A modular supercongruence for _6F_5: an Apéry-like story
aa r X i v : . [ m a t h . N T ] N ov A MODULAR SUPERCONGRUENCE FOR F :AN AP ´ERY-LIKE STORY ROBERT OSBURN, ARMIN STRAUB AND WADIM ZUDILIN
Abstract.
We prove a supercongruence modulo p between the p th Fourier co-efficient of a weight 6 modular form and a truncated F -hypergeometric series.Novel ingredients in the proof are the comparison of two rational approximationsto ζ p q to produce non-trivial harmonic sum identities and the reduction of theresulting congruences between harmonic sums via a congruence between the Ap´erynumbers and another Ap´ery-like sequence. Introduction
There has been considerable recent interest in the study of arithmetic propertiesconnecting p th Fourier coefficients of integral weight modular forms and truncatedhypergeometric series. A motivating example of this phenomenon is the modularsupercongruence [13] F „ , , , , , ˇˇˇˇ p ´ ” a p p q p mod p q , (1.1)where p is an odd prime and a p n q are the Fourier coefficients of the Hecke eigenform η p τ q η p τ q “ ÿ n “ a p n q q n (1.2)of weight 4 for the modular group Γ p q . Here and throughout, q “ e πiτ withIm τ ą η p τ q “ q { ś n “ p ´ q n q is Dedekind’s eta function and n ` F n „ a , a , . . . , a n b , . . . , b n ˇˇˇˇ z p ´ “ p ´ ÿ k “ p a q k ¨ ¨ ¨ p a n q k p b q k ¨ ¨ ¨ p b n q k z n n ! , with p a q k “ a p a ` q ¨ ¨ ¨ p a ` k ´ q , is the truncated hypergeometric series. Date : November 16, 2017.2010
Mathematics Subject Classification.
Primary 11B65; Secondary 33C20, 33F10.
Key words and phrases.
Supercongruence, Ap´ery numbers, Ap´ery-like numbers, hypergeometricfunction.
Kilbourn’s result (1.1) verifies one of 14 conjectural supercongruences betweentruncated F -hypergeometric series (evaluated at 1) corresponding to fundamentalperiods 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[25]. Two more cases have been proven in [10] and [16]. Moreover, there is now ageneral combinatorial framework [15]–[17] which not only covers these 14 cases, butalso the 8 cases in dimensions 1 and 2. In addition, (1.1) is one of van Hamme’soriginal 13 Ramanujan-type supercongruences (see (M.2) in [29]). For further detailson this and related topics we refer to [9], [12], [20], [28].The purpose of this paper is to observe that a relationship akin to (1.1) existsbetween a truncated F -hypergeometric series and a modular form of weight 6. Ourmain result is the following. Theorem 1.1.
For all odd primes p , F „ , , , , , , , , , ˇˇˇˇ p ´ ” b p p q p mod p q , (1.3) where η p τ q η p τ q ` η p τ q “ η p τ q ` η p τ q η p τ q “ ÿ n “ b p n q q n (1.4) is the unique newform in S p Γ p qq . Theorem 1.1 is of particular practical relevance due to Weil’s bounds | b p p q| ă p { ,which tell us that the values of the truncated sums modulo p are sufficient for recon-structing the Fourier coefficients b p p q , and hence the Hecke eigenform. Mortensonhas further observed numerically that (1.3) appears to hold modulo p . The technicaldifficulties in generalizing our approach to verify this observation seem considerable.It would therefore be particularly interesting whether a different approach can befound, which verifies the congruence more naturally.The paper is organized as follows. In Section 2, we provide additional historic con-text, going back to Ap´ery’s proof of the irrationality of ζ p q , and introduce Ap´ery-likesequences. This also serves to prepare for our proof of Theorem 1.1, which, interest-ingly, involves two constructions [18], [24], [31] of rational approximations