A modular supercongruence for $_6F_5$: an Ap\'ery-like story

0.2 MB·

by  Robert Osburn

A MODULAR SUPERCONGRUENCE FOR F :
6 5
AN APE´RY-LIKE STORY
ROBERT OSBURN, ARMIN STRAUB AND WADIM ZUDILIN
7
1
0 Abstract. We prove a supercongruence modulo p3 between the pth Fourier co-
2
efficient of a weight 6 modular form and a truncated 6F5-hypergeometric series.

n Novel ingredients in the proof are the comparison of two rational approximations
a to ζp3q to produce non-trivial harmonic sum identities and the reduction of the
J
resultingcongruencesbetweenharmonicsumsviaacongruencebetweentheAp´ery
5
numbers and another Ap´ery-like sequence.
1

]
T
N
1. Introduction
.
h
t There has been considerable recent interest in the study of arithmetic properties
a
m connecting pth Fourier coefficients of integral weight modular forms and truncated
[ hypergeometric series. A motivating example of this phenomenon is the modular

supercongruence [14]
1
v
1, 1, 1, 1
8 F 2 2 2 2 1 ” appq pmod p3q, (1.1)
9 4 3 1, 1, 1
0 „ ˇ p´1
ˇ
4
where p is an odd prime and apnq areˇ the Fourier coefficients of the Hecke eigenform
0 ˇ
.
1 8
0 ηp2τq4ηp4τq4 “ apnqqn (1.2)
7
n“1
1 ÿ
: of weight 4 for the modular group Γ p8q. Here and throughout, q “ e2πiτ with
v 0
i Imτ ą 0, ηpτq “ q1{24 8 p1´qnq is Dedekind’s eta function and
X n“1
ar śa , a , ..., a p´1 pa q ¨¨¨pa q zn
0 1 n 0 k n k
F z “ ,
n`1 n b , ..., b pb q ¨¨¨pb q n!
„ 1 n ˇ p´1 k“0 1 k n k
ˇ ÿ
with paq “ apa`1q¨¨¨pa`k ´1q, isˇthe truncated hypergeometric series.
k ˇ
Date: January 17, 2017.
2010 Mathematics Subject Classification. Primary 11B65; Secondary 33C20, 33F10.
Key words and phrases. Supercongruence,Ap´erynumbers, Ap´ery-likenumbers, hypergeometric
function.
1
2 ROBERTOSBURN,ARMIN STRAUBAND WADIMZUDILIN
Kilbourn’s result (1.1) verifies one of 14 conjectural supercongruences between
truncated F -hypergeometric series (evaluated at 1) corresponding to fundamental
4 3
periods of the Picard–Fuchs differential equation for Calabi–Yau manifolds of dimen-
sion 3 and the Fourier coefficients of modular forms of weight 4 and varying level [8].
Two more cases have been proven in [10] and [17]. Moreover, there is now a general
combinatorial framework [16]–[18] which not only covers these 14 cases, but also the
8 cases in dimensions 1 and 2. In addition, (1.1) is one of van Hamme’s original 13
Ramanujan-type supercongruences (see (M.2) in [29]). For further details on this
and related topics we refer to [9], [13], [21], [27].
The purpose of this paper is to observe that a relationship akin to (1.1) exists
between a truncated F -hypergeometric series and a modular form of weight 6. Our
6 5
main result is the following.
Theorem 1.1. For all odd primes p,
1, 1, 1, 1, 1, 1
F 2 2 2 2 2 2 1 ” bppq pmod p3q, (1.3)
6 5
1, 1, 1, 1, 1
„ ˇ p´1
ˇ
where ˇ
ˇ
8
ηpτq8ηp4τq4 `8ηp4τq12 “ ηp2τq12 `32ηp2τq4ηp8τq8 “ bpnqqn (1.4)
n“1
ÿ
is the unique newform in S pΓ p8qq.
