On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono

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by  Michael H. Mertens

ON CLASS INVARIANTS FOR NON-HOLOMORPHIC MODULAR
FUNCTIONS AND A QUESTION OF BRUINIER AND ONO
5
1
0 MICHAEL H. MERTENS AND LARRY ROLEN
2

g
Abstract. Recently,BruinierandOnofoundanalgebraicformulaforthepartition
u
functionintermsoftracesofsingularmoduliofacertainnon-holomorphicmodular
A
function. Inthis paper weprovethatthe rationalpolynomialhavingthese singular

4 moduli as zeros is (essentially) irreducible, settling a question of Bruinier andOno.
2
TheproofusescarefulanalyticestimatestogetherwithsomerelatedworkofDewar

and Murty, as well as extensive numerical calculations of Sutherland.
]
T
N
.
h
t
a
1. Introduction
m
[ A partition of a natural number n is a non-increasing sequence of natural num-

4 bers which sum up to n. Let p(n) denote the number of partitions of n. Despite
v its elementary definition, it is computationally infeasible to compute this number di-
3
rectly for large n. A much more efficient method for computing the partition number
4
7 was offered by Euler’s recursive formula for p(n), obtained as a consequence of his
3
pentagonal number theorem [6]. Much later, Hardy and Ramanujan [11] found the
0
. asymptotic formula
1
50 p(n) 1 eπ√23n as n ,
∼ 4n√3 → ∞
1
:
v developing and employing a device which has become essential in analytic number
i
X theory, theso-called Circle Method. Byrefining their method, Rademacher [20] found
r
a
The researchof the first author leading to these results has receivedfunding from the European
Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) /
ERC Grant agreement n. 335220 - AQSER. The second author thanks the University of Cologne
andthe DFGfortheirgeneroussupportviathe UniversityofColognepostdoc grantDFGGrantD-
72133-G-403-151001011,fundedundertheInstitutionalStrategyoftheUniversityofColognewithin
the German Excellence Initiative.
1
2 MICHAEL H. MERTENS ANDLARRYROLEN
his famous exact formula for p(n),
2π ∞ Ak(n) π√24n 1
p(n) = (24n 1)34 k I32 6k − ,
− Xk=1 (cid:18) (cid:19)
where I3 is the modified Bessel function of the first kind and Ak(n) is a certain
2
Kloosterman sum.
More recently, Bruinier and Ono [3] proved that p(n) can also be computed as a
finite sum of distinguished algebraic numbers. More precisely, they showed that
1
p(n) = P(τ ),
Q
24n 1
− QX∈Qδ
where is a set of representatives of positive definite binary integral quadratic
δ
Q
forms Q(x,y) = ax2+bxy+cy2 of discriminant δ := 24n+1 satisfying 6 a, modulo
− |
the action of the group Γ (6), where we choose a set of representatives such that
0
b 1 (mod12). Moreover, τ is the unique point in the upper half-plane H satisfying
Q
≡
Q(τ ,1) = 0, and P is a certain Maass form of weight 0 defined in (2.2).
Q
In particular, Bruinier and Ono proved that P(τ ) is an algebraic number, and in
Q
fact it is known that (24n 1)P(τ ) is an algebraic integer [16]. For related work
Q
−
studying andapplying formulasforthepartitionfunctionintermsoftracesofsingular
moduli, the interested reader is also referred to [2, 8, 15]. In this context, it is natural
to define the polynomial
(1.1) H (x) := (x P(τ )) Q[x].
δ Q
− ∈
QY∈Qδ
By an elementary calculation (see Lemma 3.7 of [4]), this factors over Q as
Hδ(x) = ε(f)h(cid:16)fδ2(cid:17)H δ (ε(f)x),
f2
f>0
fY2δ
| b
where h(d) denotes the class number of discriminant d, ε(f) = 1 if f 1 (mod12)
≡ ±
and ε(f) = 1 otherwise. We also set
−
(1.2) H (x) := (x P(τ )),
δ Q
−
QY∈Pδ
b
where istheset ofprimitive formsin , i.e., thoseformsforwhichgcd(a,b,c) = 1.
δ δ
P Q
CLASS INVARIANTS FOR NON-HOLOMORPHIC MODULAR FUNCTIONS 3
Bruinier and Ono [3] asked whether H (x) is irreducible and in [4] they produced,
δ
together with Sutherland, very strong numerical evidence for the affirmative answer.
