On Products of Eigenforms

Eknath Ghate

May 3, 2001

1 Introduction

In [5] we observed that, apart from sixteen^{1} exceptions, the product of two level one eigenforms in not an eigenform. This observation was also made independently by W. Duke [4]. The purpose of this note is to investigate whether the product of eigenforms can be an eigenform when the level is square free.

Let M_{k}(Γ_{1}(N)) denote the space of modular forms of weight k ≥ 1 and level N ≥ 1. Let S_{k}(Γ_{1}(N)) denote the subspace of cusp forms, and let E_{k}(Γ_{1}(N)) denote the subspace of those modular forms that are orthogonal to S_{k}(Γ_{1}(N)) with respect to the Petersson inner product. Thus we have

M_{k}(Γ_{1}(N)) = S_{k}(Γ_{1}(N)) ⊕ E_{k}(Γ_{1}(N)).

These spaces may be decomposed according to nebentypus:

M_{k}(Γ_{1}(N)) = ⊕_{χ}M_{k}(N, χ),
S_{k}(Γ_{1}(N)) = ⊕_{χ}S_{k}(N, χ),  (1)
E_{k}(Γ_{1}(N)) = ⊕_{χ}E_{k}(N, χ),

as χ varies through all Dirichlet characters mod N.

Each of the spaces in (1) has an explicit basis consisting of modular forms that are eigenvectors of all the Hecke operators T_{n} with n coprime to N.

1In [5] the exceptional cases were miscounted as fifteen; the relation E_{14} = E_{4}E_{10} should have been included in equation (1) of [5].

For S_{k}(N, χ) this basis is constructed out of newforms of lower level in the usual manner. Let M be a positive integer divisible by the conductor c(χ) of χ. We say that f ∈ S_{k}(M, χ) is primitive if it is a newform, it is an eigenvector of all the Hecke operators of level M, and it is normalized so that the q-term in its q-expansion is 1. Then the set of cusp forms

⋃_{M} ⋃_{Q} {f(Qz) | f is a primitive element of S_{k}(M, χ)},

where M varies through all positive integers satisfying M|N and c(χ)|M, and Q varies through all positive integers dividing N/M, forms a basis of S_{k}(N, χ) consisting of common eigenforms of all the Hecke operators T_{n} with n coprime to N.

A similar explicit basis consisting of modular forms that are eigenvectors of all the Hecke operators T_{n} for n coprime to N exists for E_{k}(N, χ) as well. We refer the reader to Theorem 2 in the text for its description.

Let B denote the explicit basis of M_{k}(Γ_{1}(N)) obtained as above. We shall say that an element of M_{k}(Γ_{1}(N)) is an almost everywhere eigenform (or a.e. eigenform for short) if, up to a scalar multiple, it is an element of B.

The purpose of this note is to shed some light on the following question which was asked by D. Prasad:

Question 1 Let k > l ≥ 1 and N ≥ 1 be integers. Say g ∈ M_{l}(Γ_{1}(N)) and h ∈ M_{k−l}(Γ_{1}(N)) are a.e. eigenforms. When is gh ∈ M_{k}(Γ_{1}(N)) an a.e. eigenform?

There are three cases to consider:

A) g ∈ S_{l}(Γ_{1}(N)) and h ∈ S_{k−l}(Γ_{1}(N)),

B) g ∈ S_{l}(Γ_{1}(N)) and h ∈ E_{k−l}(Γ_{1}(N)),

C) g ∈ E_{l}(Γ_{1}(N)) and h ∈ E_{k−l}(Γ_{1}(N)).

Since the property of being an a.e. eigenform imposes stringent conditions on the q-expansion of a modular form and since the convolution product of q-expansions is unlikely to preserve these conditions one expects that the answer to Question 1 is ‘rarely’. This is corroborated by the following theorem which may be considered a summary of the results proved in the main body of the paper (see Proposition 2 and Theorems 3 and 4).

Theorem 1 Say N is square free. Let g and h be a.e. eigenforms of weight l ≥ 3 and k − l ≥ 3, and nebentypus ψ and χ respectively. Then:

1. In case A) gh is never an a.e. eigenform.

2. In case B) assume that g is a newform and dim S_{k}^{new}(N, ψχ) ≥ 2. Then gh is not an a.e. eigenform.

3. In case C) assume that l ≠ k/2 and that ψ and χ are primitive characters. Assume that

dim S_{k}^{new}(N, ψχ) ≥ \begin{cases}1 & when gh ∈ E_{k}(N, ψχ), \\ 2 & when gh ∈ S_{k}(N, ψχ). \end{cases}

Then gh is not an a.e. eigenform.

The proof of Theorem 1 uses the Rankin-Selberg method and the fact that when N is square free the Atkin-Lehner involutions act transitively on the set of Eisenstein series.

We now make some comments on the hypotheses of Theorem 1.

The restriction that the weights of g and h be larger than 3 arises because of technical reasons (Eisenstein series of weight 1 and 2 behave slightly differently than Eisenstein series of weight larger than 3) as well as conceptual ones (the Rankin-Selberg L-function may vanish at the point of interest when the weight is 1 or 2).

In case B) the theorem is false if g is not a newform. For example if E_{4} is the unique normalized Eisenstein series of level one, Δ_{12} is the Ramanujan Delta function and Δ_{16} is the unique cusp form of level one and weight 16, then the identity

Δ_{12}(z) · E_{4}(z) = Δ_{16}(z)

of eigenforms will propagate to S_{16}(Γ_{0}(N)) for arbitrary N by simply substituting Nz for z. Thus it is makes sense to require g to be a newform.

In case C) we remind the reader that even though g and h are Eisenstein series it is possible that gh is a cusp form. For instance if g = f_{3}(z, χ_{−3}, 1) ∈ E_{3}(3, χ_{−3}) and h = f_{5}(z, 1, χ_{−3}) ∈ E_{5}(3, χ_{−3}) (here χ_{−3} is the quadratic character of conductor 3; see Theorem 2 for the other notation) then g vanishes exactly at the cusp ∞ whereas h vanishes exactly at the cusp 0. Since there are only two cusps for Γ_{1}(3) we see that gh ∈ S_{8}(Γ_{0}(3)) must be a cusp form. (It is in fact an eigenform since S_{8}(Γ_{0}(3)) has dimension 1).

