Identities between Hecke Eigenforms

0.19 MB·

by  Dianbin Bao

Identities between Hecke Eigenforms
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D. Bao
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January 13, 2017
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T Abstract
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h. In this paper, we study solutions to h = af2 + bfg + g2, where
t f,g,h are Hecke newforms with respect to Γ (N) of weight k > 2
a 1
m and a,b = 0. We show that the number of solutions is finite for
6
[ all N. Assuming Maeda’s conjecture, we prove that the Petersson

inner product f2,g is nonzero, where f and g are any nonzero cusp
1 h i
v eigenformsforSL2(Z)ofweight k and2k, respectively. Asacorollary,
9 we obtain that, assuming Maeda’s conjecture, identities between cusp
8 eigenforms for SL (Z) of the form X2 + n α Y = 0 all are forced
1 2 i=1 i i
3 by dimension considerations. We also give a proof using polynomial
0 identities between eigenforms that the j-fPunction is algebraic on zeros
.
1 of Eisenstein series of weight 12k.
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:
v 1 Introduction
i
X
r
a
Fourier coefficients of Hecke eigenforms often encode important arithmetic
information. Given an identity between eigenforms, one obtains nontrivial
relations between their Fourier coefficients and may in further obtain solu-
tions to certain related problems in number theory. For instance, let τ(n) be
the nth Fourier coefficient of the weight 12 cusp form ∆ for SL (Z) given
2
by
∞ ∞
∆ = q (1 qn)24 = τ(n)qn
−
n=1 n=1
Y X
1
and define the weight 11 divisor sum function σ (n) = d11. Then the
11 dn
Ramanujan congruence |
P
τ(n) σ (n) mod 691
11
≡
can be deduced easily from the identity
1008 756
E E2 = × ∆,
12 − 6 691
where E (respectively E ) is the Eisenstein series of weight 6 (respectively
6 12
12) for SL (Z).
2
SpecificpolynomialidentitiesbetweenHeckeeigenformshavebeenstudiedby
many authors. Duke [7] and Ghate [9] independently investigated identities
oftheformh = fg between eigenformswithrespecttothefullmodulargroup
Γ = SL (Z) and proved that there are only 16 such identities. Emmons [8]
2
extended the search to Γ (p) and found 8 new cases. Later Johnson [11]
0
studied the above equation for Hecke eigenforms with respect to Γ (N) and
1
obtained a complete list of 61 eigenform product identities. J. Beyerl and
K. James and H. Xue [2] studied the problem of when an eigenform for
SL (Z) is divisible by another eigenform and proved that this can only occur
2
in very special cases. Recently, Richey and Shutty [14] studied polynomial
identities and showed that for a fixed polynomial (excluding trivial ones),
there are only finitely many decompositions of normalized Hecke eigenforms
for SL (Z) described by a given polynomial.
2
Since the product of two eigenforms is rarely an eigenform, in this paper
we loosen the constraint and study solutions to the equation h = P(f,g),
where h,f,g are Hecke eigenforms and P is a general degree two polynomial.
The ring of modular forms is graded by weight, therefore P is necessarily
homogeneous, i.e., P(f,g) = af2 + bfg + cg2. With proper normalization,
we can assume c = 1.
The first main result of this paper is the following:
Theorem 1.1. For all N Z and a,b C , there are at most finitely
+ ×
∈ ∈
triples (f,g,h) of newforms with respect to Γ (N) of weight k > 2 satisfying
1
the equation
h = af2 +bfg +g2. (1)
2
Theorem 1.1 is obtained by estimating the Fourier coefficients of cusp eigen-
forms and its proof is presented in Section 3. In the same section, as further
motivation for studying identities between eigenforms, we also give an ele-
mentary proof via identities that:
Theorem 1.2. Let E (z) be the Eisenstein series of weight 12k, and
12k
A := ρ F E (ρ) = 0 for some k Z
12k +
{ ∈ | ∈ }
be the set of zeros for all Eisenstein series of weight 12k with k 1 in the
≥
fundamental domain F for SL (Z). If j is the modular j-function and ρ A,
2
∈
then j(ρ) is an algebraic number.
For a stronger result, which says that j(ρ) is algebraic for any zero ρ of a
meromorphic modular form f = ∞n=hanqn for SL2(Z) with an(h) = 1 for
which all coefficients lie in a number field, see Corollary 2 of the paper by
P
Bruinier, Kohnen and Ono [3].