In this paper, we settle their question. b
Theorem 1.1. The polynomial H (x) is irreducible over Q. Moreover we have that
δ
Ω is the splitting field of H (x), where δ = 24n + 1 = t2d with d a fundamental
t δ
−
discriminant and Ω the ring clasbs field of the order of conductor t in K := Q(√d).
t
b
Remark. It should be possible to give a general version of Theorem 1.1 for almost
holomorphic modular functions with rational expansions at the cusps, however, the
authors have chosen to highlight this special case for the sake of explicitness, and in
particular in order to establish the irreducibility for all polynomials, as opposed to all
but finitely many in the general case.
The paper is organized as follows. In Section 2 we recall some tools required for
the proof of Theorem 1.1 such as Masser’s formula and a convenient form of Shimura
reciprocity due to Schertz. The proof itself is the subject of Section 3.
Acknowledgements
The authors are grateful to Kathrin Bringmann and Andrew Sutherland for many
useful conversations and comments which greatly improved the exposition of this
paper. They would also like to thank the anonymous referee for helpful comments.
2. Preliminaries
2.1. Masser’s formula. Throughout, τ = u+iv H(thecomplex upper half-plane)
∈
with u,v R, and let q := e2πiτ. For a function f : H C, an integer k, and a
∈ →
matrix γ = (a b) GL (R) with detγ > 0 we define the weight k slash operator by
c d ∈ 2
aτ +b
f kγ(τ) := det(γ)k2(cτ +d)−kf .
| c+tau+d
(cid:18) (cid:19)
Denote by E the normalized weight k Eisenstein series for SL (Z) with leading
k 2
coefficient 1, and let
E3(τ)
j(τ) := 1728 4
E3(τ) E2(τ)
4 − 6
4 MICHAEL H. MERTENS ANDLARRYROLEN
be the classical j-invariant. Furthermore,
η(τ) := q214 ∞ 1 qℓ
−
ℓ=1
Y(cid:0) (cid:1)
is the Dedekind eta function. We require the function
(2.1)
E (τ) 2E (2τ) 3E (3τ)+6E (6τ)
F(τ) := 2 − 2 − 2 2 = q 1 10 29q 104q2 ...,
−
2η(τ)2η(2τ)2η(3τ)2η(6τ)2 − − − −
which is a weakly holomorphic modular form of weight 2 for Γ (6). The Maass
0
−
raising operator
1 ∂ k
R := ,
k
2πi∂τ − 4πv
maps this weakly holomorphic modular form F to a non-holomorphic modular func-
tion
(2.2) P(τ) := R (F)(τ),
2
− −
which is an eigenfunction of the hyperbolic Laplacian
∂2 ∂2
∆ := v2 +
− ∂u2 ∂v2
(cid:18) (cid:19)
with eigenvalue 2.
−
As in [16], we decompose P as
P = A+B C
·
where A and B are modular functions for Γ (6) given by
0
1 ∂ 1 F(τ)E (τ)(7j(τ) 6912)
6
(2.3) A(τ) := F(τ) E (τ)F(τ)+ − ,
2
−2πi · ∂τ − 6 6E (τ)(j(τ) 1728)
4
−
F(τ)E (τ)j(τ)
6
(2.4) B(τ) := ,
E (τ)
4
and C is a non-holomorphic modular function for SL (Z), given by
2
E (τ) 3 7j(τ) 6912
4
(2.5) C(τ) := E (τ) − .
2
6E (τ)j(τ) − πv − 6j(τ)(j(τ) 1728)
6 (cid:18) (cid:19) −
This decomposition is especially useful as Masser gives a very important formula
for the singular moduli of C in Appendix I in [18]. To state this result, we recall
CLASS INVARIANTS FOR NON-HOLOMORPHIC MODULAR FUNCTIONS 5
the modular polynomial Φ (x,y) which is defined for a discriminant D < 0 by the
D
−
relation
Φ (j(τ),y) := (y j(Mτ)).
D
− −
M
Y∈V
Here, is a system of representatives of SL (Z) Γ , where Γ denotes the set of
2 D D
V \ − −
all primitive integral 2 2-matrices of determinant D (see e.g. [13], Section IV.1.6).