6 0
Theorem1.1isofparticular practical relevance duetoWeil’s bounds|bppq| ă 2p5{2,
which tell us that the values of the truncated sums modulo p3 are sufficient for recon-
structing the Fourier coefficients bppq, and hence the Hecke eigenform. Mortenson
hasfurther observed numerically that (1.3) appearsto holdmodulo p5. The technical
difficulties in generalizing our approach to verify this observation seem considerable.
It would therefore be particularly interesting whether a different approach can be
found, which verifies the congruence more naturally.
The paper is organized as follows. In Section 2, we provide additional historic con-
text, goingbacktoAp´ery’s proofoftheirrationalityofζp3q,andintroduceAp´ery-like
sequences. This also serves to prepare for our proof of Theorem 1.1, which, interest-
ingly, involves two constructions [19], [24], [31] of rational approximations to ζp3q as
well as a congruence between the Ap´ery numbers and another Ap´ery-like sequence.
This congruence is proven in Section 3. In Section 4, we briefly review Greene’s
Gaussian hypergeometric series. A result of Frechette, Ono and Papanikolas [11]
expresses the Fourier coefficients bppq in terms of these finite field analogs of the clas-
sical hypergeometric series. The Gaussian hypergeometric functions that thus arise
have been determined modulo p3 in [20] in terms of sums involving harmonic sums.
A MODULAR SUPERCONGRUENCE FOR 6F5: AN APE´RY-LIKE STORY 3
In Section 5, we reduce the resulting congruences between sums involving harmonic
numbers, then prove Theorem 1.1. One of the challenging auxiliary congruences is
p´1
2 p´1 `k 3 p´1 3
p´1qk 2 k k2 1`3kpHp´21`k `Hp´21´k ´2Hkq
k“0 ˆ ˙ ˆ ˙
ÿ ` ˘
p´1
2 p´1 `k 2 p´1 2
” 2 2 pmod p2q. (1.5)
k k
k“0ˆ ˙ ˆ ˙
ÿ
As usual, H “ Hp1q, and Hprq denote the generalized harmonic numbers
n n n
n
1
Hprq “ .
n jr
j“1
ÿ
The fact that the right-hand side of (1.5) involves the Ap´ery numbers and the re-
lation of the latter to the irrationality of ζp3q helped us to apply some “irrational”
ingredients, in the form of two different constructions of rational approximations to
ζp3q, to complete the proof. Finally, in Section 6, we comment on the need to certify
congruences algorithmically.
2. Historic context and Ap´ery-like sequences
The Ap´ery numbers [25, A005259]
n 2 2
n n`k
Apnq “ (2.1)
k k
k“0ˆ ˙ ˆ ˙
ÿ
rose to prominence by Ap´ery’s proof [2] of the irrationality of ζp3q at the end of the
1970s and were studied by number theorists in the 1980s because of their arithmetic
significance. Prominently, for instance, Beukers conceptualized Ap´ery’s proof by
realizing that the ordinary generating function admits a parametrization by modular
forms. Beukers also established [4] a second relation to modular forms by showing
that
p´1
A ” appq pmod pq, (2.2)
2
ˆ ˙
where apnq are the Fourier coefficients of the Hecke eigenform (1.2). After some dor-
mancy, the Ap´ery numbers resurfaced when Ahlgren and Ono [1] proved Beukers’
conjecture that (2.2) holds modulo p2. In a different direction, Beukers and Za-
gier [30] initiated the exploration of generalizations, often referred to as Ap´ery-like
sequences, which also arise as integral solutions to recurrence equations like
pn`1q3Apn`1q´p2n`1qp17n2 `17n`5qApnq`n3Apn´1q “ 0, (2.3)
4 ROBERTOSBURN,ARMIN STRAUBAND WADIMZUDILIN
which is satisfied by the Ap´ery numbers Apnq and characterizes them together with
the single initial condition Ap0q “ 1.
