MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES

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1 MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES MATIJA KAZALICKI Abstract. Using the theory of Stienstra and Beukers [9], we rove various elementary congruences for the numbers ) 2 ) 2 ) 2 2i1 2i2 2ik, for k, n N. i 1,i 2...i k 0 i 1+i 2+ i k =n i 1 i 2 To obtain that, we study the arithmetic roerties of Fourier coefficients of certain weakly holomorhic) modular forms. i k 1. Introduction and statement of results Consider the family of ellitic curves given by the Legendre equation whose eriod integrals Ω 1 t) = 1 t y 2 = xx 1)x t), t C, dx xx 1)x t), Ω 2 t) = 1 dx xx 1)x t), are solutions of the differential equation of Picard-Fuch tye tt 1)Ω t) + 2t 1)Ω t) + 1 Ωt) = 0. 4 One finds that Ω 2 t) = π 2 F 1 1, 1; 1, t), where 2 2 a) n b) n 2F 1 a, b; c, t) = c) n n! tn is the Gauss hyergeometric function. This further gives identity 1 1.1) θτ) = 2 F 1 2, 1 ) ) 2 2n 2 ; 1, 16lτ) = lτ) n. n Throughout the aer, τ H, q = e πiτ, θτ) = n Z qn2 ) 2 is the classical weight 1 theta series, and lτ) = q 8q 2 +44q 3 192q 4 + is the normalized 2010 Mathematics Subject Classification. 11A07, 11B83, 11B65, 11F30. Key words and hrases. modular forms, hyergeometric functions, congruences. 1 n=0 n=0

2 2 MATIJA KAZALICKI ellitic modular lambda function hautmoduln for Γ2)). For more details see [5]. In this aer, we identify certain weakly holomorhic) modular forms fτ) = P lτ))θ k τ) dlτ) dτ, for some P t) Z[t], and k N, whose Fourier coefficients are easily understood in terms of elementary arithmetic e.g. a sliting behavior of rimes in the quadratic extensions). Using identity 1.1), we exloit the relation via formal grou theory) between the coefficients of the ower series P t) 2 F 1 1, 1; 1, 16t) k 2 2, and Fourier coefficients of fτ) to rove some elementary congruences for the numbers ) 2 ) 2 ) 2 2i1 2i2 2ik A k n) =, for k, n N. i 1,i 2...i k 0 i 1 +i 2 + i k =n i 1 i 2 Beukers and Stienstra [9] invented this aroach to study congruence roerties of Aery numbers ) ) 2 n + k n Bn) =. k k k=0 Using the formal Brauer grou of some ellitic K3-surface, they roved that for all rimes, and m, r N with m odd ) ) ) m r 1 m r 1 1 B a)b + 1) 1 m 2 2 r 2 1 B 0 mod r ), where η 6 4τ) = an)q2n. Many authors have subsequently studied arithmetic roerties of Bn) s and discovered similar three term congruence relations for other Aery like numbers. For related work see [1, 3, 6, 7, 10, 11]. In contrast to these result, the novelity of this aer is that we use families of modular forms, as well as the weakly holomorhic modular form to extract information about congruence roerties of numbers A k n). In articular, even though Foureier coefficients of weakly holomorhic modular forms do not satisfy three term relation satisfied by coefficients of Hecke eigenforms, they sometimes satisfy three term congruence relation of Atkin and Swinnerton-Dyer tye see [4]), which then give rise to the three term congruence relations satisfied by corresonding Aery like numbers. Let be the free subgrou of SL 2 Z) generated by the matrices A = ) and B = ). Note that Γ2) = {±I}, and that Γ 1 4) = ) 1/ /2 0 ) Divisors of θτ), lτ), and 1 16lτ) are suorted at cuss. For an integer i k

3 MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 3 k, we denote by M k ) and S k ) the saces of modular forms and cus forms of weight k for grou. First we identify some weakly holomorhic) modular forms for that have simle Fourier coefficients. Theorem 1.1. For n N, let h n τ) = θτ) 6n lτ)) n+1 2 lτ) 2n = a n m)q m. Then h n τ) S 6n+1 ), and for a rime 1) n+1 mod 4), we have that a n ) = 0. Theorem 1.2. a) The modular form f 1 τ) = lτ)1 16lτ))θτ) 5 = m=1 b 1 n)q n M 5 ) is the cus form with comlex multilication by Qi). In articular, for a rime > 2 we have { 2x b 1 ) = 4 12x 2 y 2 + 2y 4 if 1 mod 4), and = x 2 + y 2, 0 if 3 mod 4). b) For an integer n > 2, the Fourier coefficients c 1 n) of the weakly holomorhic modular form g 1 τ) = lτ) lτ)) 2 θτ) 5 = n=2 c 1n)q n satisfy the following congruence relation c) For integer n > 1, let b 1 n) 108c 1 n) mod n 3 ). f n τ) = θτ) 6n lτ)) n 1 2 lτ) 2n 2 f 1 τ) = b n m)q m. m=1 Then for a rime 1) n mod 4), we have that b n ) = 0. Corollary 1.3. Let > 3 be a rime, and r N. If 1 mod 4), let x and y be integers such that = x 2 + y 2. Denote by D 3 n) = A 3 n 1) 16A 3 n 2). Then the following congruences hold 1.2) ) 1 A 3 m r 1) b 1 )A 3 m r 1 1)+ 4 A 3 m r 2 1) 0 mod r ),

