Using the theory of Stienstra and Beukers, we prove various elementary
congruences for the numbers \sum
\binom{2i_1}{i_1}^2\binom{2i_2}{i_2}^2...\binom{2i_k}{i_k}^2, where k,n \in N,
and the summation is over the integers i_1, i_2, ...i_k >= 0 such that
i_1+i_2+...+i_k=n. To obtain that, we study the arithmetic properties of
Fourier coefficients of certain (weakly holomorphic) modular forms.Comment: 9 page

Kazalicki, M.

English

arXiv

MPG.PuRe

arXiv:1301.3303v1  [math.NT]  15 Jan 2013MODULAR FORMS, HYPERGEOMETRIC FUNCTIONSAND CONGRUENCESMATIJA KAZALICKIAbstract. Using the theory of Stienstra and Beukers [9], we provevarious elementary congruences for the numbers∑i1,i2...ik≥0i1+i2+···ik=n(2i1i1)2(2i2i2)2· · ·(2ikik)2, for k, n ∈ N.To obtain that, we study the arithmetic properties of Fourier coefficientsof certain (weakly holomorphic) modular forms.1. Introduction and statement of resultsConsider the family of elliptic curves given by the Legendre equationy2 = x(x− 1)(x− t), t ∈ C,whose period integralsΩ1(t) =∫1tdx√x(x− 1)(x− t), Ω2(t) =∫ ∞1dx√x(x− 1)(x− t),are solutions of the differential equation of Picard-Fuch typet(t− 1)Ω′′(t) + (2t− 1)Ω′(t) +14Ω(t) = 0.One finds that Ω2(t) = pi 2F1(12, 12; 1, t), where2F1 (a, b; c, t) =∞∑n=0(a)n(b)n(c)nn!tnis the Gauss hypergeometric function. This further gives identity(1.1) θ(τ) = 2F1(12,12; 1, 16l(τ))=∞∑n=0(2nn)2l(τ)n.Throughout the paper, τ ∈ H, q = epiiτ , θ(τ) =(∑n∈Z qn2)2is the classicalweight 1 theta series, and l(τ) = q−8q2+44q3−192q4+· · · is the normalized2010 Mathematics Subject Classification. 11A07, 11B83, 11B65, 11F30.Key words and phrases. modular forms, hypergeometric functions, congruences.12 MATIJA KAZALICKIelliptic modular lambda function (hauptmoduln for Γ(2)). For more detailssee [5].In this paper, we identify certain (weakly holomorphic) modular formsf(τ) = P (l(τ))θk(τ)dl(τ)dτ,for some P (t) ∈ Z[t], and k ∈ N, whose Fourier coefficients are easilyunderstood in terms of elementary arithmetic (e.g. a spliting behavior ofprimes in the quadratic extensions). Using identity (1.1), we exploit therelation (via formal group theory) between the coefficients of the powerseries P (t)2F1(12, 12; 1, 16t)k, and Fourier coefficients of f(τ) to prove someelementary congruences for the numbersAk(n) =∑i1,i2...ik≥0i1+i2+···ik=n(2i1i1)2(2i2i2)2· · ·(2ikik)2, for k, n ∈ N.Beukers and Stienstra [9] invented this approach to study congruenceproperties of Apery numbersB(n) =∞∑k=0(n + kk)(nk)2.Using the formal Brauer group of some elliptic K3-surface, they proved thatfor all primes p, and m, r ∈ N with m oddB(mpr − 12)−a(p)B(mpr−1 − 12)+(−1)p−12 p2B(mpr−2 − 12)≡ 0 (mod pr),where η6(4τ) =∑∞n=1 a(n)q2n.Many authors have subsequently studied arithmetic properties of B(n)′sand discovered similar three term congruence relations for other Apery likenumbers. For related work see [1, 3, 6, 7, 10, 11].In contrast to these result, the novelity of this paper is that we use fam-ilies of modular forms, as well as the weakly holomorphic modular form toextract information about congruence properties of numbers Ak(n). In par-ticular, even though Foureier coefficients of weakly holomorphic modularforms do not satisfy three term relation satisfied by coefficients of Heckeeigenforms, they sometimes satisfy three term congruence relation of Atkinand Swinnerton-Dyer type (see [4]), which then give rise to the three termcongruence