Table Of ContentARITHMETIC PROPERTIES FOR APE´RY-LIKE NUMBERS
DERMOT MCCARTHY,ROBERTOSBURNAND BRUNDABANSAHU
9
0
0 Abstract. It is known that the numbers which occur in Ap´ery’s proof of the irrationality
2
of ζ(2) have many interesting congruences properties while the associated generating function
n satisfies a second order differential equation. Weprovecongruences for numberswhich arise in
u Beukers’ and Zagier’s studyof integral solutions of Ap´ery-likedifferential equations.
J
8
1
1. Introduction
]
T
In the course of his work on proving the irrationality of ζ(2), Ap´ery introduced, for a positive
N
integer n, the following sequence of numbers [3], [25]
.
h
t n n+j n 2
a
B(n):= .
m j j
j=0(cid:18) (cid:19)(cid:18) (cid:19)
[ X
Several authors [4], [10], [11] have subsequently studied many interesting congruence properties
1 for B(n) and its generalizations. In [23], the authors were able to relate the B(n)’s to the p-th
v
Fourier coefficient of a modular form. If we define
3
1
∞
4
3 η6(4z) := a(n)qn
6. nX=1
0 where
9 ∞
0 η(z) := q214 (1−qn)
:
v nY=1
i is the Dedekind eta function, q := e2πiz and z ∈ H, then Beukers and Steinstra proved using the
X
formal Brauer group of some elliptic K3-surfaces that for all odd primes p and any nonnegative
r
a integers m, r with m odd, we have
mpr −1 mpr−1−1 p−1 mpr−2−1
(1) B −a(p)B +(−1) 2 p2B ≡ 0 (mod pr).
2 2 2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
They also conjectured the congruence
p−1
(2) B ≡ a(p) (mod p2)
2
(cid:16) (cid:17)
which does not follow from (1) and has been proven in [1] and [26]. Congruence (2) is but one
exampleofageneralphenomenacalled Supercongruences. Thistermappearedin[4]andwasthe
Date: June 18, 2009.
2000 Mathematics Subject Classification. Primary: 11A07; Secondary: 11F11.
1
2 DERMOTMCCARTHY,ROBERTOSBURNANDBRUNDABANSAHU
subject of the Ph.D. thesis of Coster [9]. It refers to the fact that congruences of this type are
stronger than the ones suggested by formal group theory. Other examples of supercongruences
have been observed in the context of number theory [13], [16], [17], [19], [30], mathematical
physics [14], [15], and algebraic geometry [7], [21]. Currently, there is no systematic explanation
for such congruences. Perhaps, as mentioned in [23], they are related to formal Chow groups
[22].
It is known that the B(n)’s satisfy the recurrence relation
(n+1)2B(n+1) = (11n2+11n+3)B(n)+n2B(n−1)
for n ≥ 1. This implies that the generating function
∞
B(t)= B(n)tn
n=0
X
satisfies the differential equation
LB(t) = 0
where
d2 d
L = t(t2+11t−1) +(3t2 +22t−1) +t−3.
dt2 dt
In [6], Beukers studies the differential equation
(3) ((t3+at2+bt)F′(t))′ +(t−λ)F(t) = 0
where a,b and λ are rational parameters and asks for which values of these parameters this
equation has a solution in Z[[t]]. This equation has a unique solution which is regular at the
origin with F(0) = 0 given by
∞
F(t) = u(n)tn
n=0
X
with u(0) =1 and satisfies the recurrence relation
b(n+1)2u(n+1)+(an2+an−λ)u(n)+n2u(n−1) = 0
where n ≥ 1. In [29], Zagier describes a search over a suitably chosen domain of 100 million
triples (a,b,λ). He finds 36 triples which yield an integral solution to (3). Of these, he classifies
six as “sporadic” cases and conjectures that, up to normalization, these are the only cases with
a 6= 0, a2 6= 4b where (3) has an integral solution. All six cases (which include B(n)) have a
binomial sum representation and a geometric origin (see [2], [29]).