Fouriercoefficients ofa normalized Hecke eigenform areall algebraicintegers.
AsforGaloissymmetry between Fouriercoefficients ofnormalizedeigenforms
in the space of cusp forms for SL (Z) of weight k 12 , Maeda (see [10]
k 2
S ≥
Conjecture 1.2 ) conjectured that:
Conjecture 1.3. The Hecke algebra over Q of is simple (that is, a single
k
S
number field) whose Galois closure over Q has Galois group isomorphic to
the symmetric group S , where n = dim .
n k
S
Maeda’sconjecturecanbeusedtostudypolynomialidentities between Hecke
eigenforms. See J.B. Conrey and D.W. Farmer [4] and J. Beyerl and K.
James and H. Xue [2] for some previous work. In this paper, we prove the
following:
Theorem 1.4. Assuming Maeda’s conjecture for and , if f ,
k 2k k
S S ∈ S
g are any nonzero cusp eigenforms, then the Petersson inner product
2k
∈ S
f2,g is nonzero.
h i
Theorem 1.4 says that, assuming Maeda’s conjecture, the square of a cusp
eigenform for SL (Z) is ”unbiased”, i.e., it does not lie in the subspace
2
spanned by a proper subset of an eigenbasis. The proof of Theorem 1.4
is presented in Section 4.
3
2 Preliminaries and Convention
A standard reference for this section is [6]. Let f be a holomorphic function
on the upper half plane = z C Im(z) > 0 . Let γ = (a b) SL (Z).
H { ∈ | } c d ∈ 2
Then define the slash operator [γ] of weight k as
k
f [γ](z) := (cz +d) kf(γz).
k −
|
If Γ is a subgroup of SL (Z), we say that f is a modular form of weight k
2
with respect to Γ if f [γ](z) = f(z) for all z and all γ Γ. Define
k
| ∈ H ∈
a b a b 1
Γ (N) := SL (Z) ∗ mod N ,
1 c d ∈ 2 c d ≡ 0 1
(cid:26)(cid:18) (cid:19) (cid:12)(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
(cid:12)
(cid:12)
(cid:12)
a b
Γ (N) := SL (Z) c 0 mod N .
0 c d ∈ 2 ≡
(cid:26)(cid:18) (cid:19) (cid:12) (cid:27)
(cid:12)
(cid:12)
Denote by (Γ (N)) the vector space o(cid:12)f weight k modular forms with
k 1
M
respect to Γ (N).
1
Let χ be a Dirichlet character mod N, i.e., a homomorphism χ : (Z/NZ)
×
→
C . By convention one extends the definition of χ to Z by defining χ(m) = 0
×
for gcd(m,N) > 1. In particular the trivial character modulo N extends to
the function
1 if gcd(n,N) = 1,
1 (n) =
N
(0 if gcd(n,N) > 1.
Take n (Z/NZ) , and define the diamond operator
×
∈
n : (Γ (N)) (Γ (N))
k 1 k 1
h i M → M
as n f := f [γ] for any γ = (a b) Γ (N) with d n mod N. Define the
h i |k c d ∈ 0 ≡
χ eigenspace
(N,χ) := f (Γ (N)) n f = χ(n)f, n (Z/NZ) .
k k 1 ×
M { ∈ M |h i ∀ ∈ }
4
Then one has the following decomposion:
(Γ (N)) = (N,χ),
k 1 k
M M
χ
M
where χ runs through all Dirichlet characters mod N such that χ( 1) =
−
( 1)k. See Section 5.2 of [6] for details.
−
Let f (Γ (N)), since (1 1) Γ (N), one has a Fourier expansion:
∈ Mk 1 0 1 ∈ 1
∞
f(z) = a (f)qn, q = e2πiz.
n
n=0
X
If a = 0 in the Fourier expansion of f [α] for all α SL (Z), then f is
0 k 2
| ∈
called a cusp form. For a modular form f (Γ (N)) the Petersson inner
k 1
∈ M
product of f with g (Γ (N)) is defined by the formula
k 1
∈ S
f,g = f(z)g(z)yk 2dxdy,
−
h i
Z /Γ1(N)
H
wherez = x+iy and /Γ (N)isthequotientspace. See[6]Section5.4.
