PERIODS OF EISENSTEIN SERIES AND SOME APPLICATIONS
ZACHARY A. KENT
Abstract. Bringmann, Guerzhoy, Ono, and the author recently developed an Eichler-Shimura
theory for weakly holomorphic modular forms by studying associated period polynomials. Here
we include the Eisenstein series in this theory.
Classically,
the periods of a weight k (k ≥ 4 will always denote an even integer) cusp form
P
f = l≥1 af (l)q l (q := e2πiτ with τ = x + iy ∈ C with y > 0) on Γ := SL2 (Z) are given by
Z i∞
(1)
rn (f ) :=
f (τ ) τ n dτ for integers n such that 0 ≤ n ≤ k − 2.
0
These periods appear in the coefficients of the period polynomial of f defined by
Z i∞
k−2
X
k−2
n
k−2
(−1)
(2)
r(f ; z) :=
f (τ )(z − τ )
dτ =
rn (f ) z k−2−n .
n
0
n=0
P
More generally, we will consider f = l−∞ af (l)q l ∈ Mk! , where Mk! denotes the space of
weakly holomorphic modular forms of weight k (these are meromorphic modular forms whose
poles, if any, are supported at cusps). In this case, the integrals in (1) and (2) may be divergent.
In [1], Bringmann, Guerzhoy, Ono, and the author found a way to avoid this obstruction by
considering the formal Eichler integral of f given by
X af (l)
ql .
Ef (z) :=
k−1
l
l−∞
l6=0
The period function of f as defined in loc. cit. is
(3)
r(f ; z) := ck Ef (z) − z k−2 Ef (−1/z) ,
Γ(k−1)
where ck := − (2πi)
k−1 . For a cusp form f , this agrees with (2). Therefore, in analogy with (2),
we define rn (f ), the periods of f ∈ Mk! , by
k−2
X
af (0)
1
k−2
k−1
n
(4)
r(f ; z) =
z
+
+
(−1)
rn (f ) z k−2−n .
k−1
z
n
n=0
The space Mk! decomposes into a direct sum of Sk! (the subspace of Mk! whose forms have
vanishing constant term) and the space generated by the weight k Eisenstein series,
X
Bk X
Gk (τ ) := −
+
σk−1 (n)q n , where σk−1 (n) =
dk−1
2k n≥1
d|n
2000 Mathematics Subject Classification. 11F67, 11F03.
1
2
ZACHARY A. KENT
and Bk is the kth Bernoulli number. It was noticed, in [1], that by restricting to forms in Sk! ,
the period functions in (4) are actually polynomials. Here, we will study period functions, so it
suffices to consider r(Gk , z) and the “extra periods” in (4) that depend on aGk (0).
With respect to previous work relating periods to L-values [1, 5, 6, 7], we would like to
interpret the “extra periods” in (4) in terms of the L-series for Gk . We do this following Zagier
in [7]. Define the L-series for Gk by
X
L(Gk , s) =
σk−1 (n)n−s = ζ(s)ζ(s + 1 − k),
n≥1
where ζ(s) is Riemann’s zeta function. Then L(Gk , s) has a meromorphic continuation to the
complex plane and satisfies the functional equation
(2π)−s Γ(s)L(Gk , s) = ik (2π)s−k Γ(k − s)L(Gk , k − s).
Furthermore, L(Gk , s) has a pole only at the point s = k, which is simple, with residue
Γ(k−1)
aGk (0) (2πi)k /Γ(k). Interpreting the binomial coefficient k−2
as the quotient Γ(n+1)Γ(k−1−n)
,
n
we see that this quotient has a simple zero at each integer n such that n 6∈ {0, 1, . . . , k − 2}, and
the first derivative at the point n = k − 1 is −1/(k − 1). This motivates the definition of the
nth normalized period of Gk to be
rbn (Gk ) := Γ(k − 1) ·
in+1
L(Gk , s + 1)
· lim
.
n+1
s→n Γ(k − 1 − s)
(2π)
Combining the above discussion with identity (3) reveals that
Z i∞ Bk
k−2
r(Gk ; z) = ck EGk (z) − z
(z − τ )k−2 dτ
EGk (−1/z) =
Gk (τ ) +
2k
0
(5)
1−n
X
L(Gk , s + 1) X
i
(−1)n rbn (Gk )z k−2−n .