relations satisfied by corresponding Apery like numbers.Let ∆ be the free subgroup of SL2(Z) generated by the matrices A = ( 1 20 1 )andB = ( 1 02 1 ). Note that Γ(2) = {±I}∆, and that Γ1(4) =(1/2 00 1)∆(1/2 00 1)−1.Divisors of θ(τ), l(τ), and 1− 16l(τ) are suported at cusps. For an integerMODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 3k, we denote by Mk(∆) and Sk(∆) the spaces of modular forms and cuspforms of weight k for group ∆.First we identify some (weakly holomorphic) modular forms for ∆ thathave “simple” Fourier coefficients.Theorem 1.1.For n ∈ N, lethn(τ) = θ(τ)6n+1(1− 16l(τ))⌊n+12⌋l(τ)2n =∞∑m=1an(m)qm.Then hn(τ) ∈ S6n+1(∆), and for a prime p ≡ (−1)n+1 (mod 4), we havethat an(p) = 0.Theorem 1.2.a) The modular formf1(τ) = l(τ)(1 − 16l(τ))θ(τ)5 =∞∑n=1b1(n)qn ∈M5(∆)is the cusp form with complex multiplication by Q(i). In particular,for a prime p > 2 we haveb1(p) ={2x4 − 12x2y2 + 2y4 if p ≡ 1 (mod 4), and p = x2 + y2,0 if p ≡ 3 (mod 4).b) For an integer n > 2, the Fourier coefficients c1(n) of the weaklyholomorphic modular form g1(τ) = l(τ)2(1−16l(τ))2θ(τ)5 =∑∞n=2 c1(n)qnsatisfy the following congruence relationb1(n) ≡ 108c1(n) (mod n3).c) For integer n > 1, letfn(τ) = θ(τ)6n−6(1− 16l(τ))⌊n−12⌋l(τ)2n−2f1(τ) =∞∑m=1bn(m)qm.Then for a prime p ≡ (−1)n (mod 4), we have that bn(p) = 0.Corollary 1.3. Let p > 3 be a prime, and r ∈ N. If p ≡ 1 (mod 4), let xand y be integers such that p = x2 + y2. Denote by D3(n) = A3(n − 1) −16A3(n− 2). Then the following congruences hold(1.2)A3(mpr−1)−b1(p)A3(mpr−1−1)+(−1p)p4A3(mpr−2−1) ≡ 0 (mod pr),4 MATIJA KAZALICKI(1.3)D3(mpr−1)−b1(p)D3(mpr−1−1)+(−1p)p4D3(mpr−2−1) ≡ 0 (mod pr−(−1p )+12 ).In particular, we have(1.4) A3(p− 1) ≡{16x4 (mod p) if p ≡ 1 (mod 4),0 (mod p) if p ≡ 3 (mod 4).(1.5) D3(p− 1) ≡{427x4 (mod p) if p ≡ 1 (mod 4),0 (mod p) if p ≡ 3 (mod 4).For integers n ≥ 1 and m ≥ 0, define the sequences Bn(m) and Cn(m)by the following identities(1− 16t)⌊n−12⌋t2n−12F1(12,12; 1, 16t)6n−1=∞∑m=0Bn(m)tm,(1− 16t)⌊n−12⌋t2n−22F1(12,12; 1, 16t)6n−3=∞∑m=0Cn(m)tm.Corollary 1.4. For n ∈ N and a prime p > 2, we have that Bn(p− 1) ≡ 0(mod p), if p ≡ (−1)n+1 (mod 4). Moreover, Cn(p − 1) ≡ 0 (mod p) ifp ≡ (−1)n (mod 4).Example. Let p > 2 be a prime. Consider the coefficients of 2F1(12, 12; 1, 16t)2.The corresponding modular forml(τ)(1− 16l(τ))2F1(12,12; 1, 16l(τ))4=∞∑k=1(−1)k (σ3(k/2)− σ3(k)) qkis an Eisenstein series whose pth Fourier coefficient is ≡ 1 mod p (if k is odd,we define σ3(k/2) to be 0). Hence Lemma 4.2 impliesp−1∑k=0(2kk)2(2(p− 1− k)p− 1− k)2≡ 1 (mod p),or equivalently(p− 1p−12)4≡ 1 (mod p).2. AcknowledgementsI would like to thank Max Planck Institute for Mathematics in Bonnfor the excelent working environment and the financial support that theyprovided for me during my stay in March of 2012.MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 53. Modular forms for ∆3.1. Preliminaries. Let(3.1) η(τ) = q112∞∏n=1(1− q2n)be a Dedekind eta function (recall q = epiiτ ). As we mentioned in theintroduction(3.2) l(τ) =η(2τ)16η(τ/2)8η(τ)24and(3.3) 1− 16l(τ) =η(τ/2)16η(2τ)8η(τ)24are modular functions for ∆. They are holomorphic onH, and l(τ) 6= 0, 1/16for all τ ∈ H.Modular curve X(2) has three cusps: 0, 1, and∞. As functions on X(2),l(τ) and 1− 16l(τ) have simples poles at ∞ and zeros of order 1 at cusps 0and 1 respectively.It is well known that(3.4) θ(τ) =η(τ)10η(τ/2)4η(2τ)4is a modular form of weight 