× −
It is a well-known fact (cf. [14], Chapter 5, Section 2) that Φ (x,y) Z[x,y]. Now
D
let Q be a quadratic form1 of discriminant D and τ the cor−respondi∈ng CM-point.
Q
Then we define numbers β (τ ) via the power series expansion
µ,ν Q
Φ (x,y) =: β (τ )(x j(τ ))µ(y j(τ ))ν.
D µ,ν Q Q Q
− − −
µ,ν
X
It is easy to see that all these numbers lie in the field Q(j(τ )) and that β (τ ) =
Q µ,ν Q
β (τ ). Masser’s formula then states that in the case that the discriminant D is not
ν,µ Q
special, i.e., not of the form D = 3d2, one has
−
2β (τ ) β (τ )
0,2 Q 1,1 Q
(2.6) C(τ ) = − .
Q
β (τ )
1,0 Q
Note in particular that for any n, δ = 24n+1 is not special. Using the formula of
−
Masser, we are able to reduce our problem of studying singular moduli for nonholo-
morphic modular functions to the study of holomorphic modular functions associated
to each discriminant, as in the following result.
Lemma 2.1. Forevery non-special discriminantD < 0, there exists a (meromorphic)
modular function M for Γ (6) such that
D 0
P(τ ) = M (τ )
Q D Q
for all quadratic forms Q of discriminant D.
Proof. Using the definition of the modular polynomial one finds explicitly that (see
also (2.9) in [4])
(2.7) β (τ ) = [y]Φ (j(τ ),y +j(τ )),
0,1 Q D Q Q
−
(2.8) β (τ ) = [y]Φ (j(τ ),y +j(τ )),
1,1 Q ′ D Q Q
−
(2.9) β (τ ) = [y2]Φ (j(τ ),y +j(τ )),
0,2 Q D Q Q
−
1From here on, the term “quadratic form” always means “positive definite integral binary qua-
dratic form” if not declared otherwise.
6 MICHAEL H. MERTENS ANDLARRYROLEN
where Φ := ∂ Φ and [yk]Φ (x,y) denotes the coefficient of yk in Φ . The
′ D ∂x D D D
right-han−d sides of−(2.7)–(2.9) c−learly also make sense if we replace the a−lgebraic
number j(τ ) by the modular function j(τ), yielding modular functions β (τ) for
Q µ,ν
SL (Z). Thus the function
2
2β (τ) β (τ)
0,2 1,1
(2.10) M (τ) := A(τ)+B(τ) −
D
β (τ)
1,0
(cid:3)
has the desired properties.
2.2. N-systems. Inthissubsection, wecitesomeresults ofSchertz [22]. Webeginby
recalling a convenient set of representatives for primitive quadratic forms introduced
in [22] for the study of class invariants known as Weber’s class invariants.
Definition 2.2. Let N N and D = t2d < 0 be a discriminant, with t N and d a
∈ ∈
fundamental discriminant. Moreover, let Q ,...,Q (Q (x,y) = a x2+b xy+c y2)
1 r j j j j
{ }
be a system of representatives of primitive quadratic forms modulo SL (Z). We call
2
the set Q ,...,Q an N-system mod t if the conditions
1 r
{ }
gcd(c ,N) = 1 and b b (modN), 1 j,ℓ r
j j ℓ
≡ ≤ ≤
are satisfied.
Remark. Schertz gave this definition (see [22], p. 329) in terms of ideal classes of
the ring class field and with switched roles of a and c. He also proved constructively
that an N-system mod t always exists (see [22], Proposition 3).
The following theorem (see [22], Theorem 4) is a key in the proof of Theorem 1.1.
Theorem 2.3 (Schertz). Let g be a modular function for Γ (N) for some N
0
∈
N whose Fourier coefficients at all cusps lie in the Nth cyclotomic field. Suppose
furthermore that g(τ) and g( 1) have rational Fourier coefficients, and let Q(x,y) =
−τ
ax2 + bxy + cy2 be a quadratic form of discriminant D = t2d, d a fundamental
discriminant, with gcd(c,N) = 1 and N a. Then we have that g(τ ) Ω unless g
Q t
| ∈
has a pole at τ .
Q
Moreover, if Q = Q ,...,Q is an N-system mod t, then
1 h
{ }
g(τ ),...,g(τ = σ(g(τ )) : σ Gal
{ Q1 Qh} { Q1 ∈ D}
where Gal denotes the Galois group of Ω /Q(√d).