In case C) the assumption that ψ and χ are primitive is made, as in case B), to exclude from consideration identities in level N which have propagated from smaller levels (example: if E_{k} is the unique normalized Eisenstein series of level one and weight k then E_{4}(Nz)E_{6}(Nz) = E_{10}(Nz) in E_{10}(Γ_{0}(N)) for all N).

The assumption that l ≠ k/2 in case C) is more serious and interesting. It arises since the (twisted) L-function of a cusp form of weight k may vanish at the center of the critical strip. In this case the method of proof of this paper fails. However one can work around this in special cases by a direct comparison of Fourier coefficients. When N = 1 this was done in [5]. When N is arbitrary we only sketch the strategy for doing this in general. As an application of this circle of ideas we prove a result which give a criterion for the the non-vanishing of certain (twisted) standard L-functions at the center of the critical strip (see Theorem 5 and its corollaries).

We end this paper with some examples of a.e. eigenforms which can be written as products of a.e. eigenforms of smaller weight.

2 Rankin-Selberg method

In this section we briefly review the Rankin-Selberg method since it is at the heart of the proof of Theorem 1. A reference is Shimura [9].

Let N ≥ 1 and k > l ≥ 1 be integers. Let χ and ψ be Dirichlet characters mod N. Say that f = ∑ a_{n}q^{n} ∈ S_{k}(N, χ) and g ∈ M_{l}(N, ψ). Let f_{c}(z) = \overline{f(−\overline{z})} = ∑_{n} \overline{a}_{n}q^{n}. Then some standard integration shows

I(s, f, g) := ∫_{0}^{∞} ∫_{0}^{1} \overline{f_{c}(z)}g(z)y^{s−1}dxdy = (4π)^{−s} Γ(s) D(s, f ⊗ g),

where

D(s, f ⊗ g) = ∑_{n=1}^{∞} \frac{a_{n}b_{n}}{n^{s}}

is the Rankin product L-function of f and g. Now let D = Γ_{0}(N)\H. Then another standard unwinding argument shows that

I(s, f, g) = ∫ ∫_{D} \overline{f_{c}(z)}g(z)E_{k−l,χψ}(s + 1 − k, z)y^{k} \frac{dxdy}{y^{2}} ,

where

E_{k,ω}(s, z) = ∑_{Γ_{0}(N)_{∞}\Γ_{0}(N)} \frac{ω(d)}{(cz + d)^{k}|cz + d|^{2s}}  (2)

is the Eisenstein series with parameter s. If f_{1} ∈ S_{k}(Γ_{1}(N) and f_{2} ∈ M_{k}(Γ_{1}(N)) let

(f_{1}, f_{2}) = ∫ ∫_{D} \overline{f_{1}(z)}f_{2}(z) y^{k} \frac{dxdy}{y^{2}}

be the usual Petersson inner product of f_{1} and f_{2}. Specializing the two expression above for I(s, f, g) at s = k − 1 yields the important identity

(4π)^{1−k}Γ(k − 1)D(k − 1, f ⊗ g) = (f_{c}, gE_{k−l,χψ}),  (3)

where E_{k−l,ω} ∈ E_{k−l}(N, \overline{ω}) is a modular form (except in the case when both k − l = 2 and ω = 1) which is described in more detail in Section 3.

3 Eisenstein series on Γ_{1}(N)

In this section we recall some facts about Eisenstein series on Γ_{1}(N) for an arbitrary integer N ≥ 1. Further details can be found in Chapters 2, 4, and 7 of Miyake’s book [8].

Let t be the number of inequivalent cusps for Γ_{1}(N). When N ≥ 3, −1 ∉ Γ_{1}(N), and so we may further classify the cusps as regular or irregular (see page 19 of [8]). Let u denote the number of regular cusps and v the number of irregular cusps (for N ≥ 3). We have (cf. [8], Theorem 4.2.9):

t = \begin{cases}1 & if N = 1, \\ 2 & if N = 2, \\ 2 & if N = 3, (0 and ∞ are both regular), \\ 3 & if N = 4, (0, ∞ are regular, 1/2 is irregular), \\ \frac{1}{2} ∑_{0<d|N} ϕ(d)ϕ(N/d) & if N ≥ 5, (all cusps are regular). \end{cases}

The number of cusps is closely connected to the dimension of the space of Eisenstein series. In fact we have (cf. [8], Theorems 2.5.2 and 2.5.3):

dim E_{k}(Γ_{1}(N)) = \begin{cases}t & if k ≥ 4 is even, \\ t − 1 & if k = 2, \\ u & if k ≥ 3 is odd and N ≥ 3, \\ \frac{u}{2} & if k = 1 and N ≥ 3, \\ 0 & if k ≥ 1 is odd and N = 1, 2. \end{cases}  (4)

Note that in the generic case, namely k ≥ 3 and N ≥ 5, the dimension is t = u = \frac{1}{2} ∑_{0<d|N} ϕ(d)ϕ(N/d).

In fact it is easy to explicitly write down a basis of a.e. eigenforms of E_{k}(Γ_{1}(N)) (see [8], Theorems 4.7.1 and 4.7.2):

Theorem 2 Let χ_{1} and χ_{2} be Dirichlet characters mod M_{1}, respectively M_{2}, such that χ_{1}χ_{2}(−1) = (−1)^{k}, with k ≥ 1. Assume that

• if k = 2 and χ_{1} = χ_{2} = 1, then M_{1} = 1 and M_{2} is a prime number,

• otherwise, χ_{1} and χ_{2} are primitive characters.

Then, there is an element f = f_{k}(z, χ_{1}, χ_{2}) = ∑_{n=0}^{∞} a_{n}q^{n} ∈ E_{k}(M, χ), with M = M_{1}M_{2} and χ = χ_{1}χ_{2}, characterized by the properties

• L(s, f) = L(s, χ_{1})L(s − k + 1, χ_{2}), and,

• the constant term of f is

a_{0} = \begin{cases}0 & if k ≥ 2 and χ_{1} ≠ 1, or, \\ & if k = 1 and both χ_{1} ≠ 1 and χ_{2} ≠ 1, \\ \frac{M−1}{24} & if k = 2 and χ_{1} = 1, χ_{2} = 1, \\ \frac{−B_{k, χ}}{2k} & otherwise. \end{cases}

The modular form f_{k}(z, χ_{1}, χ_{2}) ∈ E_{k}(M, χ) is an eigenvector for all the Hecke operators of level M. Modulo the relation f_{1}(z, χ_{1}, χ_{2}) = f_{1}(z, χ_{2}, χ_{1}) when k = 1, the set of elements

f_{k}(Qz, χ_{1}, χ_{2})

with QM_{1}M_{2}|N, and χ_{1}, χ_{2}, M_{1}, M_{2} satisfying the conditions above, form a basis of E_{k}(N, χ) consisting of common eigenform of all the Hecke operators T_{n} of level N, with n coprime to N.