to ζ p q aswell 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’sGaussian hypergeometric series. A result of Frechette, Ono and Papanikolas [8]expresses the Fourier coefficients b p p q in terms of these finite field analogs of the clas-sical hypergeometric series. The Gaussian hypergeometric functions that thus arisehave been determined modulo p in [19] in terms of sums involving harmonic sums. MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 3 In Section 5, we reduce the resulting congruences between sums involving harmonicnumbers, then prove Theorem 1.1. One of the challenging auxiliary congruences is p ´ ÿ k “ p´ q k ˆ p ´ ` kk ˙ ˆ p ´ k ˙ ` ` k p H p ´ ` k ` H p ´ ´ k ´ H k q ˘ ” p ´ ÿ k “ ˆ p ´ ` kk ˙ ˆ p ´ k ˙ p mod p q . (1.5)As usual, H n “ H p q n , and H p r q n denote the generalized harmonic numbers H p r q n “ n ÿ j “ j r . 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 ζ p q helped us to apply some “irrational”ingredients, in the form of two different constructions of rational approximations to ζ p q , to complete the proof. Finally, in Section 6, we comment on the need to certifycongruences algorithmically.2. Historic context and Ap´ery-like sequences
The Ap´ery numbers [26, A005259] A p n q “ n ÿ k “ ˆ nk ˙ ˆ n ` kk ˙ (2.1)rose to prominence by Ap´ery’s proof [2] of the irrationality of ζ p q at the end of the1970s and were studied by number theorists in the 1980s because of their arithmeticsignificance. Prominently, for instance, Beukers conceptualized Ap´ery’s proof byrealizing that the ordinary generating function admits a parametrization by modularforms. Beukers also established [4] a second relation to modular forms by showingthat A ˆ p ´ ˙ ” a p p q p mod p q , (2.2)where a p n q 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 p . In a different direction, Beukers and Za-gier [30] initiated the exploration of generalizations, often referred to as Ap´ery-likesequences, which also arise as integral solutions to recurrence equations like p n ` q A p n ` q ´ p n ` qp n ` n ` q A p n q ` n A p n ´ q “ , (2.3) ROBERT OSBURN, ARMIN STRAUB AND WADIM ZUDILIN which is satisfied by the Ap´ery numbers A p n q and characterizes them together withthe single initial condition A p q “ C p n q , [26, A183204], where C ℓ p n q “ n ÿ k “ ˆ nk ˙ ℓ ` ´ ℓk p H k ´ H n ´ k q ˘ . (2.4)The phenomenon that these sequences are integral for all positive integers ℓ has beenproved in [14, Proposition 1]. For ℓ “ , , , ,
5, these sequences were explicitly eval-uated by Paule and Schneider [21], who further ask whether C ℓ p n q can be expressedas a single sum of hypergeometric terms for ℓ ě
6. It turns out that C p n q is oneof 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 followingobservation.
Proposition 2.1.
The sequence C p n q has the binomial sum representations C p n q “ p´ q n n ÿ k “ ˆ nk ˙ ˆ n ` kk ˙ˆ kn ˙ “ n ÿ k “ p´ q k ˆ n ` n ´ k ˙ˆ n ` kk ˙ , which make the integrality of C p n q transparent. 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 [21]and [7], or can be automatically derived by an algorithm such as creative telescoping.An expression for C p n q as a variation of the first of the sums in Proposition 2.1,and hence the answer to the question of Paule and Schneider, for ℓ “
6, was alreadyobserved in [6, Entry 17 in Table 2]. No single-sum hypergeometric expressions for C ℓ p n q are known when ℓ ě A p n q and theAp´ery-like numbers C p n q , from (2.1) and (2.4), is another ingredient in our proofof Theorem 1.1. It is proved in Section 3. Lemma 2.2.