The purpose of this brief note is to prove congruences, akin to (1) and (2), in three of the
“sporadic” cases which can be expressed in terms of binomial sums and have a parametrization
in terms of modular functions. The remaining cases can be done similarly. Let
n 3
n
A(n) := ,
k
k=0(cid:18) (cid:19)
X
APE´RY-LIKE NUMBERS 3
n 2
n 2k
C(n):= ,
k k
k=0(cid:18) (cid:19) (cid:18) (cid:19)
X
and
⌊n2⌋ n 2k 2
E(n) := 4n−2k .
2k k
k=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
Our main result is the following.
Theorem 1.1. Let
1 η(z)9 8η(2z)9 ∞
(4) − − := γ(n)qn
9η(3z)3 9η(6z)3
n=0
X
and
1 η(2z)30 η(4z)4η(2z)6 ∞
(5) − +4 := λ(n)qn.
4η(z)12η(4z)12 η(z)4
n=0
X
Then for primes p ≥ 5 and integers m, r ≥ 1, we have
(6) A(mpr)−γ(p)A(mpr−1)+ −3 p3A(mpr−2) ≡ 0 (mod pr),
p
(cid:0) (cid:1)
(7) C(mpr)−γ(p)C(mpr−1)+ −3 p3C(mpr−2) ≡ 0 (mod pr),
p
and (cid:0) (cid:1)
(8) E(mpr)−λ(p)E(mpr−1)+ −4 p3E(mpr−2) ≡ 0 (mod pr)
p
where · is the Legendre symbol modulo p. (cid:0) (cid:1)
p
Usin(cid:0)g (cid:1)Theorem 1.1, we deduce the following.
Corollary 1.2. Let p ≥ 5 be a prime. We have
(9) A(p)≡ 2γ(p) (mod p2),
(10) C(p)≡ 3γ(p) (mod p2),
and
(11) E(p) ≡ 4λ(p) (mod p2).
4 DERMOTMCCARTHY,ROBERTOSBURNANDBRUNDABANSAHU
The proof for each of the congruences in Corollary 1.2 is simpler than that of (2) as it
only requires a short combinatorial argument and the fact that γ(p) and λ(p) are coefficients of
Hecke eigenforms. Toprove (2), oneuses Greene’s notion of Gaussian hypergeometric series (see
Chapter 11 of [20]). Finally, there exist morecongruences for the numbers A(n) and C(n)of the
form (1) which involve the p-th Fourier coefficients of η2(8z)η(4z)η(2z)η2(z) and η3(6z)η3(2z)
respectively; see [23]. For supercongruences concerning these modular forms, see [18].
The paper is organized as follows. In Section 2 we recall some properties of integral weight
modularformsandprove acongruenceforan alternating harmonicsum. Theproofsof Theorem
1.1 and Corollary 1.2 are given in Section 3.
2. Preliminaries
We firstbrieflyrecall somebasic facts concerning integer weight modularforms(see [20]). Let
M (Γ (N),χ) be the space of modular forms of weight k on Γ (N) with character χ. For each
k 0 0
∞
prime p we have the Hecke operator T whose action on f(z) = a(n)qn ∈ M (Γ (N),χ) is
p,χ k 0
n=0
X
given by
∞
f(z)| T = a(np)+χ(p)pk−1a n qn,
p,χ p
nX=0(cid:16) (cid:0) (cid:1)(cid:17)
where a n = 0 if p ∤ n. If f(z) is a normalized eigenform (so a(1) = 1) for all of the T such
p p,χ
that a(0) 6= 0, then
(cid:0) (cid:1)
(12) a(p) = 1+χ(p)pk−1.