In reducing the harmonic sums that we encounter in the proof of Theorem 1.1, a
crucial role is played by the sequence C pnq, [25, A183204], where
6
n ℓ
n
C pnq “ 1´ℓkpH ´H q . (2.4)
ℓ k k n´k
k“0ˆ ˙
ÿ ` ˘
The phenomenon that these sequences are integral for all positive integers ℓ has been
proved in[15, Proposition1]. Forℓ “ 1,2,3,4,5,these sequences wereexplicitly eval-
uated by Paule and Schneider [22], who further ask whether C pnq can be expressed
ℓ
as a single sum of hypergeometric terms for ℓ ě 6. It turns out that C pnq is one
6
of the sporadic Ap´ery-like sequences discovered in [7] (see also [32]), so that, for
ℓ “ 6, the question of Paule and Schneider is answered affirmatively by the following
observation.
Proposition 2.1. The sequence C pnq has the binomial sum representations
6
n 2
n n`k 2k
C pnq “ p´1qn
6
k k n
k“0ˆ ˙ ˆ ˙ˆ ˙
ÿ
n 3
3n`1 n`k
“ p´1qk ,
n´k k
k“0 ˆ ˙ˆ ˙
ÿ
which make the integrality of C pnq transparent.
6
That all three sums are equal can be verified by checking that each sequence sat-
isfies the same three-term recursion (a variation of (2.3)). These are recorded in [22]
and [7], or can be automatically derived by an algorithm such as creative telescoping.
An expression for C pnq as a variation of the first of the sums in Proposition 2.1,
6
and hence the answer to the question of Paule and Schneider, for ℓ “ 6, was already
observed in [6, Entry 17 in Table 2]. No single-sum hypergeometric expressions for
C pnq are known when ℓ ě 7.
ℓ
The following unexpected congruence between the Ap´ery numbers Apnq and the
Ap´ery-like numbers C pnq, from (2.1) and (2.4), is another ingredient in our proof
6
of Theorem 1.1. It is proved in Section 3.
Lemma 2.2. For all odd primes p,
p´1 p´1
A ” C pmod p2q. (2.5)
6
2 2
ˆ ˙ ˆ ˙
We point out that suitable modular parameterizations of the generating functions
8 Apnqzn and 8 C pnqzn convert them into weight 2 modular forms of level 6
n“0 n“0 6
ř ř
A MODULAR SUPERCONGRUENCE FOR 6F5: AN APE´RY-LIKE STORY 5
and 7, respectively [5] and [7]. We further note that the congruence (2.5) is rather
trivially complemented by the congruence
p´1 p´1
A ” D pmod pq,
2 2
ˆ ˙ ˆ ˙
which is straightforward and is only true modulo p, where
8 n 4
Dpnq “
k
n“0ˆ ˙
ÿ
is another Ap´ery-like sequence [25, A005260], associated with a modular form of
weight 2 and level 10 (see [7]).
3. Another Ap´ery number congruence
This section is concerned with proving the congruence (2.5) of Lemma 2.2 and,
thereby, collecting some basic congruences involving harmonic numbers. The form
in which we will later use this congruence is
m 2 2 m 6
m m`k m
” 1´6kpH ´H q pmod p2q. (3.1)
k k k k m´k
k“0ˆ ˙ ˆ ˙ k“0ˆ ˙
ÿ ÿ ` ˘
Here, and throughout, p is an odd prime and m “ pp ´ 1q{2. For our proof of the
congruence (3.1) it is however crucial to use the alternative representation
n 3
3n`1 n`k
C pnq “ p´1qk
6
n´k k
k“0 ˆ ˙ˆ ˙
ÿ
for the sequence C pnq provided by Proposition 2.1.