For all odd primes p , A ˆ p ´ ˙ ” C ˆ p ´ ˙ p mod p q . (2.5)We point out that suitable modular parameterizations of the generating functions ř n “ A p n q z n and ř n “ C p n q z n convert them into weight 2 modular forms of level 6 MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 5 and 7, respectively [5] and [7]. We further note that the congruence (2.5) is rathertrivially complemented by the congruence A ˆ p ´ ˙ ” D ˆ p ´ ˙ p mod p q , which is straightforward and is only true modulo p , where D p n q “ ÿ n “ ˆ nk ˙ is another Ap´ery-like sequence [26, A005260], associated with a modular form ofweight 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 formin which we will later use this congruence is m ÿ k “ ˆ mk ˙ ˆ m ` kk ˙ ” m ÿ k “ ˆ mk ˙ ` ´ k p H k ´ H m ´ k q ˘ p mod p q . (3.1)Here, and throughout, p is an odd prime and m “ p p ´ q{
2. For our proof of thecongruence (3.1) it is however crucial to use the alternative representation C p n q “ n ÿ k “ p´ q k ˆ n ` n ´ k ˙ˆ n ` kk ˙ for the sequence C p n q provided by Proposition 2.1.First, note that ˆ m ` km ˙ “ p m ` q k k ! “ p q k k ! ˆ ` p k ´ ÿ j “ j ` ` O p p q ˙ (3.2)and ˆ mk ˙ “ p´ q k p´ m q k k ! “ p´ q k p q k k ! ˆ ´ p k ´ ÿ j “ j ` ` O p p q ˙ . (3.3)Now, since k ´ ÿ j “ j ` “ k ´ ÿ j “ j ` ` p ` O p p q “ k ´ ÿ j “ j ` m ` ` O p p q “ H m ` k ´ H m ` O p p q , ROBERT OSBURN, ARMIN STRAUB AND WADIM ZUDILIN we can write the expressions (3.2) and (3.3) in the forms ˆ mk ˙ “ p´ q k p q k k ! ˆ ´ p p H m ` k ´ H m q ` O p p q ˙ , (3.4)and ˆ m ` km ˙ “ p q k k ! ˆ ` p p H m ` k ´ H m q ` O p p q ˙ “ p´ q k ˆ mk ˙ˆ ` p p H m ` k ´ H m q ` O p p q ˙ “ p´ q k ˆ mk ˙` ` p p H m ` k ´ H m q ` O p p q ˘ . (3.5)Recall that 2 m “ p ´
1, so that H m ´ k “ H p ´ ´ k ÿ j “ p ´ j “ k ÿ j “ ˆ j ` pj ˙ ` O p p q “ H k ` pH p q k ` O p p q . (3.6)By swapping k with m ´ k , we get H m ` k “ H m ´ k ` pH p q m ´ k ` O p p q , (3.7)and, in view of the invariance of ` mk ˘ under replacing k with m ´ k , we can translateformula (3.5) to ˆ m ´ km ˙ “ p´ q m ´ k ˆ mk ˙` ` p p H m ´ k ´ H m q ` O p p q ˘ “ p´ q m ´ k ˆ mk ˙` ` p p H k ´ H m q ` O p p q ˘ , (3.8)which will be useful later.On the other hand, ˆ m ` k ˙ “ ˆ m ` pk ˙ “ p´ q k p´ m ´ p q k k ! “ p´ q k p´ m q k k ! ˆ ´ p k ´ ÿ j “ ´ m ´ p ` j ` O p p q ˙ “ ˆ mk ˙` ` p p H m ´ H m ´ k q ` O p p q ˘ , so that ˆ m ` m ´ k ˙ “ ˆ mk ˙` ` p p H m ´ H k q ` O p p q ˘ . (3.9) MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 7 It follows from (3.5), (3.7) and (3.9) that ˆ m ` km ˙ ˆ mk ˙ “ ˆ mk ˙ ` ` p p H m ´ k ´ H m q ` O p p q ˘ “ ˆ mk ˙ ` ` p p H m ´ k ´ H m q ` O p p q ˘ and p´ q k ˆ m ` m ´ k ˙ˆ m ` km ˙ “ ˆ mk ˙ ` ` p p H m ´ H k q ` O p p q ˘` ` p p H m ´ k ´ H m q ` O p p q ˘ “ ˆ mk ˙ ` ` p p H m ´ k ´ H k ´ H m q ` O p p q ˘ . It remains to use the symmetry k Ø m ´ k in the form m ÿ k “ ˆ mk ˙ H m ´ k “ m ÿ k “ ˆ mk ˙ H k to conclude that the desired congruence (2.5) is indeed true modulo p .4. Gaussian hypergeometric series
