On Monomial Relations Between Eisenstein Series

Eknath Ghate

June 13, 2000

Abstract

We show that the relations E_{8} = E_{4}^{2}, E_{10} = E_{4}E_{6} and E_{14} = E_{6}E_{8} = E_{4}E_{10}, are the only relations of the form E_{k} = E_{l}E_{k−l}, where E_{k} is the Eisenstein series of weight k for the full modular group. More generally we show that, apart from sixteen exceptions, the product of two level one eigenforms is never an eigenform. The proof uses the Rankin-Selberg method.

1 Introduction

Let M_{k}(SL_{2}(Z)) denote the space of level one modular forms of (even) weight k, and let

E_{k}(z) = 2ζ(k) ∑_{m,n∈Z}′ \frac{1}{(mz + n)^{k}} = 1 − \frac{2k}{B_{k}} ∑_{n=1}^{∞} σ_{k−1}(n) q^{n} ∈ M_{k}(SL_{2}(Z))

denote the normalized Eisenstein series of level one and weight k. The reader may well be familiar with the identities:^{1}

E_{8} = E_{4}^{2}, E_{10} = E_{4}E_{6}, and E_{14} = E_{6}E_{8} = E_{4}E_{10},  (1)

1The identities (1) capture some simple but surprising relations between the arithmetical functions σ_{k}(n) for k = 3, 5, 7, 9 and 13, where σ_{k}(n) = ∑_{d|n} d^{k} is the sum of the kth powers of the positive divisors of n. For instance the first relation yields:

σ_{7}(n) = σ_{3}(n) + 120 ∑_{j=1}^{n−1} σ_{3}(j)σ_{3}(n − j), and the reader may easily write down the other two such equations.

expressing E_{k}, for k = 8, 10 and 14, as a product of Eisenstein series of lower weight. These identities are a simple consequence of the fact that for these weights dim M_{k}(SL_{2}(Z)) = 1, and that the q-expansions in all three cases force the constant of proportionality to be 1.

The first objective of this note is to show that, apart from the four exceptions (1) above, E_{k} is never the product of two level one Eisenstein series of lower weight. For instance,

E_{12} = \frac{441}{691}E_{4}^{3} + \frac{250}{691}E_{6}^{2}, and E_{16} = \frac{1617}{3617}E_{4}^{4} + \frac{2000}{3617}E_{4}E_{6}^{2} etc.^{2}

Though an elementary proof of this fact may be given by simply studying the asymptotic properties of Bernoulli numbers, the proof we give has a more conceptual flavour: we make use of the Rankin-Selberg method which yields integral expressions for modular L-functions. As we shall see, the obstruction to the existence of a relation of the form E_{k} = E_{l}E_{k−l}, is the existence of a cusp form of weight k (along with the inequality B_{k} ≠ B_{k/2} when k ≡ 0 (mod 4)). Thus the relations (1) are a happy consequence of the fact that there are no cusp forms of weights 8, 10 or 14 (and that, coincidentally, B_{8} = B_{4} = −1/30).

The second objective of this note is to confirm a suspicion of D. Prasad, who, on reading a preliminary version of this note, suggested that in general the product of two (level one) eigenforms should hardly ever be an eigenform. Indeed, one would not expect the convolution product of q-expansions to preserve the Hecke relations among Fourier coefficients, unless it was forced to do so for dimension reasons. Using the Rankin-Selberg method again, we shall show that, except for sixteen exceptions, the product of two level one eigenforms is never an eigenform (see Proposition 1).

This note was inspired by a recent paper of Conrey and Farmer [2], who were concerned with the vanishing of the central critical value of the standard Hecke L-function attached to level 1 eigenforms (in this context see also the work of Kohnen-Zagier [6] and Hida-Maeda [4]). We would like to thank the authors of [2] for sharing their paper with us while it was still in preprint form. We also thank D. Prasad for his interest in this note, and for asking the right questions. Finally, it should be noted (and we thank D. Zagier for reminding us of this) that the ideas used in this note have been more or less well known since the time of Rankin.^{3} However, it seems that the results we seek have not been stated explicitly, whence this note.

2That E_{k} can be expressed polynomially in terms of E_{4} and E_{6} follows, of course, from the familiar fact that there is a Q-basis consisting of monomial expressions of the form E_{4}^{a}E_{6}^{b} for the space of forms in M_{k}(SL_{2}(Z)) having rational Fourier expansion.

2 The proof

Let f = ∑ a_{n}q^{n} ∈ S_{k}(SL_{2}(Z)) be a level one cusp form and let g = ∑ b_{n}q^{n} ∈ M_{l}(SL_{2}(Z)) be a level one modular form of smaller even weight l, with 4 ≤ l ≤ k − 2. Then some standard integration shows

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

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 Γ = SL_{2}(Z) and D = Γ\H. Then another standard unwinding argument shows that

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

where

E_{k}(s, z) = ∑_{Γ_{∞}\Γ} \frac{1}{(cz + d)^{k}} \frac{1}{|cz + d|^{2s}} .

