will not use this fact, but details can be found in [6].

3.5. To conclude this section, we present Merel’s description of the Hecke action on the Manin symbols. Let n ≥ 1 be an integer, and let T_{n} be the associated Hecke operator (cf [5]). We denote the action of T_{n} on a modular form f by f |T_{n}. Merel defines the following operators on M which we will also denote by T_{n}, abusing notation slightly.

Theorem 3.6. [6, Theorem 2 and Proposition 10] The operator T_{n} acts on any Manin symbol x^{r}y^{s}(u, v) via

T_{n} x^{r}y^{s}(u, v) = ∑_{\begin{matrix}a>b≥0 \\ d>c≥0 \\ ad−bc=n \end{matrix}}(ax + by)^{r}(cx + dy)^{s}(au + cv, bu + dv). (1)

If n is not coprime to l, then we omit the terms for which g.c.d.(l, au+cv, bu+dv) > 1. This action is compatible with the pairing between cusp forms and Manin symbols

⟨f |T_{n}, x^{r}y^{s}(u, v)⟩ = ⟨f, T_{n} x^{r}y^{s}(u, v)⟩.

It is also proved in [6, Remark 2, p. 76] that this Hecke action is compatible with the symmetrization maps:

Proposition 3.7.

T_{n}(x^{r}y^{s}(u, v)_{±}) = (T_{n} x^{r}y^{s}(u, v))_{±}.

3.8. In this subsection we associate to every pair of integers (m, n) a certain Manin symbol R_{(m,n)}. These symbols satisfy relations analogous to those satisfied by the weight two symbols.

Definition 3.9. Let m and n be two integers. If g.c.d.(m, n, l) > 1 then we set R_{(m,n)} to be zero. Otherwise, R_{(m,n)} := (mx + ny)^{k−2}(m, n).

We remark that even though R_{(m,n)} is built out of the Manin symbol (m, n), its value depends on more than just the residues of m and n modulo l. It is straightforward to see that the symbols R_{(m,n)} obey the following relations:

1. R_{(m,n)} + R_{(−n,m)} = 0.

2. R_{(m,n)} + R_{(−m−n,m)} + R_{(n,−m−n)} = 0.

3. (R_{(m,n)})_{±} = (R_{(m,n)} ± R_{(−m,n)})/2.

We will denote (R_{(m,n)})_{±} by R_{(m,n)}^{±}.

4. (Mod l)-polynomials

4.1. We will need to work with certain functions that we call (mod l)-polynomials. We fix for the remainder of this section a positive integer l.

Definition 4.2. A function h: ℤ → ℂ is called a (mod l)-polynomial if its restriction to each coset lℤ + k is a polynomial.

One can think of a ( mod l)-polynomial as of a collection of l ordinary polynomials, one for each residue modulo l. For example, the function that equals m^{2} + m when m is even and m^{3} when m is odd is a (mod 2)-polynomial.

Proposition 4.3. The set of (mod l)-polynomials forms a ring. For any (mod l)- polynomial h, the function H defined on positive integers m by

H(m) := ∑_{0<i<m} h(i)

extends uniquely to a (mod l)-polynomial.

Proof. Only the second statement requires proof. By additivity, it suffices to consider a (mod l)-polynomial h that is zero except on a single coset lℤ + k_{0}, 0 ≤ k_{0} ≤ l − 1. Then for every coset lℤ + k_{1}, 0 ≤ k_{1} ≤ l − 1 we have

H(k + l) − H(k) = \{ \begin{matrix}h(k − k_{1} + k_{0}), & k_{1} ≤ k_{0} \\ h(k − k_{1} + k_{0} + l), & k_{1} > k_{0} \end{matrix}

for k ∈ lℤ + k_{1}, which implies that H(k) is polynomial on lℤ + k_{1}. Uniqueness is clear.

One can analogously define (mod l)-polynomials h(m, n) of two variables by requiring polynomiality on each pair of cosets (lℤ + k_{1}, lℤ + k_{2}).