In the following, we discuss some preliminaries concerning Greene’s Gaussian hy-pergeometric series [11]. Let F p denote the finite field with p elements. We extendthe domain of all characters χ of F ˆ p to F p by defining χ p q “
0. For characters A and B of F ˆ p , define ˆ AB ˙ “ B p´ q p J p A, ¯ B q , where J p χ, λ q denotes the Jacobi sum for χ and λ characters of F ˆ p . For characters A , A , . . . , A n and B , . . . , B n of F ˆ p and x P F p , define the Gaussian hypergeometricseries by n ` F n ˆ A , A , . . . , A n B , . . . , B n ˇˇˇˇ x ˙ p “ pp ´ ÿ χ ˆ A χχ ˙ˆ A χB χ ˙ ¨ ¨ ¨ ˆ A n χB n χ ˙ χ p x q , where the summation is over all characters χ on F ˆ p . ROBERT OSBURN, ARMIN STRAUB AND WADIM ZUDILIN
We consider the case where A i “ φ p , the quadratic character, for all i , and B j “ ǫ p ,the trivial character mod p , for all j , and write n ` F n p x q “ n ` F n ˆ φ p , φ p , . . . , φ p ǫ p , . . . , ǫ p ˇˇˇˇ x ˙ p for brevity. By [11], p nn ` F n p x q P Z .For λ P F p and ℓ ě X ℓ p p, λ q “ λ m m ÿ k “ p´ q ℓk ˆ m ` kk ˙ ℓ ˆ mk ˙ ℓ ` ` ℓk p H m ` k ´ H k q` ℓ k p H m ` k ´ H k q ´ ℓk p H p q m ` k ´ H p q k q ˘ λ ´ k ,Y ℓ p p, λ q “ λ m m ÿ k “ p´ q ℓk ˆ m ` kk ˙ ℓ ˆ mk ˙ ℓ ` ` ℓk p H m ` k ´ H k q´ ℓk p H m ` k ´ H m ´ k q ˘ λ ´ kp ,Z ℓ p p, λ q “ λ m m ÿ k “ ˆ kk ˙ ℓ ´ ℓk λ ´ kp . Here, as before, m “ p p ´ q{ ℓ F ℓ ´ modulo p . Precisely,we have the following. Theorem 4.1.
Let p be an odd prime, λ P F p , and ℓ ě be an integer. Then, p ℓ ´ ℓ F ℓ ´ p λ q ” ´ ` p X ℓ p p, λ q ` pY ℓ p p, λ q ` Z ℓ p p, λ q ˘ p mod p q . An analogous result holds for the opposite parity, that is, for n ` F n when n iseven. 5. Two lemmas and the proof of Theorem 1.1
Lemma 5.1.
Let p be an odd prime. Then X p p, q ´ Y p p, q ” p´ q p p ´ q{ ´ p mod p q . Proof.
Consider the rational function R p t q “ R n p t q “ ś nj “ p t ´ j q ś nj “ p t ` j q , defined for any integer n ě
0. Its partial fraction decomposition assumes the form R p t q “ n ÿ k “ ˆ A k p t ` k q ` B k t ` k ˙ , MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 9 where A k “ ` R p t qp t ` k q ˘ˇˇ t “´ k “ ˆ n ` kk ˙ ˆ nk ˙ , and, on considering the logarithmic derivative of R p t qp t ` k q , B k “ dd t ` R p t qp t ` k q ˘ˇˇˇˇ t “´ k “ ` R p t qp t ` k q ˘˜ n ÿ j “ t ´ j ´ n ÿ j “ j ‰ k t ` j ¸ˇˇˇˇˇ t “´ k “ A k ` p H k ´ H n ` k q ` p H k ´ H n ´ k q ˘ . The related partial fraction decomposition tR p t q “ n ÿ k “ ˆ A k t p t ` k q ` B k tt ` k ˙ “ n ÿ k “ ˆ A k pp t ` k q ´ k qp t ` k q ` B k pp t ` k q ´ k q