6
First, note that
m`k pm`1q p1q p k´1 1
“ k “ 2 k 1` `Opp2q (3.2)
m k! k! 2 j ` 1
ˆ ˙ ˆ j“0 2 ˙
ÿ
and
m p´1qkp´mq p1q p k´1 1
“ k “ p´1qk 2 k 1´ `Opp2q . (3.3)
k k! k! 2 j ` 1
ˆ ˙ ˆ j“0 2 ˙
ÿ
Now, since
k´1 k´1 k´1
1 1 1
“ `Oppq “ `Oppq “ H ´H `Oppq,
j ` 1 j ` 1 ` p j `m`1 m`k m
j“0 2 j“0 2 2 j“0
ÿ ÿ ÿ
6 ROBERTOSBURN,ARMIN STRAUBAND WADIMZUDILIN
we can write the expressions (3.2) and (3.3) in the forms
m p1q p
“ p´1qk 2 k 1´ pH ´H q`Opp2q , (3.4)
k k! 2 m`k m
ˆ ˙ ˆ ˙
and
m`k p1q p
“ 2 k 1` pH ´H q`Opp2q
m k! 2 m`k m
ˆ ˙ ˆ ˙
2
m p
“ p´1qk 1` pH ´H q`Opp2q
k 2 m`k m
ˆ ˙ˆ ˙
m
“ p´1qk 1`ppH ´H q`Opp2q . (3.5)
m`k m
k
ˆ ˙
` ˘
Recall that 2m “ p´1, so that
k k
1 1 p
H “ H ´ “ ` `Opp2q “ H `pHp2q `Opp2q. (3.6)
2m´k p´1 p´j j j2 k k
j“1 j“1ˆ ˙
ÿ ÿ
By swapping k with m´k, we get
H “ H `pHp2q `Opp2q, (3.7)
m`k m´k m´k
and, in view of the invariance of m under replacing k with m´k, we can translate
k
formula (3.5) to
` ˘
2m´k m
“ p´1qm´k 1`ppH ´H q`Opp2q
m k 2m´k m
ˆ ˙ ˆ ˙
m ` ˘
“ p´1qm´k 1`ppH ´H q`Opp2q , (3.8)
k m
k
ˆ ˙
` ˘
which will be useful later.
On the other hand,
3m`1 m`p p´m´pq
“ “ p´1qk k
k k k!
ˆ ˙ ˆ ˙
p´mq k´1 1
“ p´1qk k 1´p `Opp2q
k! ´m´p`j
ˆ j“0 ˙
ÿ
m
“ 1`ppH ´H q`Opp2q ,
k m m´k
ˆ ˙
` ˘
so that
3m`1 m
“ 1`ppH ´H q`Opp2q . (3.9)
m k
m´k k
ˆ ˙ ˆ ˙
` ˘
A MODULAR SUPERCONGRUENCE FOR 6F5: AN APE´RY-LIKE STORY 7
It follows from (3.5), (3.7) and (3.9) that
2 2 4
m`k m m
“ 1`ppH ´H q`Opp2q 2
m´k m
m k k
ˆ ˙ ˆ ˙ ˆ ˙
4` ˘
m
“ 1`pp2H ´2H q`Opp2q
m´k m
k
ˆ ˙
` ˘
and
3
3m`1 m`k
p´1qk
m´k m
ˆ ˙ˆ ˙
4
m
“ 1`ppH ´H q`Opp2q 1`ppH ´H q`Opp2q 3
k m k m´k m
ˆ ˙
4` ˘` ˘
m
“ 1`pp3H ´H ´2H q`Opp2q .
m´k k m
k
ˆ ˙
` ˘
It remains to use the symmetry k Ø m´k in the form
m 4 m 4
m m
H “ H
m´k k
k k
k“0ˆ ˙ k“0ˆ ˙
ÿ ÿ
to conclude that the desired congruence (2.5) is indeed true modulo p2.