D t
CLASS INVARIANTS FOR NON-HOLOMORPHIC MODULAR FUNCTIONS 7
Remark. Schertz stated the above theorem for modular functions for the group
a b
Γ0(N) := SL (Z) : b 0 (modN)
c d ∈ 2 ≡
(cid:26)(cid:18) (cid:19) (cid:27)
which is clearly isomorphic to Γ (N) via conjugation with S := (0 1). This conju-
0 1 −0
gation must be carried over to the quadratic forms as well which explains the change
of roles of the coefficients a and c compared to [22].
2.3. Poincar´e series. Inthissubsection webriefly recall some importantfactsabout
Maass-Poincar´e series. For more details, we refer to the survey in Section 8.3 of [19]
and the earlier works [7, 12].
Let M denote the usual M-Whittaker function (see e.g. [10], p. 1014) and define
ν,µ
for v > 0, k Z, and s C the function
∈ ∈
k
Ms,k(v) := v−2M−k2,s−21(v).
Using this, we construct the following Poincar´e series for Γ (N),
0
1
(2.11) P (τ) := [ (4πmv)e 2πimu] γ,
m,s,k,N s,k − k
2Γ(2s) M |
γ∈Γ∞X\Γ0(N)
where m N, τ = u + iv H, Re(s) > 1, k N, and Γ := (1 n) : n Z .
∈ ∈ ∈ − ∞ {± 0 1 ∈ }
It is not hard to check (see e.g. Proposition 2.2 in [3]) that under the Maass raising
operator, the Poincar´e series P is again mapped to a Poincar´e series
m,s,k,N
k
(2.12) R (P ) = m s+ P .
k m,s,k,N m,s,k+2,N
2
(cid:18) (cid:19)
In the special case when k < 0 and s = 1 k, the series P := P defines
− 2 m,k,N m,1−k2,k,N
a harmonic Maass form of weight k for Γ (N) whose principal part at the cusp is
0
∞
given by q m, and at all the other cusps the principal part is 0. In this situation, we
−
have the following explicit Fourier expansion at (see e.g. [19], Theorem 8.4).
∞
Proposition 2.4. For m,N N, k N, and τ H we have
∈ ∈ − ∈
(1 k)!P (τ) = (k 1)(Γ(1 k,4πmv) Γ(1 k))q m + b (ℓ,v)qℓ,
m,k,N − m,k,N
− − − − −
ℓ Z
X∈
where the coefficients b (ℓ,v) are defined as follows.
m,k,N
8 MICHAEL H. MERTENS ANDLARRYROLEN
1) If ℓ < 0, then
k 1
ℓ −2
b (ℓ,v) = 2πi2 k(k 1)Γ(1 k,4π ℓ v)
m,k,N −
− − | | m
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)K((cid:12) m,ℓ,c) 4π mℓ
(cid:12) (cid:12)− J | | ,
1 k
× c>0 c · − pc !
c 0X(modN)
≡
where
Γ(α;,x) := ∞e ttα 1dt.
− −
Zx
2) If ℓ > 0, then
bm,k,N(ℓ,v) = 2πi2−k(1 k)!ℓk−21m1−2k K(−m,ℓ,c) I1 k 4π |mℓ| .
− − c>0 c · − pc !
c 0X(modN)
≡
3) If ℓ = 0, then
K( m,0,c)
b (0,v) = (2πi)2 km1 k − .
m,k,N − − − c2 k
−
c>0
c 0X(modN)
≡
Here, I and J denote the usual I- and J-Bessel functions and K(m,ℓ,c) is the usual
s s
Kloosterman sum,
md+ℓd
K(m,ℓ,c) := exp 2πi ,
c
d (Xmodc)∗ (cid:18) (cid:18) (cid:19)(cid:19)
where d runs through the residue classes (modc) which are coprime to c and d denotes
the multiplicative inverse of d (modc).
3. Irreducibility of H (x)
δ
Nowthatwehaverecalledtherelevantknownfacts, weproceedtowardsprovingour
b
main result, Theorem 1.1. We require some information about the Fourier expansions
ofA,B,andM ,fromequations(2.3), (2.4), and(2.1)respectively, atallcusps, which
D
we provide in the following lemma.
Lemma 3.1. (1) The modular functions A and B have Fourier expansions with
rational Fourier coefficients at all cusps of Γ (6).