t ` k ˙ “ n ÿ k “ ˆ ´ kA k p t ` k q ` A k ´ kB k t ` k ` B k ˙ and the residue sum theorem imply n ÿ k “ p A k ´ kB k q “ ÿ all finite poles Res pole tR p t q “ ´ Res t “8 tR p t q“ coefficient of s in Taylor’s s -expansion of 1 s R ´ s ¯ “ coefficient of s in Taylor’s s -expansion of s ś mj “ p ´ js q ś mj “ p ` js q “ “ A , from which ř nk “ p A k ´ kB k q “ n ÿ k “ ˆ n ` kk ˙ ˆ nk ˙ ` ´ k p H k ´ H n ` k ´ H n ´ k q ˘ “ , (5.1)which played a crucial role in [1] and [13]. Notice that (5.1) implies Y p p, q “ . (5.2)Equality (5.1) and its derivation above follow the approach of Nesterenko from [18]of proving Ap´ery’s theorem (see also [31]). We can perform a similar analysis for the rational function r R p t q “ r R n p t q “ ś nj “ p t ´ j q ś nj “ p t ` j q “ n ÿ k “ ˆ r A k p t ` k q ` r B k p t ` k q ` r C k t ` k ˙ . As before, we get r A k “ ` r R p t qp t ` k q ˘ˇˇ t “´ k “ p´ q n ` k ˆ n ` kk ˙ ˆ nk ˙ , r B k “ r A k p H k ´ H n ` k ´ H n ´ k q , r C k “ r A k p H k ´ H n ` k ´ H n ´ k q ´ r A k p H p q n ` k ´ H p q k ´ H p q n ´ k q and by considering the sum of the residues of the rational functions R p t q , tR p t q and t R p t q , we deduce that n ÿ k “ r C k “ n ÿ k “ p r B k ´ k r C k q “ n ÿ k “ p r A k ´ k r B k ` k r C k q “ . We only record the first and last equalities for our future use: n ÿ k “ p´ q k ˆ n ` kk ˙ ˆ nk ˙ ` p H k ´ H n ` k ´ H n ´ k q ´p H p q n ` k ´ H p q k ´ H p q n ´ k q ˘ “ n ÿ k “ p´ q k ˆ n ` kk ˙ ˆ nk ˙ ` ´ k p H k ´ H n ` k ´ H n ´ k q ` k p H k ´ H n ` k ´ H n ´ k q ´ k p H p q n ` k ´ H p q k ´ H p q n ´ k q ˘ “ p´ q n . (5.4)Recall that, throughout, m “ p p ´ q{
2. Now, taking n “ m in (5.4) and applying H m ´ k ” H m ` k p mod p q and H p q m ´ k ” ´ H p q m ` k p mod p q , we obtain X p p, q “ m ÿ k “ p´ q k ˆ m ` kk ˙ ˆ mk ˙ ` ´ k p H k ´ H m ` k q (5.5) ` k p H k ´ H m ` k q ´ k p H p q m ` k ´ H p q k q ˘ ” p´ q m p mod p q . The result then follows after combining (5.2) with (5.5). (cid:3)
Lemma 5.2.
Let p be an odd prime. Then Y p p, q ” Z p p, q p mod p q . MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 11 Proof.
Consider the rational function p R p t q “ p R n p t q “ n ! p t ` n q ś nj “ p t ´ j q ¨ ś nj “ p t ` n ` j q ś nj “ p t ` j q “ n ÿ k “ ˆ p A k p t ` k q ` p B k p t ` k q ` p C k p t ` k q ` p D k t ` k ˙ . Then p A k “ p´ q n pp n ´ k q ´ k q ˆ n ` kn ˙ˆ n ´ kn ˙ˆ nk ˙ , p B k “ p´ q n ˆ n ` kn ˙ˆ n ´ kn ˙ˆ nk ˙ ` ` p n ´ k q ` ´p H n ` k ´ H k q` p H n ´ k ´ H n ´ k q ´ p H n ´ k ´ H k q ˘˘ . An important consequence of a hypergeometric transformation due to W. N. Bailey[3], [33] (see also [24] and [31] for the links with rational approximations to ζ p q ) isthe equality A p n q “ n ÿ k “ p B k “ p´ q n n ÿ k “ ˆ n ` kn ˙ˆ n ´ kn ˙ˆ nk ˙ ˆ ` ` p n ´ k qp H k ´ H n ´ k ´ H n ` k ` H n ´ k q ˘ . (5.6)Now, take n “ m (recall that m “ p p ´ q{