We leave it to the reader to check (cf. pages 179-181 of [8]) that, by an explicit counting argument, one recovers^{2} the formula (4).

Not all the Eisenstein series f_{k}(Qz, χ_{1}, χ_{2}) ∈ E_{k}(N, χ) of Theorem 2 appear in the Rankin-Selberg method. In fact those that do are essentially those coming from the cusp at ∞. Let us make this more precise. Fix an integer k ≥ 1 and a Dirichlet character ω mod N. Then the Eisenstein series that appear in the Rankin-Selberg method are of the form:

E_{k, ω}(z) = ∑_{Γ_{0}(N)_{∞}\Γ_{0}(N)} \frac{ω(d)}{(cz + d)^{k}} = \frac{1}{2} ∑_{{c≡0 (N) \atop (c,d)=1}} \frac{ω(d)}{(cz + d)^{k}} .  (5)

Except for the case k = 2 and ω = 1, E_{k, ω}(z) is a holomorphic function of z, and so defines an element

E_{k, ω} ∈ M_{k}(N, \overline{ω}).

Actually, when k = 1 or 2 (and ω is arbitrary) the series (5) may not converge absolutely. It is defined instead by a well known trick of Hecke: one considers the Eisenstein series in (2)

E_{k, ω}(s, z) = ∑_{Γ_{0}(N)_{∞}\Γ_{0}(N)} \frac{ω(d)}{(cz + d)^{k} |cz + d|^{2s}} ,

shows that it has an analytic continuation to a right half plane containing s = 0, and then one sets

E_{k, ω}(z) = E_{k, ω}(0, z).

When k = 2 and ω = 1, the specialization E_{2, 1}(z) so obtained, is nonholomorphic in z. However one knows that for each prime p,

E_{2}^{(p)}(z) = E_{2, 1}(z) − pE_{2, 1}(pz).

2There are some minor typos in the formula for the dimension of E_{k}(Γ_{1}(N)) on page 181 of [8]. The second entry should read k = 2, rather than k ≥ 2. Moreover the first two entries of the formula do not make sense when N = 4, and should be replaced by

dim E_{k}(Γ_{1}(4)) = \begin{cases}3 & if k ≥ 4 is even, \\ 2 & if k ≥ 3 is odd, \\ 2 & if k = 2, \end{cases} as predicted by the formula (4).

is a holomorphic function of z, and so defines an element of M_{2}(N, 1) for each p|N. Further details can be found in Chapter 7 of [8].

We now identify the Eisenstein series that appear in the Rankin-Selberg method in terms of the basis of a.e. eigenforms given in Theorem 2:

Lemma 1 1. Assume that k ≥ 3. We have

E_{k, ω}(z) = −\frac{2k}{B_{k,\overline{ω}}} · f_{k}(Qz, χ_{0}, \overline{χ_{2}}),

where χ_{0} is the principal character of level M_{1} = 1, χ_{2} is the primitive character of conductor M_{2} associated to ω, and Q = N/M_{2}.

2. When k = 2 and ω = 1, we have

E_{2}^{(p)}(z) = −8π^{2} · f_{2}(z, χ_{0}, 1_{p})

where 1_{p} is the trivial character of level M_{2} = p.

Proof: Assume momentarily that ω is primitive of level N. By (7.1.13) and (7.1.30) of [8], we see that

E_{k, ω}(z) = −\frac{2k}{B_{k,\overline{ω}}} · f_{k}(z, χ_{0}, \overline{ω}),

as desired. The constant of proportionality was computed using Theorem 3.3.4(1) of [8]. More generally, if ω is associated to the primitive character χ_{2} of level M_{2}, then substituting c = c′Q with Q = \frac{N}{M_{2}} in (5) above, we have

E_{k, ω}(z) = \frac{1}{2} ∑_{{c′≡0 (M_{2}) \atop (c′,d)=1}} \frac{χ_{2}(d)}{(c′Qz + d)^{k}} = E_{k, χ_{2}}(Qz) = −\frac{2k}{B_{k,\overline{χ_{2}}}} · f_{k}(Qz, χ_{0}, \overline{χ_{2}})

since (Q, d) ≠ 1 ⟹ (N, d) ≠ 1 ⟹ ω(d) = 0. This proves the first statement. Actually the proof given here should work for all k ≥ 1 except the case when k = 2 and ω = 1 but we do not pursue this point here.

For the second statement we simply refer the reader to Lemma 7.2.19 of [8]. □

Let us close this section with the following corollary to Theorem 2:

Corollary 1 The tuples (k, N, χ) for which dim E_{k}(N, χ) = 1, along with an explicit generator, are as follows (here p denotes a prime):

Weight  Level  Nebentypus  Generator
k = 1  N = p^{m}, N ≠ 2 χ is primitive, odd  E_{1, χ}
k = 2  N = p  χ = 1  E_{2}^{(p)}
k ≥ 4 is even  N = 1  χ = 1  E_{k, χ_{0}}
Table 1: Tuples (k, N, χ) with dim E_{k}(N, χ) = 1.

4 Atkin-Lehner involutions

Let N be an arbitrary integer, and let Q|N be such that

(Q, N/Q) = 1.  (6)

Then there is an Atkin-Lehner involution w_{Q} defined by

w_{Q} = (\begin{matrix}Qx & y \\ Nv & Qw \end{matrix})  (7)

where x, y, v, w ∈ Z satisfy x ≡ 1 mod N/Q, y ≡ 1 mod Q and det(w_{Q}) = Q. w_{Q} acts on the space M_{k}(N, χ) in the usual way: if f ∈ M_{k}(N, χ) then

f|_{w_{Q}}(z) = det(w_{Q})^{k/2}j(w_{Q}, z)^{−k}f(w_{Q}z).