·
lim
=
Γ(k − 1) ·
=
n+1 s→n Γ(k − 1 − s)
(2π)
n∈Z
n∈Z
In general, for a weakly holomorphic modular form f ∈ Mk! , we combine (4) and (5) to define
rbn (f ) (for n ∈ Z), the normalized periods of f , by
X
(6)
r(f ; z) =
(−1)n rbn (f )z k−2−n .
n∈Z
Combining (4), (5), and (6), we find that the normalized periods of f are given by

k−2

for 0 ≤ n ≤ k − 2,
 n rn (f )
(7)
rbn (f ) := −af (0)/(k − 1) for n = −1, k − 1, and

0
for n < −1 or n > k − 1.
It is natural to consider an Eichler-Shimura isomorphism in this context extending the work
of [1, 5, 7]. In this direction, let Vk be the space of polynomials in z of degree at most k − 2, and
b k := z −1 · Vk+2 . There is an action of γ = ( a b ) ∈ Γ on any function
define the larger space V
c d
g given by (g|w γ) (z) := (cz + d)−w g az+b
.
Furthermore,
we
can
extend
this action linearly to
cz+d
the group ring Z[Γ], allowing us to define the period function and period polynomial spaces by
n
o
c k := P ∈ V
b k : P |2−k (1 + S) = P |2−k (1 + U + U 2 ) = 0 ,
W
PERIODS OF EISENSTEIN SERIES AND SOME APPLICATIONS
3
c k , respectively, where S := ( 0 −1 ) and U := ( 1 −1 ) are the standard
and Wk := Vk ∩ W
1 0
1 0
generators for Γ of order 2 and 3. The authors in [1] showed that r(f ; z) ∈ Wk for any
c k \ W. Combining these results, it follows that
f ∈ Sk! . Zagier showed in [7] that r(Gk ; z) ∈ W
!
c k for every f ∈ M .
r(f ; z) ∈ W
k
c k are
Remark. Explicitly, the relations defining W
k−1
X
n
c k ⇔ an = (−1)n−1 ak−2−n , and a−1 + (−1)n
an X ∈ W
k−1
X
cmn am = 0
m=−1
n=−1
where
(
−1 ≤ n ≤ k − 1, and cmn =
m
,
n
k−2−m
,
k−2−n
if m ≥ n,
if m ≤ n.
b k splits as a direct sum V
bk = V
b+ ⊕ V
b − of even and odd functions. This splitting
The space V
k
k
ck = W
c+ ⊕ W
c − and Wk = W+ ⊕ W− . Now
induces further splittings of the subspaces W
k
k
k
k
c ± . Denote the
for any f ∈ Mk! , we may write r(f ; z) = r+ (f ; z) + r− (f ; z) with r± (f ; z) ∈ W
k
space of weight k holomorphic modular (resp. cusp) forms by Mk (resp. Sk ). Then the classical
Eichler-Shimura isomorphism (see [5]) says that we have isomorphisms r± : Sk → Wk± . In [7],
c ± are isomorphisms. Here, we extend
Zagier extended this to show that the maps r± : Mk → W
k
things further to produce an Eichler-Shimura isomorphism theorem for Mk! making use of the
1 d
differential operator D := 2πi
which satisfies, by a well known identity of Bol (see Theorem
dz
1.2 of [2]),
(8)
!
Dk−1 : M2−k
−→ Sk! .
c k be the period function map. Then we have an exact sequence
Theorem 1. Let r : Mk! → W
r
!
c k −→ 0.