1 for ∆. It has a zero at cusp 1 of order 1/2(cusp 1 is irregular).We will later need the following dimension formula for the spaces of cuspforms. Let Γ be a finite index subgroup of SL2(Z) of genus g such that−I /∈ Γ. For k odd, [8, Theorem 2.25] gives the following formula for thedimension of Sk(Γ)(3.5) dimSk(Γ) = (k − 1)(g − 1) +12(k − 2)r1 +12(k − 1)r2 +j∑i=1ei − 12ei,where r1 is the number of regular cusp, r2 is the number of irregular cusps,and e′is are the orders of elliptic points. Since ∆ has no elliptic points (it isa free group), it follows that dimS5(∆) = 1.3.2. Identities. In this subsection we list some technical facts and identi-ties which will be used later in the proofs of the theorems. The proofs ofthese statements are straightforward, so detail are ommited.6 MATIJA KAZALICKILemma 3.1. LetΨ(τ) = (1− 16l(τ))1/2l2(τ)θ(τ)6 = q2 +∞∑m=2a(m)q2m.Then, Ψ(τ) ∈ S6(Γ0(4)) is a newform. Moreover a(2) = 0, hence the co-efficients of the Fourier expansion of Ψ(τ) in q are supported at integerscongruent to 2 mod 4.Lemma 3.2. The following identity holdsη(τ/2 + 1/2)η(τ/2)=η(τ)3η(τ/2)2η(2τ).Proof. It follows from the product formula (3.1) for Dedekind eta function.The following lemma follows from the previous lemma and equations (3.3)and (3.4).Lemma 3.3. We have thatθ(τ + 1)θ(τ)= (1− 16l(τ))1/2.Denote by D = 1piiddτ= q ddq. We will need the following curious identities.Lemma 3.4. The following equality holds−112D4(1θ(τ)3)= (1− 16l(τ))l(τ)θ(τ)5 − 108(1− 16l(τ))2l(τ)2θ(τ)5.Proof. By Bol’s theorem [2], the lefthand side of the equality is a weaklyholomorphic modular form of weight 5 for group ∆. One checks that theinitial Fourier coefficients of the both sides of the equality agree, hence thelemma follows. Lemma 3.5. The following identity holdsD(l(τ)) = l(τ)(1− 16l(τ))θ(τ)2.Proof. By Bol’s theorem [2], D(l(τ)) is a weakly holomorphic modular formof weight 2. The initial Fourier coefficients of both forms agree, hence theclaim follows. 4. Proofs of the theorems4.1. Proof of Theorem 1.1 and Theorem 1.2.MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 7Proof of Theorem 1.1. It is easy to check that hn(τ) vanishes at cusps (seeSection 3.1). Denote by ν(τ) = θ(τ)12(1 − 16l(τ))l(τ)4. If n = 2k is even,then hn(τ) = θ(τ)ν(τ)k. From Lemma 3.1, it follows that ν(τ) has Fouriercoefficients supported at integers that are congruent 0 mod 4. Since θ(τ) =∑∞i=0 r2(i)qi has a property that r2(4j− 1) = 0 for j ∈ N, the claim follows.If n = 2k + 1 is odd, then hn(τ) = h1(τ)ν(τ)k. Therefore, it is enoughto prove the statement of the theorem for h1(τ). Since, by Lemma 3.1,the Fourier coefficients of θ(τ)6(1−16l(τ))1/2l(τ)2 are supported at integersthat are congruent to 2 mod 4, it follows that the coefficients of θ(τ)7(1 −16l(τ))1/2l2(τ) are supported at integers congruent to 3 mod 4. The sameis true for the coefficients of h1(τ) since Lemma 3.1 and Lemma 3.3 implythath1(τ) = θ(τ + 1)7(1− 16l(τ + 1))1/2l(τ + 1)2.Proof of Theorem 1.2. a) It is easy to check that f1(τ) is a cusp form (seeSection 3.1). The space of cusp forms S5(∆) is one dimensional (see equation(3.5)). Since it contains a CM modular form with the stated properties, theclaim follows.b) It follows from Lemma 3.4.c) Same as the proof of Theorem 1.1. 