Decompose χ = χ_{Q} · χ_{N/Q} into its Q and N/Q parts. It is well known (see Proposition 1.1 of [1]) that the Atkin-Lehner involution w_{Q} maps

w_{Q} : M_{k}(N, χ_{Q} · χ_{N/Q}) → M_{k}(N, \overline{χ_{Q}} · χ_{N/Q})

and that it takes cusp forms to cusp forms. Moreover, since w_{Q} commutes, up to a constant, with T_{n} for all n with (n, N) = 1, it takes a.e. eigenforms to a.e. eigenforms (see Proposition 1.2 of [1]). In fact w_{Q} takes primitive cusp forms (of level N) to primitive cusp forms (up to multiplication by a constant). On the other hand the following Proposition describes how the a.e. eigenforms of Theorem 2 are permuted under the w_{Q}.

Proposition 1 Let k ≥ 3. Let f_{k}(z, χ_{1}, χ_{2}) ∈ E_{k}(N, χ) be an a.e. eigenform as in Theorem 2. Let w_{Q} be the Atkin-Lehner involution defined in (7). Write Q_{1} = (Q, M_{1}) and Q_{2} = (Q, M_{2}) and decompose

χ_{1} = χ_{Q_{1}}χ_{M_{1}/Q_{1}} and χ_{2} = χ_{Q_{2}}χ_{M_{2}/Q_{2}}

into their Q and prime-to-Q parts. Set

α = \frac{χ_{Q_{2}}(−M_{2}/Q_{2})χ_{M_{2}/Q_{2}}(Q_{2})}{χ_{Q_{1}}(M_{2}/Q_{2})χ_{M_{1}/Q_{1}}(Q_{2})}.

Then

f_{k}(z, χ_{1}, χ_{2})|_{w_{Q}} = Q^{k/2}Q_{2}^{−k} · α · f_{k} (\frac{Qz}{Q_{1}Q_{2}} , \overline{χ_{Q_{2}}}χ_{M_{1}/Q_{1}}, \overline{χ_{Q_{1}}}χ_{M_{2}/Q_{2}}) .

Proof: For the primitive characters χ_{1} and χ_{2} as above consider the Eisenstein series

E_{k}(z, χ_{1}, χ_{2}) = ∑_{c,d} \frac{χ_{1}(c)χ_{2}(c)}{(cz + d)^{k}},

where the sum is over all non-zero integers c and d. By (7.1.13) of [8] we have

f_{k}(z, χ_{1}, χ_{2}) = (const) · E_{k}(M_{2}z, χ_{1}, \overline{χ_{2}}).  (8)

We compute f_{k}(z, χ_{1}, \overline{χ_{2}})|_{w_{Q}}

= Q^{k/2}j(w_{Q}, z)^{−k}f(w_{Q}z, χ_{1}, \overline{χ_{2}})
= Q^{k/2}(const) · (Nvz + Qw)^{−k}E_{k}(M_{2}w_{Q}z, χ_{1}, χ_{2})
= Q^{k/2}(const) · (Nvz + Qw)^{−k} ∑_{c,d} \frac{χ_{1}(c)χ_{2}(d)}{(cM_{2} \frac{Qxz+y}{Nvz+Qw} + d)^{k}}

= Q^{k/2}(const) · ∑_{c,d} \frac{χ_{1}(c)χ_{2}(d)}{Q_{2}^{k} ((cxQ_{2} + d\frac{NQ_{2}}{M_{2}Q}v) (\frac{M_{2}Qz}{Q_{2}^{2}} ) + (c\frac{M_{2}}{Q_{2}} y + d\frac{Q}{Q_{2}} w))^{k}} .

Make the change of variables

(\begin{matrix}c′ \\ d′ \end{matrix}) = (\begin{matrix}Q_{2}x & \frac{NQ_{2}}{M_{2}Q}v \\ \frac{M_{2}}{Q_{2}} y & \frac{Q}{Q_{2}} w \end{matrix}) (\begin{matrix}c \\ d \end{matrix}) .

Then the change of variables matrix has determinant Qxw− \frac{N}{Q} vy = 1. Moreover

c′ ≡ cQ_{2} (M_{1}/Q_{1}) since M_{1}/Q_{1}|(NQ_{2})/(QM_{2}) and x ≡ 1 (M_{1}/Q_{1}),
c′ ≡ −dQ_{2}/M_{2} (Q_{2}) since − Nv/Q ≡ 1 (Q_{2}),
d′ ≡ cM_{2}/Q_{2} (Q_{1}) since Q_{1}|Q/Q_{2} and y ≡ 1 (Q_{1}), and,
d′ ≡ d/Q_{2} (M_{2}/Q_{2}) since Qw ≡ 1 (M_{2}/Q_{2}).

The above congruences imply that

χ_{1}(c) = χ_{Q_{1}}(c) · χ_{M_{1}/Q_{1}}(c)
= χ_{Q_{1}}(d′)χ_{Q_{1}}(M_{2}/Q_{2})^{−1} · χ_{M_{1}/Q_{1}}(c′)χ_{M_{1}/Q_{1}}(Q_{2})^{−1},
χ_{2}(d) = χ_{Q_{2}}(d) · χ_{M_{2}/Q_{2}}(d)
= χ_{Q_{2}}(c′)χ_{Q_{2}}(−M_{2}/Q_{2}) · χ_{M_{2}/Q_{2}}(d′)χ_{M_{2}/Q_{2}}(Q_{2}).

Thus we have f_{k}(z, χ_{1}, \overline{χ_{2}})|_{w_{Q}}

= Q^{k/2}Q_{2}^{−k}α(const) · ∑_{c′,d′} \frac{χ_{Q_{1}}(d′)χ_{M_{1}/Q_{1}}(c′)χ_{Q_{2}}(c′)χ_{M_{2}/Q_{2}}(d′)}{(c′ \frac{M_{2}Qz}{Q_{2}^{2}} + d′)^{k}}

= Q^{k/2}Q_{2}^{−k}α(const) · E_{k} (\frac{QM_{2}z}{Q_{2}^{2}} , χ_{Q_{2}}χ_{M_{1}/Q_{1}}, χ_{Q_{1}}χ_{M_{2}/Q_{2}})
= Q^{k/2}Q_{2}^{−k}α · f_{k} (\frac{Qz}{Q_{1}Q_{2}} , χ_{Q_{2}}χ_{M_{1}/Q_{1}}, \overline{χ_{Q_{1}}} \overline{χ_{M_{2}/Q_{2}}}) ,

where the last line follows from (8). Replacing χ_{2} with \overline{χ_{2}} yields the Proposition. □

5 Case A

In this section we observe that when the level is square free the product of two cuspidal a.e. eigenforms is never an a.e. eigenform. We have:

Proposition 2 Say that k > l ≥ 1 are integers and that and N ≥ 1 is a square free integer. Say g ∈ S_{l}(Γ_{1}(N)) and h ∈ S_{k−l}(Γ_{1}(N)) are a.e. eigenforms. Then gh is NOT an a.e. eigenform.