4. Gaussian hypergeometric series
In the following, we discuss some preliminaries concerning Greene’s Gaussian hy-
pergeometric series [12]. Let F denote the finite field with p elements. We extend
p
the domain of all characters χ of Fˆ to F by defining χp0q “ 0. For characters A
p p
and B of Fˆ, define
p
A Bp´1q
“ JpA,B¯q,
B p
ˆ ˙
where Jpχ,λq denotes the Jacobi sum for χ and λ characters of Fˆ. For characters
p
A ,A ,...,A and B ,...,B of Fˆ and x P F , define the Gaussian hypergeometric
0 1 n 1 n p p
series by
A , A , ..., A p A χ A χ A χ
F 0 1 n x “ 0 1 ¨¨¨ n χpxq,
n`1 nˆ B1, ..., Bn ˇ ˙p p´1 χ ˆ χ ˙ˆB1χ˙ ˆBnχ˙
ˇ ÿ
where the summation is overˇˇall characters χ on Fˆ.
p
8 ROBERTOSBURN,ARMIN STRAUBAND WADIMZUDILIN
Weconsider thecase where A “ φ , thequadratic character, forall i, andB “ ǫ ,
i p j p
the trivial character mod p, for all j, and write
φ , φ , ..., φ
F pxq “ F p p p x
n`1 n n`1 n ǫ , ..., ǫ
p p
ˆ ˇ ˙p
ˇ
for brevity. By [12], pn F pxq P Z. ˇ
n`1 n ˇ
For λ P F and ℓ ě 2 an integer, we now define the quantities
p
m ℓ ℓ
m`k m
X pp,λq “ λm p´1qℓk 1`4ℓkpH ´H q
ℓ m`k k
k k
k“0 ˆ ˙ ˆ ˙
ÿ `
`2ℓ2k2pH ´H q2 ´ℓk2pHp2q ´Hp2qq λ´k,
m`k k m`k k
m ℓ ℓ
m`k m ˘
Y pp,λq “ λm p´1qℓk 1`2ℓkpH ´H q
ℓ k k m`k k
k“0 ˆ ˙ ˆ ˙
ÿ `
´ℓkpH ´H q λ´kp,
m`k m´k
m 2ℓ
Z pp,λq “ λm 2k 16´ℓkλ´k˘p2.
ℓ
k
k“0ˆ ˙
ÿ
Here, as before, m “ pp´1q{2.
The main result in [20] provides an expression for F modulo p3. Precisely,
2ℓ 2ℓ´1
we have the following.
Theorem 4.1. Let p be an odd prime, λ P F , and ℓ ě 2 be an integer. Then,
p
p2ℓ´1 F pλq ” ´ p2X pp,λq`pY pp,λq`Z pp,λq pmod p3q.
2ℓ 2ℓ´1 ℓ ℓ ℓ
` ˘
5. Two lemmas and the proof of Theorem 1.1
Lemma 5.1. Let p be an odd prime. Then
X pp,1q´Y pp,1q ” p´1qpp´1q{2 ´1 pmod pq.