0
CLASS INVARIANTS FOR NON-HOLOMORPHIC MODULAR FUNCTIONS 9
(2) If D < 0 is a non-special discriminant, then the Fourier coefficients of M at
D
all cusps lie in the field Q(ζ6), where ζ6 := e26πi, and the Fourier coefficients
of M (τ) and M ( 1) are rational.
D D −τ
Proof. (1) The denominator of the function F lies in the one-dimensional space
S (Γ (6)), and hence is an eigenfunction of all Atkin-Lehner operators, the eigenvalue
4 0
being always 1. The numerator, which we denote by F , is a weight two modular form
1
for Γ (6) and is an eigenfunction of all Atkin-Lehner involutions as well. This can
0
be seen by a short and direct calculation: For the convenience of the reader, we give
some details. Namely, note that the Atkin-Lehner operators obtained by slashing
with one of the following matrices
0 1 3 1 2 1
W := − , W := , W := − ,
6 6 0 3 6 3 2 6 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
map the cusp to 0, 1, 1, (respectively). One finds by a direct calculation of the
∞ 2 3
Smith normal form of the matrices W , d = 2,3,6, that
d
F W = F , F W = F , and F W = F .
1 2 6 1 1 2 3 1 1 2 2 1
| | − | −
Therefore, F is an eigenfunction of all Atkin-Lehner involutions of level 6.
Using this,we can directly calculate the Fourier expansion of F at all cusps (cor-
recting a typo in (3.2) of [5]). To this end we choose the following 12 right coset
representatives of SL (Z)/Γ (6) (where T := (1 1)),
2 0 0 1
1 0
γ := ,
0 1
∞
(cid:18) (cid:19)
1 0
γ13,r := 3 1 Tr for r = 0,1,
(3.1) (cid:18) (cid:19)
1 1
γ12,s := 2 3 Ts for s = 0,1,2,
(cid:18) (cid:19)
0 1
γ := − Tt for t = 0,1,2,3,4,5.
0,t 1 0
(cid:18) (cid:19)
Using the relation W = γ V , where V := (d 0), we find that
6 0,0 6 d 0 1
1
F(τ) = F W (τ) = F γ (6τ).
2 6 2 0,0
| 6 |
10 MICHAEL H. MERTENS ANDLARRYROLEN
Thus we find the following Fourier expansions of F at the cusp 0,
(3.2) F γ (τ) = 6 ∞ a ζtmqm6 ,
|2 0,t m 6
m= 1
X−
where F(τ) =: ∞ a qm. Similarly, we have
m
m= 1
−
P
1 0
W = γ V γ 1, W = γ 1V T 1γ 1 ,
2 0,0 2 −1,0 3 0−,0 3 − −1,0 6 1
2 3
(cid:18) (cid:19)
which yields
τ τ
F(τ) = −F|2W2(τ) = −6F|2V2γ−21,10 6 = −3F|2γ21,0 3 .
(cid:16) (cid:17) (cid:16) (cid:17)
Hence, at 1, we have the Fourier expansions
2
(3.3) F|2γ21,s(τ) = 3 ∞ amζ63+2msqm3 .
m= 1
X−
A similar calculation yields the Fourier expansions at 1:
3
(3.4) F|2γ31,r(τ) = 2 ∞ am(−1)mrqm2 .
m= 1
X−
Thus, it is clear that B has a Fourier expansion with rational coefficients at all cusps,
since B is a weight 2 meromorphic modular form for SL (Z) which has rational (in
F 2
fact integral) Fourier coefficients at .
∞
The Serre derivative of F, which is given by 1 ∂ F(τ) + 1E (τ)F(τ) =: A (τ)
2πi∂τ 6 2 1
has a rational leading term at , namely 7, and integer coefficients otherwise (note
∞ 6
that for n > 0 the n-th coefficient of E is divisible by 24). The same is true for the
2
function
F(τ)E (τ)(7j(τ) 6912)
6
A (τ) := −
2
6E (τ)(j(τ) 1728)
4
−
(again, note that all but the zeroth Fourier coefficient of E are divisible by 504 =
6
6 84). Therefore, A = A + A , has integer coefficients at . Since the Serre
1 2
· − ∞
derivative commutes with the action of SL (Z) and A is the product of F and a level
2 2
1 form, the same argument as for B yields the claim.