Proof: Say that

gh = f  (9)

is an a.e. eigenform. Write f(z) = f_{0}(Qz) where M|N, Q|(N/M) and f_{0} ∈ S_{k}(M, χ) is a primitive element. Since N is square free, condition (6) is automatically satisfied for any divisor Q of N. Let w_{Q} be the Atkin-Lehner involution defined in (7). We compute

f|_{w_{Q}} = Q^{k/2}(Nvz + Qw)^{−k}f (\frac{Qxz + y}{Nvz + Qw} )
= Q^{−k/2}(Nvz/Q + w)^{−k}f_{0} (\frac{Qxz + y}{Nvz/Q + w} )
= Q^{−k/2}χ(w)f_{0}(z),

since

(\begin{matrix}Qx & y \\ Nv/Q & w \end{matrix}) ∈ Γ_{0}(M).

Thus, applying w_{Q} to (9) we get g|_{w_{Q}}h|_{w_{Q}} = f_{0} up to a constant. Clearly this is impossible: both g|_{w_{Q}} and h|_{w_{Q}} are cusp forms so that the q-expansion of their product begins with q^{2}; on the other hand since f_{0} is primitive its q-expansion begins with q. □

6 Case B

Fix integers k > l ≥ 1 and N ≥ 1, and mod N characters ψ and ω. Suppose that g ∈ S_{l}(N, ψ) and h ∈ E_{k−l}(N, χ) are a.e. eigenforms. We wish to know whether gh is an a.e. eigenform. We first investigate the case where h = E_{k−l,ω} is the Eisenstein series in (5).

Proposition 3 Suppose that l ≥ 1 and k − l ≥ 3. Say that g ∈ S_{l}(N, ψ) is an a.e. eigenform and that h = E_{k−l, ω} ∈ E_{k−l}(N, \overline{ω}). If g is a newform and dim S_{k}^{new}(N, ψ\overline{ω}) ≥ 2, then gh is NOT an a.e. eigenform.

Proof: First note that the map

f(z) = ∑_{n} a_{n}q^{n} ↦ f_{c}(z) = \overline{f(−\overline{z})} = ∑_{n} \overline{a}_{n}q^{n}  (10)

induces a bijection between the primitive forms in S_{k}(N, \overline{ψ}ω) and those in S_{k}(N, ψ\overline{ω}). Now suppose, towards a contradiction, that gh is an a.e. eigenform. Then, by the hypothesis on the dimension of S_{k}(N, ψ\overline{ω}), we may pick a primitive form f_{c} ∈ S_{k}(N, ψ\overline{ω}) distinct from gh. The Rankin-Selberg method (see (3)) yields

Γ(k − 1)(4π)^{1−k}D(k − 1, f ⊗ g) = (f_{c}, gE_{k−l, ω})

where D(s, f ⊗ g) is the Rankin product L-function of f and g. The right hand side vanishes, by choice of f_{c}. On the other hand the left hand side does not. Indeed, since f and g are primitive, D(s, f, g) has an Euler product decomposition which converges absolutely in the region ℜ(s) > \frac{k+l}{2} and k −1 is in this region since k − l ≥ 3. □

Theorem 3 Suppose that l ≥ 3 and k − l ≥ 3. Let g ∈ S_{l}(N, ψ) and h ∈ E_{k−l}(N, χ) be a.e. eigenforms. Assume that N is square free and that g is a newform. If dim S_{k}^{new}(N, ψχ) ≥ 2 then gh is NOT an a.e. eigenform.

Proof: In the notation of Theorem 2 we may write h = f_{k−l}(Qz, χ_{1}, χ_{2}), where Q|N/M_{1}M_{2}. Since N is square free, condition (6) is automatically satisfied for any divisor of N. In particular let w = w_{N/QM_{2}} be the Atkin-Lehner involution defined in Section 4. By Proposition 1 and Lemma 1 we have

h|_{w} = f_{k−l}(Qz, χ_{1}, χ_{2})|_{w_{N/QM_{2}}} = f_{k−l} (\frac{Nz}{M_{1}M_{2}} , χ_{0}, \overline{χ_{1}}χ_{2}) = E_{k−l,χ_{1}\overline{χ_{2}}}, (11)

where χ_{0} is the principal character and all equalities hold up to a constant.

The formula (11) allows us to reduce the proof of the Theorem to Proposition 3. Indeed, suppose, towards a contradiction, that gh is an a.e. eigenform. Since the Atkin-Lehner involutions take a.e. eigenforms to a.e. eigenforms, g|_{w}h|_{w} is an a.e. eigenform. Admittedly, it lies in the twisted space S_{k}(N, ψ_{QM_{2}} \overline{ψ_{N/QM_{2}}} · \overline{χ_{1}}χ_{2}), which nonetheless still satisfies:

dim S_{k}^{new}(N, ψ_{QM_{2}} \overline{ψ_{N/QM_{2}}} · \overline{χ_{1}}χ_{2}) ≥ 2.

Applying Proposition 3 (with g replaced by g|_{w}, ψ replaced by ψ_{QM_{2}} ·\overline{ψ_{N/QM_{2}}}, and ω = χ_{1}\overline{χ_{2}}) this is impossible since g|_{w} is a newform. This completes the proof of the Theorem. □

7 Case C

Fix integers k > l ≥ 1, and mod N Dirichlet character ψ and ω. Suppose that g ∈ E_{l}(N, ψ) and h ∈ E_{k−l}(N, χ) are a.e. eigenforms. We wish to know whether gh is an a.e. eigenform.