3 2
Proof. Consider the rational function
n pt´jq2
Rptq “ R ptq “ j“1 ,
n n pt`jq2
śj“0
defined for any integer n ě 0. Its partial fraśction decomposition assumes the form
n
A B
k k
Rptq “ ` ,
pt`kq2 t`k
k“0ˆ ˙
ÿ
A MODULAR SUPERCONGRUENCE FOR 6F5: AN APE´RY-LIKE STORY 9
where
2 2
n`k n
A “ Rptqpt`kq2 “ ,
k t“´k k k
ˆ ˙ ˆ ˙
and, on considering the lo`garithmic der˘iˇˇvative of Rptqpt`kq2,
d
B “ Rptqpt`kq2
k
dt
ˇt“´k
` ˘ˇ
n n
ˇ 1 1
“ 2 Rptqpt`kq2 ˇ ´
t´j t`j
˜ ¸ˇ
` ˘ jÿ“1 jjÿ‰“k0 ˇˇt“´k
ˇ
“ 2Ak pHk ´Hn`kq`pHk ´Hn´kq . ˇ
The related partial fraction`decomposition ˘
n n
A t B t A ppt`kq´kq B ppt`kq´kq
k k k k
tRptq “ ` “ `
pt`kq2 t`k pt`kq2 t`k
k“0ˆ ˙ k“0ˆ ˙
ÿ ÿ
n
kA A ´kB
k k k
“ ´ ` `B
pt`kq2 t`k k
k“0ˆ ˙
ÿ
and the residue sum theorem imply
n
pA ´kB q “ Res tRptq “ ´Res tRptq
k k pole t“8
k“0 allfinitepoles
ÿ ÿ
1 1
“ coefficient of s in Taylor’s s-expansion of R
s s
´m ¯p1´jsq2
“ coefficient of s in Taylor’s s-expansion of s j“1
m p1`jsq2
śj“0
“ 1 “ A ,
0 ś
from which n pA ´kB q “ 0 follows. The resulting identity is then
k“1 k k
n 2 2
ř n`k n
1´2kp2H ´H ´H q “ 1, (5.1)
k k k n`k n´k
k“0ˆ ˙ ˆ ˙
ÿ ` ˘
which played a crucial role in [1] and [14]. Notice that (5.1) implies
Y pp,1q “ 1. (5.2)
2
Equality (5.1) and its derivation above follow the approach of Nesterenko from [19]
of proving Ap´ery’s theorem (see also [31]).
10 ROBERTOSBURN,ARMIN STRAUBAND WADIMZUDILIN
We can perform a similar analysis for the rational function
n pt´jq3 n A B C
Rptq “ R ptq “ j“1 “ k ` k ` k .
n n pt`jq3 pt`kq3 pt`kq2 t`k
śj“0 k“0ˆ ˙
ÿ r r r
As beforer, we getr ś
3 3
n`k n
A “ Rptqpt`kq3 “ p´1qn`k ,
k t“´k k k
ˆ ˙ ˆ ˙
` ˘ˇ
Brk “ 3Arkp2Hk ´Hnˇ`k ´Hn´kq,
9 3
C “ A p2H ´H ´H q2 ´ A pHp2q ´2Hp2q ´Hp2q q
rk 2rk k n`k n´k 2 k n`k k n´k
and by considering the sum of the residues of the rational functions Rptq, tRptq and
r r r
t2Rptq, we deduce that
n n n
C “ pB ´kC q “ 0 and pA ´2kB `k2C q “ 1.
k k k k k k
k“0 k“0 k“0
ÿ ÿ ÿ
We only recrord the firrst andrlast equalities for our furture user: r
n 3 3
n`k n
p´1qk 3p2H ´H ´H q2´pHp2q ´2Hp2q´Hp2q q “ 0 (5.3)
k k k n`k n´k n`k k n´k
k“0 ˆ ˙ ˆ ˙
ÿ ` ˘
and
n 3 3
n`k n
p´1qk 1´6kp2H ´H ´H q` 9k2p2H ´H ´H q2
k k k n`k n´k 2 k n`k n´k
k“0 ˆ ˙ ˆ ˙
ÿ `
´ 3k2pHp2q ´2Hp2q ´Hp2q q “ p´1qn. (5.4)
2 n`k k n´k
Recall that, throughout, m “ pp ´1q{2. Now, taking n “ m in (5˘.4) and applying
H ” H pmod pq and Hp2q ” ´Hp2q pmod pq, we obtain
m´k m`k m´k m`k
m 3 3
m`k m
X pp,1q “ p´1qk 1´12kpH ´H q (5.5)
3 k k k m`k
k“0 ˆ ˙ ˆ ˙
ÿ `
`18k2pH ´H q2 ´3k2pHp2q ´Hp2qq ” p´1qm pmod pq.
k m`k m`k k
(cid:3)
The result then follows after combining (5.2) with (5.5). ˘
Lemma 5.2. Let p be an odd prime. Then
Y pp,1q ” Z pp,1q pmod p2q.
3 2