2) and let b p m, k q denote the summandin (5.6). Note that b p m, k q “ b p m, m ´ k q and substituting of (3.5) and (3.8) impliesthat b p m, k q “ ˆ mk ˙ ` ` p p H m ` k ´ H m q ` O p p q ˘` ` p p H k ´ H m q ` O p p q ˘ ˆ ` ` p m ´ k qp H k ´ H m ´ k ´ H m ` k ` H m ´ k q ˘ “ ˆ mk ˙ ` ` p p H k ` H m ´ k ´ H m q ` O p p q ˘ ˆ ` ` p m ´ k q ` H k ´ H m ´ k ` pH p q k ´ pH p q m ´ k ` O p p q ˘˘ “ ˆ mk ˙ ` ` p m ´ k qp H k ´ H m ´ k q ` p p H k ` H m ´ k ´ H m q` p p m ´ k qp H k ´ H m ´ k q ´ p p m ´ k qp H k ´ H m ´ k q H m ` p p m ´ k qp H p q k ´ H p q m ´ k q ` O p p q ˘ . (5.7) Moreover, it follows from the symmetry k Ø m ´ k in the form m ÿ k “ ˆ mk ˙ H m “ m ÿ k “ ˆ mk ˙ H m ´ k as well as Lemma 2.2, (3.1) and (5.6) that m ÿ k “ ˆ mk ˙ ` ` p m ´ k qp H k ´ H m ´ k q ˘ “ m ÿ k “ ˆ mk ˙ ` ´ k p H k ´ H m ´ k q ˘ ” m ÿ k “ b p m, k q p mod p q . Substitution of the expansion (5.7) into the latter congruence results, after simplifi-cations, in m ÿ k “ ˆ mk ˙ ` p H k ` H m ´ k ´ H m q ` p m ´ k qp H k ´ H m ´ k q´ p m ´ k qp H k ´ H m ´ k q H m ` p m ´ k qp H p q k ´ H p q m ´ k q ˘ ” p mod p q . (5.8)From a different source, namely, from the equality (5.3) applied with n “ m andreduced modulo p , we obtain m ÿ k “ ˆ mk ˙ ` p H k ´ H m ´ k q ` p H p q k ` H p q m ´ k q ˘ ” p mod p q . (5.9)Furthermore, denote c p m, k q “ p´ q k ˆ m ` kk ˙ ˆ mk ˙ ` ` k p H m ` k ` H m ´ k ´ H k q ˘ , the summand of Y p p, q . Then, with the help of (3.5), we obtain c p m, k q “ ˆ mk ˙ ` ` p p H m ` k ´ H m q ` O p p q ˘ ˆ ` ` k p H m ` k ` H m ´ k ´ H k q ˘ “ ˆ mk ˙ ` ´ k p H k ´ H m ´ k q ` p p H m ´ k ´ H m q´ pk p H k ´ H m ´ k qp H m ´ k ´ H m q ` pkH p q m ´ k ` O p p q ˘ and thus m ÿ k “ c p m, k q “ m ÿ k “ r c p m, k q , (5.10) MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 13 where r c p m, k q “ c p m, k q ` c p m, m ´ k q “ ˆ mk ˙ ` ` p m ´ k qp H k ´ H m ´ k q` p p H k ` H m ´ k ´ H m q ´ pmH k H m ´ k ´ p p m ´ k qp H k ´ H m ´ k q H m ` p p m ´ k q H k ` pkH m ´ k ` p p m ´ k q H p q k ` pkH p q m ´ k ` O p p q ˘ . (5.11)Finally, from (3.2) and (3.3), we have ˆ kk ˙ ´ k “ p { q k k ! ” p´ q k ˆ m ` km ˙ˆ mk ˙ p mod p q , and so Z p p, q ” A p m q p mod p q . (5.12)Therefore, by (5.6), (5.7) and (5.10)–(5.12), Y p p, q ´ Z p p, q “ m ÿ k “ c p m, k q ´ m ÿ k “ b p m, k q“ p m ÿ k “ ˆ mk ˙ ` p H k ` H m ´ k ´ H m q ´ mH k H m ´ k ´ p m ´ k qp H k ´ H m ´ k q H m ` p m ´ k qp H k ` H p q k q` p m ` k qp H m ´ k ` H p q m ´ k q ˘ ` O p p q . The latter sum is seen to be half of the sum in (5.8) plus m times the sum in (5.9).Thus, the result follows. (cid:3) We now prove our main result.
Proof of Theorem 1.1.