As in case B) we begin our discussion with the case when h = E_{k−l, ω} is the Eisenstein series in (5).

Proposition 4 Suppose that k > l ≥ 1. Let g = f_{l}(z, ψ_{1}, ψ_{2}) ∈ E_{l}(N, ψ) be an a.e. eigenform as in Theorem 2 with ψ = ψ_{1}ψ_{2} primitive. Let h = E_{k−l, ω} ∈ E_{k−l}(N, \overline{ω}) with ω ≠ 1 when k − l = 2. Assume that

dim S_{k}^{new}(N, ψ\overline{ω}) ≥ \begin{cases}1 & if gh ∈ E_{k}(N, ψ\overline{ω}), \\ 2 & if gh ∈ S_{k}(N, ψ\overline{ω}). \end{cases}  (12)

Assume additionally that one of the following conditions hold:

1. k − l ≥ l + 1,

2. k ≥ 3 and there is a primitive form f ∈ S_{k}(N, \overline{ψ}ω)) such that L(k/2, f⊗ ψ_{2}) ≠ 0,

3. k = 2 and there is a primitive form f ∈ S_{2}(N, \overline{ψ}ω)) such that both L(1, f ⊗ ψ_{1}) ≠ 0 and L(1, f ⊗ ψ_{2}) ≠ 0.

Then gh ∈ M_{k}(N, ψ\overline{ω}) is NOT an a.e. eigenform.

Proof: Suppose, towards a contradiction, that gh ∈ M_{k}(N, ψ\overline{ω}) is an a.e. eigenform. Then it would have to lie in either E_{k}(N, ψ\overline{ω}) or in S_{k}(N, ψ\overline{ω}). In either case, the hypothesis (12) on the dimension of S_{k}^{new}(N, ψ\overline{ω}) allows us to pick a primitive eigenform f_{c} ∈ S_{k}(N, ψ\overline{ω}) distinct from gh, which is the complex conjugate of a primitive eigenform f ∈ S_{k}(N, \overline{ψ}ω)). As usual (see (3)) we have the identity

Γ(k − 1)(4π)^{1−k}D(k − 1, f, g) = (f_{c}, gE_{k−l, ω}).  (13)

Since we have assumed that ψ is primitive, g is a common eigenform of all the Hecke operators. Thus D(s, f, g) has an Euler product expansion which yields

D(k − 1, f, g) = \frac{L(k − 1, f ⊗ ψ_{1})L(k − l, f ⊗ ψ_{2})}{L(k − l, ω)},  (14)

where the equality is up to some bad Euler factors.

For reasons of parity k − l = 1 ⟹ ω ≠ 1. So the denominator of the right hand side of (14) is a non-zero finite number.

An Euler product argument shows that L(s, f ⊗ θ) ≠ 0 in the region ℜ(s) > \frac{k+1}{2} for any mod N Dirichlet character θ. By Proposition 2 of [9] one can actually extend this slightly: one has that L(\frac{k+1}{2}, f ⊗ θ) ≠ 0 as well.

Using this we see that if any of the hypothesis 1, 2 or 3 of the Proposition hold, then the numerator in the right hand side of (14) is non-zero as well, for a suitable choice of f. For instance k − l ≥ l + 1 implies that k − l ≥ \frac{k+1}{2} and that k − 1 ≥ \frac{k+1}{2} so that both the L-values in the numerator do not vanish. This yields the usual desired contradiction since the right hand side of (13) vanishes. □

Theorem 4 Suppose that l ≥ 3 and k − l ≥ 3. Let g = f_{l}(z, ψ_{1}, ψ_{2}) ∈ E_{l}(N, ψ) and h = f_{k−l}(z, χ_{1}, χ_{2}) ∈ E_{k−l}(N, χ) be a.e. eigenforms as in Theorem 2 with ψ and χ primitive. Assume that N is square free and that l ≠ k/2. If

dim S_{k}^{new}(N, ψχ) ≥ \begin{cases}1 & when gh ∈ E_{k}(N, ψχ), \\ 2 & when gh ∈ S_{k}(N, ψχ), \end{cases}

then gh is NOT an a.e. eigenform.

Proof: As in the proof of Theorem 3 we may twist h by the Atkin-Lehner involution w = w_{N/M_{2}} (note that now Q = 1) to assume that h = E_{k−l, χ_{1}\overline{χ_{2}}}. If k − l ≥ l + 1, then since g|_{w} would continue to have primitive nebentypus, the theorem would follows from Proposition 4. If k − l ≱ l + 1, switch the roles of g and h (note we are assuming that both χ and ψ are primitive) so that we would be done in the case l ≥ k − l + 1 as well. This leaves only the case k − l = l which we have excluded by the hypothesis l ≠ k/2. □

Note that Theorem 4 gives no information when l = k/2. However, as was done in [5] when N = 1, one could attempt to give a direct ad hoc argument when l = k/2. We sketch the strategy now assuming for simplicity that l ≥ 3. The question is whether, in the notation of Theorem 2, one can have a relation of the form (up to a constant)

f_{k/2}(Q′z, ψ_{1}, ψ_{2})f_{k/2}(Qz, χ_{1}, χ_{2}) = f,  (15)

with f an a.e. eigenform. There are two cases to consider now. Either both the Eisenstein series on the left hand side of (15) do not vanish at a common cusp (in which case f is an Eisenstein series) or not (in which case f is a cusp form).

In the former case, we may always twist by an Atkin-Lehner involution (at least if N is square free) to assume that the two Eisenstein series above do not vanishes at ∞. Thus we may assume that ψ_{1} = χ_{1} = 1. Normalizing all the q-expansions so that the constant terms of each of the eigenforms in (15) is 1, an inspection of the q term yields

\frac{1}{B_{k/2,ψ}} + \frac{1}{B_{k/2,χ}} = \frac{2}{B_{k,ψχ}} .  (16)

Presumably, this relation does not hold very often, in which case neither can the relation (15). For instance when N = 1 the relation (16) becomes

B_{k} = B_{k/2}  (17)

with k ≡ 0 mod 4 which only holds for k = 8 (B_{8} = B_{4}) explaining why E_{4}^{2} = E_{8} is the unique identity of the form E_{k/2}^{2} = E_{k} where E_{k} is the unique normalized Eisenstein series of weight k.