4 4 MATIJA KAZALICKI 1.3) ) 1 D 3 m r 1) b 1 )D 3 m r 1 1)+ 4 D 3 m r 2 ) 1) 0 mod +1 r 1 2 ). In articular, we have 1.4) A 3 1) 1.5) D 3 1) { 16x 4 mod ) if 1 mod 4), 0 mod ) if 3 mod 4). { 4 27 x4 mod ) if 1 mod 4), 0 mod ) if 3 mod 4). For integers n 1 and m 0, define the sequences B n m) and C n m) by the following identities 1 16t) n t 2n 1 2F 1 2, 1 ) 6n 1 2 ; 1, 16t = B n m)t m, 1 16t) n 1 2 t 2n 2 2F 1 1 2, 1 2 ; 1, 16t ) 6n 3 = m=0 C n m)t m. Corollary 1.4. For n N and a rime > 2, we have that B n 1) 0 mod ), if 1) n+1 mod 4). Moreover, C n 1) 0 mod ) if 1) n mod 4). Examle. Let > 2 be a rime. Consider the coefficients of 2 F 1 1, 1; 1, 16t) The corresonding modular form 1 lτ)1 16lτ)) 2 F 1 2, 1 ) 4 2 ; 1, 16lτ) = 1) k σ 3 k/2) σ 3 k)) q k is an Eisenstein series whose th Fourier coefficient is 1 mod if k is odd, we define σ 3 k/2) to be 0). Hence Lemma 4.2 imlies 1 ) 2 ) 2 2k 2 1 k) 1 mod ), k 1 k k=0 ) 4 1 or equivalently 1 1 mod ). 2 k=1 2. Acknowledgements I would like to thank Max Planck Institute for Mathematics in Bonn for the excelent working environment and the financial suort that they rovided for me during my stay in March of m=0

5 MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 5 3. Modular forms for 3.1. Preliminaries. Let 3.1) ητ) = q q 2n ) be a Dedekind eta function recall q = e πiτ ). As we mentioned in the introduction 3.2) lτ) = η2τ)16 ητ/2) 8 ητ) 24 and 3.3) 1 16lτ) = ητ/2)16 η2τ) 8 ητ) 24 are modular functions for. They are holomorhic on H, and lτ) 0, 1/16 for all τ H. Modular curve X2) has three cuss: 0, 1, and. As functions on X2), lτ) and 1 16lτ) have simles oles at and zeros of order 1 at cuss 0 and 1 resectively. It is well known that 3.4) θτ) = ητ) 10 ητ/2) 4 η2τ) 4 is a modular form of weight 1 for. It has a zero at cus 1 of order 1/2 cus 1 is irregular). We will later need the following dimension formula for the saces of cus forms. Let Γ be a finite index subgrou of SL 2 Z) of genus g such that I / Γ. For k odd, [8, Theorem 2.25] gives the following formula for the dimension of S k Γ) 3.5) dim S k Γ) = k 1)g 1) k 2)r k 1)r 2 + j i=1 e i 1 2e i, where r 1 is the number of regular cus, r 2 is the number of irregular cuss, and e is are the orders of ellitic oints. Since has no ellitic oints it is a free grou), it follows that dim S 5 ) = Identities. In this subsection we list some technical facts and identities which will be used later in the roofs of the theorems. The roofs of these statements are straightforward, so detail are ommited.