It was conjectured by Koike and proven by Frechette, Onoand Papanikolas that the Fourier coefficients b p p q of (1.4) can be represented interms of Gaussian hypergeometric series. Specifically, we have (see Corollary 1.6 in[8]) b p p q “ ´ p F p q ` p F p q ` ` ´ φ p p´ q ˘ p . We now apply Theorem 4.1 with ℓ “ ℓ “
3, respectively, and simplify to obtain b p p q ” p ` X p p, q ´ Y p p, q ` ´ p´ q p p ´ q{ ˘ ` p ` Y p p, q ´ Z p p, q ˘ ` Z p p, q p mod p q . As Z p p, q “ p p ´ q{ ÿ n “ p { q n n ! ” p ´ ÿ n “ p { q n n ! p mod p q , since the summands for p p ´ q{ ă n ď p ´ p , the result followsfrom Lemmas 5.1 and 5.2. (cid:3) A ” B wanted At the time of Ap´ery’s proof it was by no means trivial to verify identities A “ B like the ones in Proposition 2.1 by verifying that both sides, A and B , satisfy thesame recurrence. For instance, van der Poorten’s beautiful article [23] describes thedifficulty in checking Ap´ery’s claim that the Ap´ery numbers A p n q satisfy the recur-rence (2.3), and principally attributes to Cohen and Zagier the clever insight to provethe claim using creative telescoping. Since then, Wilf and Zeilberger, with subse-quent support by many others, have developed creative telescoping into a pillar of arich computer algebraic theory devoted to automatically proving identities between,for instance, holonomic functions and sequences. We refer to [22] for a superb in-troduction to these ideas. Among the more recent developments is Schneider’s work[27], which extends the scope from holonomic sequences to a class of sequences thatalso includes nested sums of terms involving harmonic numbers. For instance, usingSchneider’s computer algebra package SIGMA, it is routine to verify that, for allintegers n ě n ÿ k “ ˆ nk ˙ ˆ n ` kk ˙ ` ´ k p H k ´ H n ` k ´ H n ´ k q ˘ “ , which we derived earlier as (5.1) and which played a crucial role in Ahlgren and Ono’sproof [1] of Beuker’s conjecture as well as Kilbourn’s proof [13] of the supercongruence(1.1).Building on these ideas, proving our main result (1.3) modulo p , instead of p , ismuch more straightforward as this corresponds to verifying Lemma 5.2 modulo p only,a task that can be performed in many different ways (for example, using Kilbourn’sstrategy from [13, Section 4]). Working modulo higher powers of p is considerablymore difficult. In the course of the derivation of Theorem 1.1 we encountered severaltechnical difficulties that were finally resolved by an intelligent cast of hypergeometricidentities. Specifically, in order to compute the congruence (1.3) we required theidentities of Proposition 2.1 as well as the equalities (5.1), (5.3), (5.4) and (5.6),reduced modulo a suitable power of p . Note that all these identities can, nowadays,be easily resolved by using computer algebraic techniques like the algorithms from[22] and [27] mentioned above. We are, however, very restricted in this productionbecause certain congruences (are expected to) remain not derivable this way. For MODULAR SUPERCONGRUENCE FOR F : AN AP´ERY-LIKE STORY 15 example, establishing (1.3) modulo p (or even p ) by using appropriate intermediateidentities sounds to us like a real challenge!There is therefore a natural need for an algorithmic approach to directly certifyingcongruences A ” B , say, when the terms A and B are holonomic. Specifically, itwould be great if such an approach could handle congruences such as (1.5), or evenjust (2.5) in the form n ÿ k “ ˆ nk ˙ ˆ n ` kk ˙ ” p´ q n n ÿ k “ ˆ nk ˙ ˆ n ` kk ˙ˆ kn ˙ p mod p q , where n “ p p ´ q{ p is an odd prime. Acknowledgements
The first and third authors would like to thank the organizers of the workshop“Modular forms in String Theory” (September 26–30, 2016) at the Banff Interna-tional Research Station, Alberta (Canada). The three authors thank the Max PlanckInstitute for Mathematics in Bonn (Germany), where part of this research was per-formed. The third author would like to thank Ling Long for several helpful insightson links between finite and truncated hypergeometric functions.
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