A function of the form
f(z) = ηrδ(δz)
δ|N
Y
where N ≥ 1 and r ∈ Z is called an eta-quotient. If f(z) is an eta-quotient with
δ
1
k := r ∈ Z,
δ
2
δ|N
X
δr ≡ 0 (mod 24),
δ
δ|N
X
r
δ
N ≡ 0 (mod 24),
δ
δ|N
X
and f(z) is holomorphic at the cusps of Γ (N), then f(z) ∈ M (Γ (N),χ) where
0 k 0
(−1)ks
χ(d) := , s := δrδ.
d
(cid:16) (cid:17) Yδ|N
Since η(z) is analytic and does not vanish on H, f(z) is holomorphic at the cusps of Γ (N)
0
if the associated orders of vanishing are nonnegative. If c and d are positive integers with
gcd(c,d) = 1 and d| N, then the order of vanishing of f(z) at the cusp c is
d
APE´RY-LIKE NUMBERS 5
N gcd(d,δ)2r
δ
.
24dgcd(d, N) δ
d δ|N
X
Wealsoneedafolkloreresult. Ift(z)isamodularfunctionwithrespecttoadiscretesubgroup
of SL (R), commensurable with SL (Z), and F is modular form of weight k with respect to
2 2
the same group, then F as a function of t satisfies a linear differential equation of order k+1
with coefficients algebraic in t. This result appeared in [24] and recently Yang [28] has given an
elegant proof whereby the coefficients in the differential equation can be explicitly computed.
For k = 1, this equation is of the form
D2F +p (t)D F +p (t)F = 0
t 1 t 2
where
d d
D := t , D := q ,
t q
dt dq
D t D F
q q
G := , G := ,
1 2
t F
and
D G −2G G /k D G −G2/k
p (t):= q 1 1 2 , p (t) := − q 2 2 .
1 G2 2 G2
1 1
We now recall the following result, due to Verrill [27], which generalizes Theorem 4 in [5].
Theorem 2.1. Let t(z) be a modular function for Γ (N), f(z) ∈ M (Γ (N),χ), and L ∈
0 k 0
M (Γ (N),χ) be a Hecke eigenform. Write
k+2 0
∞ ∞
f(t)= c(n)tn, L(q)= γ(n)qn.
n=1 n=1
X X
If for some integers a , d|M for some integer M, we have the following equality
d
qdt
f(q) dq = a L(qd),
d
t
d|M
X
then for primes p 6 |NM, and integers m, r ≥ 1, we have
c(mpr)−γ(p)c(mpr−1)+χ(p)pk+1c(mpr−2) ≡ 0 (mod pr).
Finally, we require a congruence concerning an alternating harmonic sum. Recall that for
n ∈ N, the harmonic sum is defined by
n
1
H := .
n
k
k=1
X
Proposition 2.2. Let p be an odd prime. Then
p−1 p−1
2 1 −1/2 2 2 (−1)k
≡ −4 (mod p).
k k k
k=1 (cid:18) (cid:19) k=1
X X
6 DERMOTMCCARTHY,ROBERTOSBURNANDBRUNDABANSAHU
Proof. Noting that u+k = (−1)k −1−u , we get
k k
(cid:0) (cid:1) (cid:0) (cid:1)
p−1 +k p−1 −1 − p −1 + p −1 2
(13) 2 2 = (−1)k 2 2 2 2 ≡ (−1)k 2 (mod p).
k k k k k
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
By (13) and the fact that
p−1
p−1 (−1)k 2 (−1)k
Hp−1 = Hp−1+ ≡ 2 (mod p),
2 k k
k=1 k=1
X X
the result follows upon taking n = p−1 in the identity
2
n (−1)k n+k n
= −2H
n
k k k
k=1 (cid:18) (cid:19)(cid:18) (cid:19)
X
and reducing modulo p.
(cid:3)
3. Proofs of Theorem 1.1 and Corollary 1.2
Proof of Theorem 1.1. We consider the modular function for Γ (6)
0
η(z)3η(6z)9
t(z) =
η(2z)3η(3z)9
and claim that the equality
∞ η(2z)η(3z)6
(14) f(t):= A(n)tn =
η(z)2η(6z)3
n=0
X
holds. In order to see this, note that (as A(n) is #5 on Zagier’s list) f(t)satisfies the differential
equation
7 1 7 1 1
(15) t3+ t2− t f′′(t)+ 3t2+ t− f′(t)+ t+ f(t)= 0.