0 −→ Dk−1 (S2−k
) −→ Mk! −→ W
Proof. The surjectivity of the restricted map r : Sk! → Wk was established in [1]. This together
c k \ Wk gives us exactness of the sequence at W
ck.
with the observation in [7] that r(Gk ; z) ∈ W
k−1
!
Exactness at D (S2−k ) follows from Bol’s identity (8).
Now, let f ∈ Mk! . A simple calculation shows that r(f ; z) = ck α(z k−2 − 1) for some α ∈ C if
!
!
and only if Ef + α ∈ M2−k
, i.e. f ∈ Dk−1 (M2−k
). Thus the kernel of the period function map is
!
Dk−1 (S2−k
), which proves exactness at Mk! .
c ± in [7], we have
Combining Theorem 1 with the isomorphisms r± : Mk → W
k
(9)
!
ck ∼
Mk! /Dk−1 (S2−k
)∼
=W
= Mk ⊕ Mk .
In contrast, we may start with the following isomorphism considered in [1] and [3]
!
Sk! /Dk−1 (M2−k
)∼
= Sk ⊕ Sk ,
from which (9) follows easily by a dimension argument. This suggests that the “multiplicity two”
Hecke theory in [1] can be extended to include the Eisenstein series Gk and a one dimensional
subspace of Sk! whose nonzero forms have the same Hecke eigenvalues as Gk . This motivates us
to refine the notion of Hecke eigenform.
4
ZACHARY A. KENT
Let m ≥ 2 be an integer, and let T (m) be the usual mth weight k Hecke operator. We call
f ∈ Mk! a Hecke eigenform if for each T (m) there is a unique complex number λm such that
!
.
(f |k T (m) − λm f ) (z) ∈ Dk−1 S2−k
Two remarks.
1) Our definition includes classical (holomorphic) Hecke eigenforms and Hecke eigenforms with
!
!
)
) as defined in [1]. Also, it is easily seen that the space Dk−1 (S2−k
respect to Sk! /Dk−1 (M2−k
does not contain any Hecke eigenforms because for every m, the choice of λm will not be unique.
2) Manin defined an action of Hecke operators on period polynomials in [6] which is compatible
with the Hecke action on the associated modular forms. This action can be extended to period
functions by using the more general definition of Hecke eigenform given here.
A routine calculation now gives us our next result, from which the extension of the “multiplicity two” theorem (Theorem 1.5 in [1]) immediately follows.
!
with nonzero constant term. Then f = Dk−1 (F ) ∈ Sk! is a Hecke
Proposition. Let F ∈ M2−k
eigenform with the same eigenvalues as the Eisenstein series Gk .
Two remarks.
1) Forms satisfying our hypothesis always exist by Proposition 2.3 of [1].
!
2) Choose F ∈ M2−k
with constant term equal to ζ(k − 1). If fk = Dk−1 (F ) − Gk , then a simple
calculation reveals that the eigenforms fk and Gk are orthogonal with respect to the extended
Petersson inner product (introduced below).
To state our last result, we introduce Petersson’s inner product which is the Hecke equivariant
(i.e. (f , g|k T (m) = (f |k T (m) , g)) and hermitian (i.e. (f , g) = (g , f )) scalar product defined
by
Z
(f , g) :=
f (τ ) g(τ ) y k−2 dx dy
H/Γ
for cusp forms f and g. Haberland’s identity expresses Petersson’s inner product as a bilinear
form in periods and was derived by Kohnen and Zagier in [5]. In [1], the identity was formally
extended to the space Mk! , and here, using the normalized periods in (6), we are able to express
Haberland’s identity in a much more concise and beautiful way.
Theorem 2. Let f, g ∈ Mk! . Then the extended Petersson inner product (as in [1]) may be
expressed as
(f , g) =
1
3(2i)k−1
X
m<n
m6≡n (mod 2)
δ(n)δ(m)
Γ(n + 1)Γ(k − 1 − m)
rbn (f ) rbm (g)
Γ(n + 1 − m)Γ(k − 1)
where δ(x) := 1 + δ−1,x + δk−1,x and δy,x is Kronecker’s delta function.