Specializing at g = E_{l} and s = k − 1 yields the identity

D(k − 1, f ⊗ E_{l})(4π)^{1−k}Γ(k − 1) = < f, E_{l}E_{k−l} >,  (2)

where, if f is a weight k cusp form, and h an arbitrary weight k form, then

< f, h >= ∫ ∫_{D} \overline{f(z)}h(z) y^{k} \frac{dxdy}{y^{2}}

3For instance Shimura [8] has used the Rankin-Selberg method to obtain information about the critical values of the standard Hecke L-function attached to modular forms of arbitrary level.

is the usual Petersson inner product.

On the other hand, there is a well known formula of Ramanujan (see [1], or pages 155-156 of [5]), which in our case yields

D(k − 1, f ⊗ E_{l}) = −\frac{2l}{B_{l}} · \frac{L(k − 1, f)L(k − l, f)}{ζ(k − l)},  (3)

where

L(s, f) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

is the standard Hecke L-function of f. Thus since neither L(k − 1, f) nor ζ(k−l) vanishes (the points of evaluation are in the region of absolute convergence), (3) tells us that D(k−1, f⊗E_{l}) = 0 if and only if L(k−l, f) = 0. Combining this with (2), we see that this happens exactly when < f, E_{l}E_{k−l} >= 0.

Now suppose, towards a contradiction, that we have a relation of the form

E_{k} = E_{l}E_{k−l}  (4)

other than the ones listed in (1) above.^{4} Then, k = 12 or k > 14, and so we are guaranteed the existence of a cusp form f ∈ M_{k}(SL_{2}(Z)), which we now pick and fix. Since E_{k} is orthogonal to (every cusp form) f with respect to the Petersson inner product, the above remarks, in conjunction with (4), imply that L(k − l, f) = 0. By symmetry (the functional equation of L(s, f) relates l and k − l) we may assume l ≥ k/2. Since L(s, f) does not vanish for Re(s) > \frac{k+1}{2} (this is the region of absolute convergence) we see that we get a contradiction, unless l = k/2.

So assume now that l = k/2. Then the alleged relation reads

E_{k} = E_{k/2}^{2},

with k ≡ 0 (mod 4). Unfortunately, we must now resort to an ad hoc (but easy) argument: a quick inspection of the q term on both sides yields that

B_{k} = B_{k/2},

which is false for k > 8. In fact |B_{2m+2}| > |B_{2m}| for all m ≥ 3, as can be easily deduced from the facts:

4Note that here we may assume that 4 ≤ l ≤ k − 4, l even, because there is no (holomorphic) Eisenstein series of weight 2 and level one.

1. The Riemann zeta function, ζ(s), is a decreasing function on the ray Re(s) > 1, s ∈ R, and,

2. For m = 1, 2, 3, …, we have

ζ(2m) = \frac{(−1)^{m+1}(2π)^{2m}B_{2m}}{2(2m)!} > 1.

Thus we see that there can be no monomial relation of the form (4) unless k = 8 (and B_{8} = B_{4}), or k = 10 or 14. That is, there are no such relations other than those in (1) above. □

A natural question posed by D. Prasad is whether it is possible, more generally, for the product of two (level one) eigenforms g and h to be an eigenform. We now show that in general the answer is NO.

The discussion so far has treated the case when both g and h are Eisenstein series. When g and h are cusp forms, gh can never be an eigenform, since the q-expansion of gh starts with q^{2}. This leaves the case when g is a cusp form, say of weight l ≥ 12, and h = E_{k−l} is an Eisenstein series of weight k − l ≥ 4.

The above reasoning easily allows us to dispose of this case as well. Indeed, if the weight k of gE_{k−l} satisfies k ≥ 24, k ≠ 26, then gE_{k−l} is never an eigenform. This is because, for such k, the dimension of the space of cusp forms is at least two, and therefore one can always pick an auxiliary cuspidal eigenform f of weight k such that (f, gE_{k−l}) = 0. On the other hand,

D(k − 1, f, g) ≠ 0,

since k − l ≥ 4 implies that s = k − 1 is in the region of absolute convergence, Re(s) > \frac{k+l}{2}. Now a formula similar to one such as (2), obtained by taking g to be a cusp form instead of the Eisenstein series E_{l}, then yields a contradiction.