4.2. Proof of Corollary 1.3 and Corollary 1.4. We need the followingresult of Beukers [1].Proposition 4.1 (Beukers). Let p be a prime andω(t) =∞∑n=1bntn−1dta differential form with bn ∈ Zp. Let t(q) =∑∞n=1Anqn,An ∈ Zp, andsupposeω(t(q)) =∞∑n=1cnqn−1dq.Suppose there exist αp, βp ∈ Zp with p|βp such thatbmpr − αpbmpr−1 + βpbmpr−2 ≡ 0 (mod pr), ∀m, r ∈ N.Thencmpr − αpcmpr−1 + βpcmpr−2 ≡ 0 (mod pr), ∀m, r ∈ N.Moreover, if A1 is p-adic unit then the second congruence implies the first.8 MATIJA KAZALICKILemma 4.2. Define a differential formω(t) =∞∑n=1bntn−1dt,where bn ∈ Zp. Let t(q) =∑∞n=1Anqn, with An ∈ Zp and A1 = 1. Defineω(t(q)) =∞∑n=1cnqn−1dq.For a prime p, we have that bp ≡ cp (mod p).Proof. We have that∞∑n=1cnqn−1 =ddq(∞∑n=1bnnt(q)n).Hence cp mod p is equal to the (p− 1)th coefficient ofddq(bppt(q)p), whichis bp. Proof of Corollary 1.3. Since f1(τ) is an eigenform for the Hecke operatorTp, the assumptions of Proposition 4.1 are satisfied for cn = b1(n), bn =A3(n− 1), αp = b1(p), βp =(−1p)p4, and t(q) = 16l(τ). The formula (1.2)follow from identityD(l(τ))2F1(12,12; 1, 16l(τ))3= f1(τ),(which is a consequence of equation (1.1) and Lemma 3.5).To prove (1.3) we use Theorem 1.2 b) to establish three term congruencerelation from Proposition 4.1 between coefficients c1(npr) (because there isone satisfied by b1(npr)’s). Note that assumptions of Proposition 4.1 aresatisfied if we take, when p ≡ 3 (mod 4), cn = c1(n), bn = D3(n − 1),αp = b1(p), βp =(−1p)p4, and t(q) = 16l(τ) (since then b(p) = 0).If p ≡ 1 (mod 4), we need to take cn = pc1(n), bn = pD3(n − 1), αp =b1(p), βp =(−1p)p4, and t(q) = 16l(τ), hence we get a congruence relationweaker for one power of p.Formulas (1.4) and (1.5) follow from Lemma 4.2 with the choice of pa-rameters as above.Proof of Corollary 1.4. We apply Lemma 4.2 to cusp forms fn(τ) and hn(τ),where we choose t(q) to be the inverse of l(q) under composition. The claimfollows from an(p) = bn(p) = 0. MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 9References[1] F. Beukers, Another congruence for the Ape´ry numbers, J. Number Theory, 25(1987), pp. 201–210.[2] G. Bol, Invarianten linearer differentialgleichungen, Abh. Math. Sem. Univ. Ham-burg, 16 (1949), pp. 1–28.[3] F. Jarvis and H. A. Verrill, Supercongruences for the Catalan-Larcombe-Frenchnumbers, Ramanujan J., 22 (2010), pp. 171–186.[4] M. Kazalicki and A. J. Scholl, Modular forms, de Rham cohomology and con-gruences, preprint.[5] M. Kontsevich and D. Zagier, Periods, in Mathematics unlimited—2001 andbeyond, Springer, Berlin, 2001, pp. 771–808.[6] D. McCarthy, R. Osburn, and B. Sahu, Arithmetic properties for Ape´ry-likenumbers, preprint.[7] R. Osburn and B. Sahu, Congruences via modular forms, Proc. Amer. Math.Soc., 139 (2011), pp. 2375–2381.[8] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Pub-lications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers,Tokyo, 1971. Kanoˆ Memorial Lectures, No. 1.[9] J. Stienstra and F. Beukers, On the Picard-Fuchs equation and the formalBrauer group of certain elliptic K3-surfaces, Math. Ann., 271 (1985), pp. 269–304.[10] H. A. Verrill, Congruences related to modular forms, Int. J. Number Theory, 6(2010), pp. 1367–1390.[11] D. Zagier, Integral solutions of Ape´ry-like recurrence equations, in Groups andsymmetries, vol. 47 of CRM Proc. Lecture Notes, Amer. Math. Soc., Providence,RI, 2009, pp. 349–366.Department of Mathematics, University of Zagreb, Croatia, Zagreb, Bi-jenicˇka cesta 30E-mail address : mkazal@math.hr

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