8 8 4 8 4
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
η(2z)η(3z)6
Let F = . The function field of modular functions on Γ (6) is generated by
η(z)2η(6z)3 0
η(z)4η(6z)8
g =
η(2z)8η(3z)4
and the values at the cusps are given by
g(i∞) = 0, g(0) = 1/9, g(1/2) = ∞, g(1/3) = 1.
The modular functions t and g satisfy the relation
g
(16) t = .
1−g
APE´RY-LIKE NUMBERS 7
Let G(z) be the the Eisenstein series of weight 2 on SL (Z). Then the Eisenstein series of
2
weight 2 associated with the cusps of Γ (6) are given by
0
1
P = (G(z)−4G(2z)−9G(3z)+36G(6z)),
∞
24
3
P = (G(z)−G(2z)−G(3z)+6G(6z)),
0
2
3
P = − (G(z)−4G(2z)−G(3z)+4G(6z)),
1/2
8
1
P = − (G(z)−G(2z)−9G(3z)+9G(6z)).
1/3
6
The value of the Eisenstein series P at the cusp a is 1 and at other cusps is 0. We find that
a
D t
q
G := = P −1/2P
1 ∞ 1/3
t
and
D F
q
G := = 1/2P .
2 F 1/3
We now let
A = G(z)−2G(2z), A = G(z)−3G(3z), A = G(z)−6G(6z).
2 3 6
Comparing the first few terms of D G −2G G with those of A2, A2, A2, A A and A A ,
q 1 1 2 2 3 6 2 3 2 6
we have
1
(17) D G −2G G = (−29A2+51A2 +3A2+20A A −58A A )
q 1 1 2 2 3 6 2 3 2 6
384
and
1
(18) D G −G2 = (−2A2+3A2 −2A A ).
q 2 2 2 3 2 6
48
We now express (D G −2G G )/G2 and (D G −G2)/G2 in terms of t. Our first step is to
q 1 1 2 1 q 2 2 1
find expressions for (D G −2G G )/P2, G2/P2 and (D G −G2)/P2 where
q 1 1 2 1 q 2 2
1 1
P := P − P
1/2 1/3
3 2
is a modular form of weight 2 with zeros at ∞ and 0 and no zeros elsewhere. To express G /P
1
in term of g, we note that G /P is of the form
1
a +a g+a g2
0 1 2
b +b g+b g2
0 1 2
where b +b g+b g2 = (g−g(i∞))(g−g(0)) = g(g−1/9). We observe that G /P starts from
0 1 2 1
q−1 and
G (0) = 0, G (1/2) = 0, G (∞)= 1, G (1/3) = 1/2.
1 1 1 1
8 DERMOTMCCARTHY,ROBERTOSBURNANDBRUNDABANSAHU
This implies that G /P = 1/g. Similarly, we have
1
A −1−18g+27g2
2
= ,
P g(1−9g)
A −2−12g+18g2
3
= ,
P g(1−9g)
A −5+6g−9g2
6
= .
P g(1−9g)
Substituting these expressions into (17) and (18), we have
D G −2G G g(−7+54g+81g2)
q 1 1 2
=
G2 1−18g+81g2
1
and
D G −G2 −2g+12g2 +54g3
q 2 2 = .
G2 1−18g+81g2
1
By (16),
D G −2G G −7t+40t2+128t3
q 1 1 2
p (t) := =
1 G2 1−15t+48t2 +64t3
1
and
D G −G2 −2t+8t2+64t3
p (t):= q 2 2 = .
2 G2 1−15t+48t2 +64t3
1
By the folklore result and some simplification, F as a function of t satisfies the differential
equation
F′′+q (t)F′+q (t)F = 0
1 2
where
p (t)+1 1−22t+88t2 +192t3
1
q (t) := =
1 t t(1−15t+48t2 +64t3)
and
−2+8t+64t2
q (t) := .