This argument fails if the space of cusp forms of weight k is one dimensional. Indeed, in this case, gE_{k−l} is forced to be a cuspidal eigenform. A complete list of the 12 exceptional identities may easily be written down:

Δ_{16} = E_{4}Δ_{12},
Δ_{18} = E_{6}Δ_{12},
Δ_{20} = E_{4}Δ_{16} = E_{8}Δ_{12},  (5)
Δ_{22} = E_{4}Δ_{18} = E_{6}Δ_{16} = E_{10}Δ_{12},
Δ_{26} = E_{4}Δ_{22} = E_{6}Δ_{20} = E_{8}Δ_{18} = E_{10}Δ_{16} = E_{14}Δ_{12},

where Δ_{k} is the unique normalized cuspidal eigenform of weight k for k = 12, 16, 18, 20, 22, and 26. □

We summarize by stating the following result whose proof we have just given:

Proposition 1 With the exception of the sixteen relations listed in (1) and (5) above, the product of two level one eigenforms is NEVER an eigenform. In fact if g is a modular eigenform of weight l, and h = E_{k−l} has weight k−l, and we write

gh = δE_{k} + ∑_{f} c_{f}f  (6)

as a linear combination over a basis of normalized^{5} eigenforms, with δ = 1 (respectively 0), if g = E_{l} (respectively a cusp form), then each

c_{f} = \frac{(f, gh)}{(f, f)} = \frac{Γ(k − 1)(4π)^{1−k}D(k − 1, f, g)}{(f, f)}  (7)

is NON-ZERO, except possibly in the situation when g = h = E_{k/2} and k ≡ 0 (mod 4), in which case AT LEAST ONE c_{f} ≠ 0 if k > 8.

Remark 1 In the case of E_{k/2}^{2}, with k > 8 and k ≡ 0 (mod 4), it is a mysterious open problem as to whether or not the coefficient c_{f} of f in the expansion (6) ever vanishes.

5Cuspidal eigenforms are normalized in the usual way so that the Fourier coefficient of the q term is 1. However, this is not the normalization we have chosen for the Eisenstein series E_{k}. Rather we have ‘normalized’ E_{k} by taking the constant term in the Fourier expansion to be 1.

It is known that if the Hecke algebra of level one and weight k (over Q) is simple then each c_{f} ≠ 0. Here is why (see [6] and [2]). Applying σ ∈ Gal(\overline{Q}/Q) to (6), and noting that σ fixes both E_{k/2}^{2} and E_{k}, one gets the relation: c_{f^{σ}} = c_{f}^{σ}. Now the simplicity of the Hecke algebra implies that the action of the Galois group on the basis of eigenforms is transitive. Thus since some c_{f} ≠ 0, all the c_{f} are non-zero. Various authors (see [4] and [2]) have checked the simplicity of the level one Hecke algebra numerically: it is known to be simple (at least) for k ≤ 400.

Another criterion for the vanishing of c_{f} has been given by Waldspurger and Kohnen-Zagier [6]. Before we state it, let us first note that since the sign in the functional equation of L(s, f) is (−1)^{k/2}, L(k/2, f) always vanishes when k ≡ 2 (mod 4). However, in the case of interest to us, namely for eigenforms f of weight k ≡ 0 (mod 4), one expects that L(k/2, f) ≠ 0 in general. Indeed, (3) and (7) above show that

L(k/2, f) = 0 ⟺ c_{f} = 0,

and so the vanishing of L(s, f) at the center of the critical strip would contradict the simplicity of the level one Hecke algebra. Now, the Waldspurger-Kohnen-Zagier theorem tells us that L(k/2, f) is essentially the square of the leading Fourier coefficient, a(1, f′), of f′, where f′ is the cuspidal eigenform of weight \frac{k+1}{2} corresponding to f under the Shimura correspondence

f ↔ f′.

Thus the vanishing of c_{f} is controlled by the vanishing of a(1, f′). As Zagier has pointed out to us, if a(1, f′) = 0, then a(n, f′) = 0 for all square values of n, and this does not mean that f′ = 0. More details and exact formulas can be found in [6].

Remark 2 Proposition 1 is in conformity with results of D. Prasad in [7], where it is proved that the tensor product of a principal series representation with any other infinite dimensional representation of GL_{2}(k), k a local field, contains all the irreducible representations of GL_{2}(k) with the obvious constraint on the central character.

Remark 3 More generally, D. Prasad has asked if it is possible for the product of two outside-N eigenforms of arbitrary level N ≥ 1 to be an outside-N eigenform. Here a modular form of level N is an outside-N eigenform, if it is a simultaneous eigenvector for all the Hecke operators T_{n}, with (n, N) = 1. This question will be investigated in [3].

In a different direction, Hida has asked:

Question: Can the product of two mod p eigenforms be an eigenform?

References

[1] D. Bump. The Rankin-Selberg method: a survey, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 49-109 (1989).

[2] 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).

[3] E. Ghate. On products of eigenforms, in preparation.

[4] H. Hida and Y. Maeda. Non-abelian base change for totally real fields, Pacific J. Math. (Special volume in honor of Olga-Taussky Todd), 189-217 (1997).

[5] H. Hida. Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, 26. Cambridge University Press, Cambridge (1993).

[6] 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).

[7] D. Prasad. Trilinear forms for representations of GL(2) and local epsilon factors, Compositio Math. 75, 1-46 (1990).

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

Current Address: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

email:  eghate@math.tifr.res.in