4.4. Let f be a cusp form on Γ_{1}(l). Define a function h: ℤ_{>0} → ℂ by the formula

h(m) := ⟨f, R_{(m,0)}^{+} + 2 ∑_{m>i>0} R_{(m,m−i)}^{+}⟩.

Proposition 4.5. The above function h extends to a ( mod l)-polynomial. Moreover,

h(−m) = −h(m).

Proof. The first statement follows from Proposition 4.3 and the definition of R_{(m,i)}^{+}. To prove the second statement, we use the symmetries of R^{+} to rewrite h(m) as

⟨f, ∑_{−m<i<m} R_{(m,i)}^{+}⟩.

Let P (m, n) be the two-variable (mod l)-polynomial defined by

P (m, n) := ∑_{i=−n+1}^{n}⟨f, R_{(m,i)}^{+}⟩

for n ≥ 0, and by polynomiality elsewhere. This polynomial satisfies the recursive relations

P (m, n + 1) − P (m, n) = ⟨f, R_{(m,−n)}^{+} + R_{(m,n+1)}^{+}⟩, P (m, 0) = 0.

Consider as well the two-variable (mod l)-polynomial Q defined by

Q(m, n) := −P (−m, −n).

It is easy to check that the symmetries of the symbols R^{+} imply that Q(m, n) satisfies the same recursive relations as P (m, n). Therefore Q = P, which implies P (−m, −m) = −P (m, m). This gives the desired symmetry of h, since R_{(±m,±m)}^{+} = 0 implies P (m, m) = h(m) for positive m, and hence for all m.

5. Main theorem

5.1. Fix a weight k ≥ 3 and a level l ≥ 5. In this section we define an endomorphism of the space 𝒮 (l) of cusp forms of weight k with respect to Γ_{1}(l), and prove that its image contains all cuspidal newforms. This definition is a generalization of [1, Definition 5.2] to k > 2.

Definition 5.2. Let ρ: 𝒮 (l) → 𝒮 (l) be the linear map

ρ(f) = ∑_{n=1}^{∞} (∫_{0}^{i∞}(f |T_{n})(s)ds) q^{n}.

Proposition 5.3. The form ρ(f) is a cusp form with nebentypus equal to that of f.

Proof. The statement follows from [6, Theorem 6]; see also [1, Proposition 5.3].

The map ρ was used in [1] because its image contains all newforms of weight two with non-vanishing L-function. The analogous statement for higher weights is the following:

Proposition 5.4. The image of ρ contains all newforms.

Proof. One needs to show that for any newform f

∫_{0}^{i∞} f(τ) dτ ≠ 0,

which is equivalent to L(f, 1) ≠ 0. Without loss of generality, we may assume that f is a Hecke eigenform. For weight k > 3, this is a value of L(f, s) outside the critical strip, and so L(f, 1) ≠ 0 by absolute convergence of the Euler product of L(f, s). For k = 3, this is a value of L(f, s) on the boundary of the critical strip. By Theorem 1.3 of [4], it is known that such a value cannot vanish.

5.5. Fix a cuspform f. We can express ρ(f) in terms of modular symbols as

ρ(f) = ∑_{n=1}^{+∞} q^{n}⟨f, T_{n}y^{k−2}(0, 1)⟩ = ∑_{n=1}^{+∞} q^{n}⟨f, T_{n}R_{(0,1)}^{+}⟩ = ∑_{n=1}^{∞} q^{n}⟨f, ∑_{\begin{matrix}a>b≥0 \\ d>c≥0 \\ ad−bc=n \end{matrix}} R_{(c,d)}^{+}⟩.