1
H
We define the space (Γ (N)) of weight k Eisenstein series with respect
k 1
E
to Γ (N) as the orthogonal complement of (Γ (N)) with respect to the
1 k 1
S
Petersson inner product, i.e., one has the following orthogonal decomposi-
tion:
(Γ (N)) = (Γ (N)) (Γ (N)).
k 1 k 1 k 1
M S ⊕E
We also have decomposition of the cusp space and the space of Eisenstein
series in terms of χ eigenspaces:
(Γ (N)) = (N,χ),
k 1 k
S S
χ
M
(Γ (N)) = (N,χ).
k 1 k
E E
χ
M
5
See Section 5.11 of [6].
For two Dirichlet characters ψ modulo u and ϕ modulo v such that uv N
|
and (ψϕ)( 1) = ( 1)k define
− −
∞
Eψ,ϕ(z) = δ(ψ)L(1 k,ϕ)+2 σψ,ϕ(n)qn, (2)
k − k 1
−
n=1
X
where δ(ψ) is 1 if ψ = 1 and 0 otherwise,
1
B
k,ϕ
L(1 k,ϕ) =
− − k
is the special value of the Dirichlet L function at 1 k, B is the kth
k,ϕ
−
generalized Bernoulli number defined by the equality
tk N−1 ϕ(a)teat
B := ,
k,ϕk! eNt 1
k 0 a=0 −
X≥ X
and the generalized power sum in the Fourier coefficient is
σψ,ϕ(n) = ψ(n/m)ϕ(m)mk 1.
k 1 −
−
mn
mX>|0
See page 129 in [6].
Let A be the set of triples of (ψ,ϕ,t) such that ψ and ϕ are primitive
N,k
Dirichlet characters modulo u and v with (ψϕ)( 1) = ( 1)k and t Z
+
− − ∈
such that tuv N. For any triple (ψ,ϕ,t) A , define
N,k
| ∈
Eψ,ϕ,t(z) = Eψ,ϕ(tz). (3)
k k
Let N Z and let k 3. The set
+
∈ ≥
Eψ,ϕ,t : (ψ,ϕ,t) A
{ k ∈ N,k}
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gives a basis for (Γ (N)) and the set
k 1
E
Eψ,ϕ,t : (ψ,ϕ,t) A ,ψϕ = χ
{ k ∈ N,k }
gives a basis for (N,χ) (see Theorem 4.5.2 in [6]).
k
E
For k = 2 and k = 1 an explicit basis can also be obtained but is more
technical. Therefore we assume k 3 when dealing with Eisenstein series.
≥
For details about the remaining two cases see Chapter 4 in [6].
Now we define Hecke operators. For a reference see page 305 in [12] . Define
∆ (N) to be the set
m
a b
γ = M (Z) c 0 mod N,detγ = m .
c d ∈ 2 | ≡
(cid:26) (cid:18) (cid:19) (cid:27)
Then ∆ (N) is invariant under right multiplication by elements of Γ (N).
m 0
One checks that the set
a b
S = M (Z) a,d > 0,ad = m,b = 0,...,d 1
0 d ∈ 2 | −
(cid:26)(cid:18) (cid:19) (cid:27)
forms a complete system of representatives for the action of Γ (N). One
0
defines the Hecke operator T End( (N,χ)) as:
m k
∈ M
f f T = mk/2 1 χ(a )f σ,
k m − σ k
7→ | |
σ
X
where σ = aσ bσ S and gcd(m,N) = 1.
cσ dσ ∈
(cid:0) (cid:1)
One can compute the action of T explicitly in terms of the Fourier expan-
m
sion:
∞
f|kTm = a0 χ(m1)m1k−1 + χ(m1)m1k−1amn/m21qn.
mX1|m Xn=1m1X|(m,n)
The multiplication rule for weight k operators T is as follows:
m
7
TmTn = χ(m1)m1k−1Tmn/m21. (4)
m1X|(m,n)
An eigenform f (N,χ) is defined as the simultaneous eigenfunction of
k
∈ M
all T with gcd(m,N) = 1.
m
If f (N,χ) is an eigenform for the Hecke operators with character χ,
k
∈ M
i.e., if
f T = λ (m)f (gcd(m,N) = 1), (5)
k m f
|
then equation (4) implies that
λ (m)λ (n) = χ(m )mk 1λ (mn/m2).