Remark. The Γ quotient above suggests that certain summands may be undefined when there are
poles. To interpret the above formula correctly, we must consider the Γ quotient together with
the corresponding normalized periods. In this way, the number of zeros will always outnumber
the amount of poles that appear and so we interpret such summands as vanishing.
PERIODS OF EISENSTEIN SERIES AND SOME APPLICATIONS
5
Proof. In identity (1.5) of [1], periods were defined slightly differently than we did here in
identities (4) and (7). Therefore, we write the relation among the various types of periods
k−2
n+1 k − 2
(10)
rbn (f ) =
rn (f ) = i
ṙn (f )
n
n
R∞
where 0 ≤ n ≤ k − 2 and ṙn (f ) := 0 f (it)tn dt (i.e. periods of f as defined in [1]).
Haberland’s identity in (4.3) of [1] is written as
X
n
k−1
n+1−m k − 2
3·2
(f , g) =
ṙn (f ) ṙk−2−m (g)
i
n
m
0≤m<n≤k−2
m6≡n
(mod 2)
X
+ 2
ik−n
0≤n≤k−2
n≡0 (mod 2)
k−1
n+1
ag (0)
af (0)
+ ṙn (f )
ṙn (g)
k−1
k−1
!
.
Making use of (10), we rewrite the above identity as
−1
X
n
k−2
k−1
δ(m)δ(n)
i1−k rbn (f ) rbm (g)
3·2
(f , g) =
m
m
0≤m<n≤k−2
m6≡n
(mod 2)
X
+
δ(m)δ(k − 1)
0≤m≤k−2
m6≡k−1 (mod 2)
X
+
δ(−1)δ(n)
0≤n≤k−2
n6≡−1 (mod 2)
k − 1 1−k
i
rbk−1 (f ) rbm (g)
m+1
k − 1 1−k
i
rbn (f ) rb−1 (g).
n+1
Now apply the substitution m → k − 2 − m to the second sum above. Then use the period
relations, rbk−2−m (g) = rbm (g), and rewrite the binomial coefficients and fractions in terms of the
Γ function to obtain
X
Γ(n + 1)Γ(k − 1 − m)
3(2i)k−1 (f , g) =
δ(n)δ(m)
rbn (f ) rbm (g).
Γ(n
+
1
−
m)Γ(k
−
1)
−1≤m<n≤k−1
m6≡n
(mod 2)
Extending the summation over all integers m and n gives the result.
Acknowledgements
The author would like to thank Pavel Guerzhoy, Ken Ono, Don Zagier, and the referee for
several suggestions that improved this paper.
References
1. K. Bringmann, P. Guerzhoy, Z. Kent, and K. Ono, Eichler-Shimura theory for mock modular forms, preprint.
2. J. Bruinier, K. Ono, and R. Rhoades, Differential operators for harmonic weak Maass forms and the vanishing
of Hecke eigenvalues, Math. Ann. 342 (2008), 673–693.
3. P. Guerzhoy, Hecke operators for weakly holomorphic modular forms and supersingular congruences, Proc.
Amer. Math. Soc. 136 (2008), 30513059.
4. K. Haberland, Perioden von Modulformen einer Variabler and Gruppencohomologie. I, II, III, Math. Nachr.
112 (1983), 245–282, 283–295, 297–315.
6
ZACHARY A. KENT
5. W. Kohnen and D. Zagier, Modular forms with rational periods. Modular forms (Durham, 1983), 197–249,
Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984.
6. Y. Manin, Periods of parabolic forms and p-adic Hecke series, Math. USSR Sbornik 21 (1973), 371-393.
7. D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991), 449–465.
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia,
30322
E-mail address: [email protected]