Now consider the following linear combination of toric quasimodular forms:

ρ_{1}(f) = ∑_{r+s=k−2} ∑_{m,n=0}^{l−1} \frac{(r + s)!}{r!s!}s̃_{m/l}^{(r+1)}s̃_{n/l}^{(s+1)}⟨f, (x+y)^{r}y^{s}(m, m+n)−(x−y)^{r}y^{s}(m, m−n)⟩

= C+∑_{r=0}^{k−2} ∑_{m=0}^{l−1} c_{r,m}s̃_{m/l}^{(r+1)}(q)+∑_{D>0} q^{D} ∑_{m,n,r,s} A⟨f, (x+y)^{r}y^{s}(m, m+n)−(x−y)^{r}y^{s}(m, m−n)⟩.

Here C, c_{r,m} are some constants whose explicit values will not be needed, and A = A(r, s, m, n, D) is defined by

A := \frac{(r + s)!}{r!s!} ∑_{\begin{matrix}m_{1},k_{1},m_{2},k_{2}>0 \\ m_{1}k_{1}+m_{2}k_{2}=D \end{matrix}} k_{1}^{r}k_{2}^{s}(δ_{k_{1}}^{mmod l} + (−1)^{r−1}δ_{k_{1}}^{−mmod l})(δ_{k_{2}}^{nmod l} + (−1)^{s−1}δ_{k_{2}}^{−nmod l}).

This formula for ρ_{1}(f) can be simplified to yield

ρ_{1}(f) = C + ∑_{r=0}^{k−2} ∑_{m=0}^{l−1} c_{r,m}s̃_{m/l}^{(r+1)}(q) + 4 ∑_{D>0} q^{D}⟨f, ∑_{\begin{matrix}m_{1},k_{1},m_{2},k_{2}>0 \\ m_{1}k_{1}+m_{2}k_{2}=D \end{matrix}}(R_{(k_{1},k_{1}+k_{2})}^{+} − R_{(k_{1},k_{1}−k_{2})}^{+})⟩.

The quasimodular form ρ_{1}(f) and the modular form ρ(f) are closely related as follows:

Proposition 5.6.

ρ_{1}(f)(q) − 12ρ(f)(q) = C + ∑_{r=0}^{k−2} ∑_{m=0}^{l−1} c_{r,m}s̃_{m/l}^{(r+1)}(q)

+4 ∑_{n>0} q^{n}( ∑_{0<k_{1}|n}(\frac{2n}{k_{1}} + 1)⟨f, R_{(k_{1},0)}^{+}⟩ + 2 ∑_{k_{1}|n,k_{1}>k_{2}>0}⟨f, R_{(k_{1},k_{1}−k_{2})}^{+}⟩). (2)

Proof. This is the consequence of the identity

∑_{\begin{matrix}m_{1}k_{1}+m_{2}k_{2}=n, \\ m_{1},k_{1},m_{2},k_{2}>0 \end{matrix}}(R_{(k_{1},k_{1}−k_{2})}^{+} − R_{(k_{1},k_{1}+k_{2})}^{+}) = −3 ∑_{\begin{matrix}ad−bc=n, \\ a>b≥0,d>c≥0 \end{matrix}} R_{(c,d)}^{+}

− ∑_{0<k_{1}|n}(\frac{2n}{k_{1}} + 1)R_{(k_{1},0)}^{+} − 2 ∑_{k_{1}|n,k_{1}>k_{2}>0} R_{(k_{1},k_{1}−k_{2})}^{+}.

This identity with k = 2 appears as an intermediate step of the proof of Theorem 5.8 in [1]. However, its proof only uses relations among R_{(m,n)}^{+} that are independent of the weight k.

5.7. Our goal now will be to show that the terms of (2) can be written as a toric quasimodular form. First of all, we will deal with the term

∑_{n>0} q^{n} ∑_{0<k_{1}|n} \frac{2n}{k_{1}}⟨f, R_{(k_{1},0)}^{+}⟩ = 2 ∑_{n>0} q^{n} ∑_{m|n} nm^{k−3}⟨f, x^{k−2}(m, 0)_{+}⟩.