2 t(1−15t+48t2+64t3)
This is equivalent to (15) and thus by uniqueness, (14) follows. We now take L(q) to be
equal to (4) as it is a Hecke eigenform in M (Γ (6), −3 ) (one need only use the results on
3 0 d
eta-quotients in Section 2 and check a finite number of primes [12]) and note that
(cid:0) (cid:1)
qdt
dq
f(q) = L(q).
t
An application of Theorem 2.1 then yields (6). If we now take the modular function for Γ (6)
0
APE´RY-LIKE NUMBERS 9
η(z)4η(6z)8
t(z) =g = ,
η(2z)8η(3z)4
then we claim that
∞ η(2z)6η(3z)
(19) f(t):= C(n)tn = .
η(z)3η(6z)2
n=0
X
We have that f(t) satisfies (C(n) is #8 on Zagier’s list)
9 1 9 1 1
(20) t3− t2+ t f′′(t)+ 3t2− t+ f′(t)+ t− f(t)= 0.
10 9 5 9 3
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
η(2z)6η(3z)
Let F = . In this case, we have
η(z)3η(6z)2
D (t)
q
G := = P −1/3P
1 ∞ 1/2
t
and
D (F)
q
G := = 1/3P .
2 1/2
F
Comparing the first few terms of D G −2G G with those of A2, A2, A2, A A and A A ,
q 1 1 2 2 3 6 2 3 2 6
we have
1
(21) D G −2G G = (57A2 −3A2+9A2+20A A −62A A )
q 1 1 2 2 3 6 2 3 2 6
288
and
1
(22) D G −G2 = (9A2−A2+A2−6A A ).
q 2 2 96 2 3 2 2 6
Proceedingasinthefirstcase,wegetG /P = (1−g)/g.SubstitutingvaluesofG /P,A /P,A /P,A /P
1 1 2 3 6
into (21) and (22), we have
D G −2G G −10t+118t2 −270t3 +162t4
q 1 1 2
p (t):= =
1 G2 (1−t)2(1−9t)2
1
and
D G −G2 −3t+39t2−117t3 +81t4
p (t) := q 2 2 = .
2 G2 (1−t)2(1−9t)2
1
By the folklore result and simplification, F as a function of t satisfies the differential equation
F′′+q (t)F′+q (t)F = 0
1 2
where
10 DERMOTMCCARTHY,ROBERTOSBURNANDBRUNDABANSAHU
p (t)+1 1−30t+236t2−450t3 +243t4
1
q (t) := =
1 t t(1−t)2(1−9t)2
and
−3t+39t2−117t3 +81t4
q (t):= .
2 (1−t)2(1−9t)2
This is equivalent to (20) and thus by uniqueness, (19) follows. If we take L(q) to be (4), then
qdt
dq
f(q) = L(q).
t
Applying Theorem 2.1 yields (7). As the proof of (8) is similar, we only mention that if we
choose the modular function on Γ (8)
0
η(z)4η(4z)2η(8z)4
t(z) = ,
η(2z)10
then
∞ η(2z)10
f(t):= E(n)tn =
η(z)4η(4z)4
n=0
X
satisfies (E(n) is #10 on Zagier’s list)
3 1 3 1 1
t3− t2+ t f′′(t)+ 3t2− t+ f′(t)+ t− f(t)= 0
8 32 4 32 8
and we take L((cid:16)q) to be (5) whi(cid:17)ch is a He(cid:16)cke eigenform(cid:17)in M (Γ(cid:16)(4), −(cid:17)4 ). We then have
3 0 d
qdt (cid:0) (cid:1)
f(q) dq = L(q2).
t
(cid:3)
pr
Proof of Corollary 1.2. Since ≡ 0 (mod pr) for k 6≡ 0 (mod p) and r ≥ 1,
k
(cid:18) (cid:19)
(23) A(p2)≡ 2 (mod p2).
By Theorem 1.1 (with m = 1 and r = 2) and the fact that (4) is a weight 3 normalized Hecke
eigenform, then (12) and (23) imply (9). Equation (9) also follows by taking k = 3 and n = 1
in Theorem 4.2 of [8]. A similar argument yields (10) since
C(p2) ≡ 3 (mod p2)
and (5) is a weight 3 normalized Hecke eigenform. Finally, we claim that
(24) E(p) ≡ 4 (mod p2).
Since