In the latter case (ψ_{1} ≠ 1 or χ_{1} ≠ 1) a similar argument might be carried out by looking at, say, the q^{2} term of (15)

8 Non-vanishing of L-functions

As an application of the some of the arguments presented in the previous sections we give a criterion for the non-vanishing of twists of standard Lfunctions at the center of the critical strip. More precisely we prove:

Theorem 5 Let N ≥ 1 be an arbitrary integer and let k ≥ 6 be an even integer. Assume that ψ and χ are mod N Dirichlet characters such that ψ and ψχ are primitive. If

E_{k/2,ψ̄} · E_{k/2,χ̄} ∉ E_{k}(N, ψχ),  (18)

then there is a primitive form f ∈ S_{k}(N, ψχ) such that

L(k/2, f ⊗ ψ) ≠ 0.

Proof: This follows immediately from the proof of Proposition 4. Set g = E_{k/2,ψ̄} = f_{k/2}(z, 1, ψ). Then, the hypothesis (18) and the fact that ψχ is primitive implies that dim S_{k}^{new}(N, χψ) ≥ 1. Further (18) implies that there exits a primitive form f_{c} ∈ S_{k}^{new}(N, χψ) (for some primitive form f ∈ S_{k}^{new}(N, χ̄ψ̄)) such that (f_{c}, gE_{k/2,χ̄}) ≠ 0. Then (13) and (14) imply that

L(k/2, f ⊗ ψ) ≠ 0,

as desired. □

In special cases (18) is easy to check. For instance, various authors have already observed that the level one version of Theorem 5 can be used to prove:

Corollary 2 ([7], [3]). If k ≡ 0 mod 4 and k > 8 then there exists a primitive cusp form of level N = 1 and weight k such that L(k/2, f) ≠ 0.

Proof: As already noted, (17) fails for such k implying that (18) holds. □

Explicit versions of Theorem 5 may be deduced for arbitrary level. The following corollary gives an example when N is prime.

Corollary 3 Say that N = p is prime and that k ≥ 6 is an even integer. Let ψ and α be distinct primitive characters of level p. Suppose that the system of equations

[\begin{matrix}−\frac{B_{k,α}}{2k} & 0 \\ 1 & 1 \\ 1 + α(2)2^{k−1} & α(2) + 2^{k−1} \end{matrix}] [\begin{matrix}x \\ y \end{matrix}] =

[\begin{matrix}\frac{B_{k/2,ψ}}{k} & \frac{B_{k/2,ψ̄α}}{k} \\ −\frac{B_{k/2,ψ}}{k} − \frac{B_{k/2,ψ̄α}}{k} \\ 1 − \frac{B_{k/2,ψ}}{k} (1 + ψ̄α(2)2^{k/2−1}) − \frac{B_{k/2,ψ̄α}}{k/2} (1 + ψ(2)2^{k/2−1}) \end{matrix}] (19)

is inconsistent. Then there exists a primitive cusp form f ∈ S_{k}(p, α) such that L(k/2, f ⊗ ψ) ≠ 0.

Proof: We apply Theorem 5 with χ replaced by ψ̄α. The Eisenstein series f_{k}(z, 1, α) and f_{k}(z, α, 1) give a basis of E_{k}(N, α). An inspection of the first three terms in the q-expansion of the relation

E_{k/2,ψ̄} · E_{k/2,ψᾱ} = x · f_{k}(z, 1, α) + y · f_{k}(z, α, 1)

gives the system of equations (19) for unknowns x and y. Thus if the system (19) is inconsistent, (18) holds, and the corollary follows. □

9 Examples

In this section we give examples of a.e. eigenforms that are products of a.e. eigenforms of smaller weight. We do not impose any restriction on the weights of the eigenforms or on their level since we expect that the results proven in this paper should generalize. In particular we do not expect any such relations in case A) even for arbitrary level.

So we start with case B). Let us assume that g is a newform. Recall that Theorem 3 does not apply when the dimension of the space S of cusp form in which the product lies has new part of dimension one. In particular it does not apply if S space has dimension one. Table 2 lists all the spaces S_{k}(N, χ) of dimension one, for k > 1, along with a decomposition, if possible, of a generator Δ_{k}(N, χ) of S_{k}(N, χ) into a product of an a.e. eigen-cusp form and an a.e. eigen-Eisenstein series of smaller weight.

Here are some remarks on reading Table 2:

• We have written Δ_{k}(N) for the generator Δ_{k}(N, χ) when χ = 1. When possible we have identified it in terms of the Ramanujan Delta function Δ(z).

• All the identities hold only up to a constant (for which see Theorem 2).

• The notation used for the nebentypus χ is the same as that used in the modular forms calculator HECKE (more about HECKE below) and is self explanatory. Warning: this notation only identifies χ up to Galois conjugacy, but this suffices for our purposes since dim S_{k}(N, χ) = dim S_{k}(N, χ^{σ}), for σ ∈ Gal(\overline{Q}/Q).

• Even when the Galois conjugacy class of χ is larger than one, we have listed only the identity for χ; the identities for the conjugate characters are obtained in an obvious manner.

• Since there are no general formulas for the dimensions of the weight 1 spaces (and we are not aware of any tables in the literature) we have omitted these spaces from the table. For this reason, we have also not listed the weight 2 spaces of dimension one (there are twenty-nine such spaces up to Galois conjugacy whose levels range between N = 11 and N = 49).