We start with the formula of [1]

s_{α}^{(2)} + s_{α}^{2} = (2πi)^{−2}(\frac{ϑ_{zz}(α, τ)}{ϑ(α, τ)} − \frac{ϑ_{zzz}(0, τ)}{3ϑ_{z}(0, τ)}) = \frac{1}{6} − 2 ∑_{n>0} q^{n} ∑_{m|n} \frac{n}{m}(e^{2πiαm} + e^{−2πiαm}),

8 LEV A. BORISOV AND PAUL E. GUNNELLS

where

s_{α}^{(k)}(q) = (2πi)^{−k} (\frac{∂^{k}}{∂z^{k}} )_{z=0} log (\frac{zϑ(z + α, τ)ϑ′(0, τ)}{ϑ(z, τ)ϑ(α, τ)}) ,

and differentiate it k − 2 times with respect to α. The proof of Proposition 2.2 shows that s_{a/l}^{(k)} are linear combinations of s̃_{a/l}^{(k)}. The definition of s_{α}^{(k)} implies that derivatives with respect to α of s_{α}^{(k)} are quasimodular forms. Indeed,

(2πi)^{−1} \frac{∂}{∂α}s_{α}^{(k)} = s_{α}^{(k+1)} − (2πi)^{−k−1} (\frac{∂^{k+1}}{∂z^{k+1}} )_{z=0} log (\frac{zϑ′(0, τ)}{ϑ(z, τ)}) = s_{α}^{(k+1)} − 2E_{k+1},

where E_{odd} := 0. Then one observes that ⟨f, x^{k−2}(m, 0)_{+}⟩ is odd or even in m depending on the parity of k, which allows one to express

∑_{n>0} q^{n} ∑_{m|n} nm^{k−3}⟨f, x^{k−2}(m, 0)_{+}⟩

as a linear combination of s̃_{a/l}^{(m)}s̃_{b/l}^{(n)}, s̃_{a/l}^{(k)} and E_{k} up to a constant. Moreover, the following lemma allows us to express s̃_{a/l}^{(k)} and E_{k} in terms of s̃_{a/l}^{(m)}s̃_{b/l}^{(n)}.

Lemma 5.8. For k > 1, modular forms s̃_{a/l}^{(k)} and E_{k} can be written as a linear combination of s̃_{a/l}^{(m)}s̃_{b/l}^{(n)} for m, n > 0, m + n = k. Proof. We start with the following identity proved in [1, Proposition 4.7]:

s_{α}s_{β} + s_{β}s_{−α−β} + s_{−α−β}s_{α} = −\frac{1}{2} ((s_{α})^{2} + (s_{β})^{2} + (s_{−α−β})^{2} + s_{α}^{(2)} + s_{β}^{(2)} + s_{−α−β}^{(2)}) .

We differentiate the above identity (k − 2) times with respect to α, and then plug in α = −a/l, β = −b/l. We get

s_{a/l}^{(k)} + (−1)^{k−2}s_{−(a+b)/l}^{(k)} − 4E_{k} = 0 mod s̃_{a/l}^{(m)}s̃_{b/l}^{(n)}.

Together with s_{−a/l}^{(k)} = (−1)^{k}s_{a/l}^{(k)}, this finishes the proof.

Together with Proposition 5.6, Lemma 5.8 and the above computations imply the following statement. Proposition 5.9. For any cusp form f, one can subtract from the form ρ(f) a linear combination of toric quasimodular forms s̃_{a/l}^{(m)}s̃_{b/l}^{(n)} to obtain a form g(q) with qexpansion

g(q) = C + ∑_{n>0} q^{n} ∑_{m|n} h(m),

where h(m) is a (mod l)-polynomial with h(m) = −h(−m).

Proof. We will use the result of Proposition 5.6. First of all, one observes that each form s̃_{a/l}^{(m)} can be written as g(q) of the above type. The part of equation (2) that contains 2n/k_{1} has been discussed above. Moreover, the remaining terms of the equation (2)

⟨f, R_{(k_{1},0)}^{+}⟩ + 2 ∑_{k_{1}>k_{2}>0}⟨f, R_{(k_{1},k_{1}−k_{2})}^{+}⟩

are (mod l)-polynomial in k_{1} and are odd in k_{1} by Proposition 4.5.