f f 1 1− f 1
m1X|(m,n)
Comparing the Fourier coefficients in equation (5), one sees that
a χ(m )mk 1 = λ (m)a , (6)
0 1 1− f 0
mX1|m
χ(m1)m1k−1amn/m21 = λf(m)an.
m1X|(m,n)
For n = 1 one then has
a = λ (m)a . (7)
m f 1
Therefore if a = 0 and we normalize f such that a = 1, then a =
1 1 m
6
λ (f).
m
Eisenstein series are eigenforms. Let Eψ,ϕ,t be the Eisenstein series defined
k
by equation (3). Then
T Eψ,ϕ,t = (ψ(p)+ϕ(p)pk 1)Eψ,ϕ,t
p k − k
8
if uv = N or p ∤ N. See Proposition 5.2.3 in [6].
Let M N and d N/M. Let α = (d 0). Then for any f : C
| | d 0 1 H →
(f [α ])(z) = dk 1f(dz)
k d −
|
defines a linear map [α ] taking (Γ (M)) to (Γ (N)).
d k 1 k 1
S S
Definition 2.1. Let d be a divisor of N and let i be the map
d
i : ( (Γ (Nd 1)))2 (Γ (N))
d k 1 − k 1
S → S
given by
(f,g) f +g [α ].
k d
7→ |
The subspace of oldforms of level N is defined by
(Γ (N)) := i (( (Γ (Np 1)))2)
k 1 p k 1 −
S S
p|N
pXprime
and the subspace of newforms at level N is the orthogonal complement with
respect to the Petersson inner product,
new old
(Γ (N)) = ( (Γ (N)) ) .
k 1 k 1 ⊥
S S
See Section 5.6 in [6].
Let ι be the map (ι f)(z) = f(dz). The main lemma in the theory of
d d
newforms due to Atkin and Lehner [1] is the following:
Theorem 2.2. (Thm. 5.7.1 in [6]) If f S (Γ (N)) has Fourier expansion
k 1
∈
f(z) = a (f)qn with a (f) = 0 whenever gcd(n,N) = 1, then f takes the
n n
form f = ι f with each f (Γ (N/p)).
P pN p p p ∈ Sk 1
|
DefinitionP2.3. A newform is a normalized eigenform in (Γ (N))new.
k 1
S
By Theorem 2.2, if f (N,χ) is a Hecke eigenform with a (f) = 0, then
k 1
∈ M
f is not a newform.
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3 Identities between Hecke eigenforms
In this section we study solutions to equation (1) with f,g,h being newforms
with respect to Γ (N). The first observation is that if a (0) = a (0) = 0
1 f g
then a (1) = 0, which implies h = 0 by equation (7), hence without loss
h
of generality we can assume that a (0) = 0. We normalize g such that
g
6
a (0) = 1. In the following we talk about two cases according to whether or
g
not a (0) = 0.
f
Lemma 3.1. Assume that g,f,h satisfies equation (1) with f and g linearly
independent over C, and let χ ,χ ,χ be the Dirichlet characters associated
g f h
with g,f,h respectively. Then χ = χ2 = χ2 = χ χ .
h g f f g
Proof. Take the diamond operator d and act on the identity h = af2 +
h i
bfg +g2 to obtain
χ (d)h = aχ2(d)f2 +bχ (d)χ (d)fg+χ2(d)g2.
h f f g g
We then substitute (1) for h to obtain
a(χ (d) χ2(d))f2+b(χ (d) χ (d)χ (d))fg+(χ (d) χ2(d))g2 = 0. (8)
h − f h − g f h − g
Note that since f and g are linearly independent over C, so are f2,fg and
g2, which one can easily prove by considering the Wronksian. Then equation
(8) implies that
a(χ (d) χ2(d)) = b(χ (d) χ (d)χ (d)) = χ (d) χ2(d) = 0.
h − f h − g f h − g
Since a,b = 0, one finds
6
χ (d) = χ2(d) = χ (d)χ (d) = χ2(d).
h f f g g
Since d is arbitrary we are done.
Proposition 3.2. Suppose that the triple of newforms (f,g,h) is a solution
to (1), a (0) = 0 and a (0) = 0. Then f = g.
f g
6 6
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