Weight  Level  Nebentypus Product of cusp form and Eisenstein series?
k = 1  N =??
k = 2 Twenty-nine  cases
k = 3  N = 7  χ = χ_{2}
k = 3  N = 8  χ = χ_{2, 2}
k = 3  N = 9  χ = χ_{6}
k = 3  N = 10  χ = χ_{4}
k = 3  N = 11  χ = χ_{2}  Δ_{3}(11, χ_{2}) = Δ_{2}(11)E_{1, χ_{2}}
k = 3  N = 11  χ = χ_{10} Δ_{3}(11, χ_{10}) = Δ_{2}(11)E_{1, χ_{10}}
k = 3  N = 12  χ = χ_{1, 2}
k = 3  N = 13  χ = χ_{12} Δ_{3}(13, χ_{12}) = Δ_{2}(13, χ_{6})E_{1, χ_{4}}
k = 3  N = 16  χ = χ_{2, 1} Δ_{3}(16, χ_{2, 1}) = Δ_{2}(16, χ_{1, 4})E_{1, \overline{χ_{2, 4}}}
= Δ_{2}(16, \overline{χ_{1, 4}})E_{1, χ_{2, 4}}
k = 4  N = 5  χ = 1  Δ_{4}(5) = (Δ(z)Δ(5z))^{1/6}
k = 4  N = 6  χ = 1  Δ_{4}(6) = (Δ(z)Δ(2z)Δ(3z)Δ(6z))^{1/12}
k = 4  N = 7  χ = 1  Δ_{4}(7) = Δ_{3}(7, χ_{2})E_{1, χ_{2}}
k = 4  N = 7  χ = χ_{3}  Δ_{4}(7, χ_{3}) = Δ_{3}(7, χ_{2})E_{1, χ_{6}}
k = 4  N = 8  χ = 1  Δ_{4}(8) = Δ_{3}(8, χ_{2, 2})E_{1, χ_{2, 2}}
k = 4  N = 9  χ = 1  Δ_{4}(9) = Δ_{3}(9, χ_{6})E_{1, \overline{χ_{6}}}
k = 5  N = 4  χ = χ_{2}
k = 5  N = 5  χ = χ_{4}
k = 5  N = 7  χ = χ_{2}  Δ_{5}(7, χ_{2}) = Δ_{3}(7, χ_{2})E_{2}^{(7)}
k = 6  N = 3  χ = 1  Δ_{6}(3) = (Δ(z)Δ(3z))^{1/4}
k = 6  N = 4  χ = 1  Δ_{6}(4) = Δ_{5}(4, χ_{2})E_{1, χ_{2}}
k = 6  N = 5  χ = 1  Δ_{6}(5) = Δ_{4}(5)E_{2}^{(5)}
k = 7  N = 3  χ = χ_{2}  Δ_{7}(3, χ_{2}) = Δ_{6}(3)E_{1, χ_{2}}
k = 8  N = 2  χ = 1  Δ_{8}(2) = (Δ(z)Δ(2z))^{1/3}
k = 8  N = 3  χ = 1  Δ_{8}(3) = Δ_{7}(3, χ_{2})E_{1, χ_{2}} = Δ_{6}(3)E_{2}^{(3)}
k = 10  N = 2  χ = 1  Δ_{10}(2) = Δ_{8}(2)E_{2}^{(2)}
k ≥ 12  N = 1  χ = 1  Twelve identities: see display (5) of [5]

Table 2: Tuples (k, N, χ) with dim S_{k}(N, χ) = 1.

Table 2 was compiled, in part, manually, using well known formulas for the dimensions of spaces of cusp forms. It was completed using the modular forms calculator HECKE designed by W. Stein which computes dimensions using the trace formula in Hijikata [6]. Stein has also tabulated the dimensions of the spaces S_{k}(N, χ) for N ≤ 400 using the alternative formula given in Cohen-Osterlé [2]. The package HECKE as well as various tables can be found at Stein’s home page [10].

Let us now turn our attention to case C). We impose the restriction that both g and h have primitive nebentypus (but allow the case l = k/2). Recall that Theorem 4 does not treat the following exceptional cases:

i) gh ∈ S_{k}(N, ψ\overline{ω}) and dim S_{k}^{new}(N, ψ\overline{ω}) ≤ 1,

ii) gh ∈ E_{k}(N, ψ\overline{ω}) and dim S_{k}^{new}(N, ψ\overline{ω}) = 0.

Let us discuss each of these cases in turn.

In case i) let us suppose that dim S_{k}(N, ψ\overline{ω}) = 1. With this restriction clearly gh is an a.e. eigenform. Many of the entries in Tables 2 can be written as a product of Eisenstein series. It would be too time consuming to give an exhaustive list of these relations. We content ourselves with an example: the cusp form Δ_{6}(3), which could not be written as a product of an a.e. eigen-cusp form and an a.e. eigen-Eisenstein series within the limits of Table 2, can be written as the product of two a.e. eigen-Eisenstein series. Namely, we have

Δ_{6}(3) = f_{3}(z, χ_{2}, 1)f_{3}(z, 1, χ_{2}).

As for case ii), one would have to go through the (finite) list of tuples (k, N, χ) for which there are no primitive cusp forms, identifying by inspection which elements of E_{k}(N, χ) can be written as products of Eisenstein series of lower weight. Again it would be too time consuming to tabulate all such relations completely. We merely point out that Table 1 gives an example of such a relation when the level is larger than one. Indeed, if p ≡ 3 (mod 4) and S_{2}(Γ_{0}(p)) = 0, that is when p = 3 or 7, we have:

E_{1, χ_{−p}}^{2} = E_{2}^{(p)},

up to a constant. Here χ_{−p} is the odd quadratic character of conductor p. In the case p = 3, this was already reflected in the entry (8, 3, 1) of Table 2 (there χ_{−3} was called χ_{2}).

References

[1] A. O. L. Atkin and Wen Ch’ing Winnie Li. Twists of newforms and pseudo-eigenvalues of W -operators, Invent. Math. 48, 221-243 (1978)

[2] H. Cohen and J. Oesterlé. Dimensions des espaces de formes modulaires, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pp. 69–78. Lecture Notes in Math., Vol. 627, Springer, Berlin, 1977

[3] B. Conrey and D. Farmer. Hecke Operators and the Non-vanishing of L-Functions, Topics in number theory (University Park, PA 1997), Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 143-150 (1999).

[4] W. Duke. When is the product of two Hecke eigenforms an eigenform? In Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 737-741 (1999)

[5] E. Ghate. On monomial relations between Eisenstein series, J. Ramanujan Math. Soc. 15, 71-79 (2000)

[6] H. Hijikata. Explicit formula of the traces of Hecke operators for Γ_{0}(N), J. Math. Soc. Japan 26, 56-82 (1974)

[7] W. Kohnen and D. Zagier. Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64, 175-198 (1981).

[8] T. Miyake. Modular Forms, Springer-Verlag (1989).

[9] G. Shimura. The special values of the zeta functions associated with cusp forms, Comm. Pure. Appl. Math. 29, 783-804 (1976).

[10] W. Stein. HECKE, http://modular.fas.harvard.edu.

Eknath Ghate School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai, 400005 India

email:  eghate@math.tifr.res.in