5.10. We are now ready to prove our main theorem.

Theorem 5.11. All cusp forms of weight three or more are toric. Moreover, any such cusp form can be written as a linear combination of the modular forms s̃_{a/l}^{(r)}s̃_{b/l}^{(s)}.

Proof. One can easily see that lifts of the forms s̃_{a/l}^{(r)} can be written as linear combinations of s̃ for the new level. Therefore, it is enough to consider a newform f. Then ρ(f) is a non-zero multiple of f, and by Proposition 5.9, it can be written up to a constant as a linear combination of toric quasimodular forms s̃_{a/l}^{(m)}s̃_{b/l}^{(n)} for m + n = k

and s̃_{a/l}^{(n)} for smaller n. Then transformation properties under Γ_{1}(l) insure that all

lower weight forms come with zero coefficients. Moreover, no s̃_{0/l}^{(2)} is actually present, for the same reason.

Corollary 5.12. If l ≥ 5, then any cusp form of weight k ≥ 3 can be written as a degree k homogeneous polynomial in weight one Eisenstein series.

Proof. This follows from Theorem 5.11 and [2, Theorem 4.11].

Corollary 5.13. The multiplication map

ℳ_{k_{1}}(l) ⊗ ℳ_{k_{2}}(l) ⟶ ℳ_{k_{1}+k_{2}}(l)

is surjective for all k_{1} ≥ k_{2} ≥ 1, except for k_{1} = k_{2} = 1.

Proof. Theorem 5.11 assures that the image of the above map contains all cuspforms, so it is enough to insure that the forms of the image take arbitrary values at the cusps. To obtain a form which vanishes at all but one cusp p we multiply a form in ℳ_{k_{1}}(l) that vanishes at all cusps except p and perhaps one other cusp q (relevant only if k_{1} = 2) by a form in ℳ_{k_{2}}(l) that vanishes at q but not at p.

Remark 5.14. A slightly weaker statement can be proved directly by using the fact that the ring of modular forms is Cohen-Macaulay. However, we are not aware of any other proofs for (k_{1}, k_{2}) = (2, 1) or (2, 2).

Remark 5.15. One can also ask which Eisenstein series are toric. It is easy to see that for a prime level p all Eisenstein series are toric. But the situation may be different for composite levels. In particular, for l = 25 toric Eisenstein series of weight two form a subspace of codimension one in the space of all Eisenstein series of weight two. We do not know any similar examples for higher weight.

Theorem 5.16. For every level l there exists N such that the ring of toric forms coincides with the ring of modular forms for weights k ≥ N. When l is prime, one can take N = 3.

Proof. In view of Theorem 5.11, one needs to show that all Eisenstein series are eventually contained in the ring of toric forms. Because the ring of toric forms is Hecke stable (see [2]), it is enough to show that the values of toric forms at the cusps eventually span (#cusps)-dimensional space. For this one needs to show that the values of s_{a/l} for two different cusps are not proportional. This is accomplished by a direct calculation that we leave to the reader.

References

1. L. A. Borisov and P. E. Gunnells, Toric modular forms and nonvanishing of L-functions,

  math.NT/9910141.

2. , Toric varieties and modular forms, math.NT/9908138.

3. K. Chandrasekharan, Elliptic functions, Springer-Verlag, 1985.

4. H. Jacquet and J. A. Shalika, A non-vanishing theorem for zeta functions of GL_{n}, Invent. Math. 38 (1976/77), no. 1, 1–16.

5. S. Lang, Introduction to modular forms, Springer-Verlag, 1976.

6. L. Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2dimensional representations, Springer, Berlin, 1994, pp. 59–94.

Department of Mathematics, Columbia University, New York, NY 10027

  E-mail address: lborisov@math.columbia.edu

Department of Mathematics, Columbia University, New York, NY 10027

  E-mail address: gunnells@math.columbia.edu