6 6 MATIJA KAZALICKI Lemma 3.1. Let Ψτ) = 1 16lτ)) 1/2 l 2 τ)θτ) 6 = q 2 + am)q 2m. Then, Ψτ) S 6 Γ 0 4)) is a newform. Moreover a2) = 0, hence the coefficients of the Fourier exansion of Ψτ) in q are suorted at integers congruent to 2 mod 4. Lemma 3.2. The following identity holds ητ/2 + 1/2) ητ/2) = m=2 ητ) 3 ητ/2) 2 η2τ). Proof. It follows from the roduct formula 3.1) for Dedekind eta function. The following lemma follows from the revious lemma and equations 3.3) and 3.4). Lemma 3.3. We have that θτ + 1) θτ) = 1 16lτ)) 1/2. Denote by D = 1 d = q d. We will need the following curious identities. πi dτ dq Lemma 3.4. The following equality holds 1 ) 1 12 D4 = 1 16lτ))lτ)θτ) lτ)) 2 lτ) 2 θτ) 5. θτ) 3 Proof. By Bol s theorem [2], the lefthand side of the equality is a weakly holomorhic modular form of weight 5 for grou. One checks that the initial Fourier coefficients of the both sides of the equality agree, hence the lemma follows. Lemma 3.5. The following identity holds Dlτ)) = lτ)1 16lτ))θτ) 2. Proof. By Bol s theorem [2], Dlτ)) is a weakly holomorhic modular form of weight 2. The initial Fourier coefficients of both forms agree, hence the claim follows. 4. Proofs of the theorems 4.1. Proof of Theorem 1.1 and Theorem 1.2.

7 MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 7 Proof of Theorem 1.1. It is easy to check that h n τ) vanishes at cuss see Section 3.1). Denote by ντ) = θτ) lτ))lτ) 4. If n = 2k is even, then h n τ) = θτ)ντ) k. From Lemma 3.1, it follows that ντ) has Fourier coefficients suorted at integers that are congruent 0 mod 4. Since θτ) = i=0 r 2i)q i has a roerty that r 2 4j 1) = 0 for j N, the claim follows. If n = 2k + 1 is odd, then h n τ) = h 1 τ)ντ) k. Therefore, it is enough to rove the statement of the theorem for h 1 τ). Since, by Lemma 3.1, the Fourier coefficients of θτ) lτ)) 1/2 lτ) 2 are suorted at integers that are congruent to 2 mod 4, it follows that the coefficients of θτ) lτ)) 1/2 l 2 τ) are suorted at integers congruent to 3 mod 4. The same is true for the coefficients of h 1 τ) since Lemma 3.1 and Lemma 3.3 imly that h 1 τ) = θτ + 1) lτ + 1)) 1/2 lτ + 1) 2. Proof of Theorem 1.2. a) It is easy to check that f 1 τ) is a cus form see Section 3.1). The sace of cus forms S 5 ) is one dimensional see equation 3.5)). Since it contains a CM modular form with the stated roerties, the claim follows. b) It follows from Lemma 3.4. c) Same as the roof of Theorem Proof of Corollary 1.3 and Corollary 1.4. We need the following result of Beukers [1]. Proosition 4.1 Beukers). Let be a rime and ωt) = b n t n 1 dt a differential form with b n Z. Let tq) = A nq n,a n Z, and suose ωtq)) = c n q n 1 dq. Suose there exist α, β Z with β such that Then b m r α b m r 1 + β b m r 2 0 mod r ), m, r N. c m r α c m r 1 + β c m r 2 0 mod r ), m, r N. Moreover, if A 1 is -adic unit then the second congruence imlies the first.

8 8 MATIJA KAZALICKI Lemma 4.2. Define a differential form ωt) = b n t n 1 dt, where b n Z. Let tq) = A nq n, with A n Z and A 1 = 1. Define ωtq)) = c n q n 1 dq. For a rime, we have that b c mod ). Proof. We have that c n q n 1 = d ) b n dq n tq)n. ) Hence c mod is equal to the 1) th d b coefficient of dq tq), which is b. Proof of Corollary 1.3. Since f 1 τ) is an eigenform for the Hecke oerator T, the assumtions of Proosition ) 4.1 are satisfied for c n = b 1 n), b n = 1 A 3 n 1), α = b 1 ), β = 4, and tq) = 16lτ). The formula 1.2) follow from identity Dlτ)) 2 F 1 1 2, 1 2 ; 1, 16lτ) ) 3 = f 1 τ), which is a consequence of equation 1.1) and Lemma 3.5). To rove 1.3) we use Theorem 1.2 b) to establish three term congruence relation from Proosition 4.1 between coefficients c 1 n r ) because there is one satisfied by b 1 n r ) s). Note that assumtions of Proosition 4.1 are satisfied if we take, when 3 mod 4), c n = c 1 n), b n = D 3 n 1), α = b 1 ), β = 1 ) 4, and tq) = 16lτ) since then b) = 0). If 1 mod ) 4), we need to take c n = c 1 n), b n = D 3 n 1), α = b 1 ), β = 4, and tq) = 16lτ), hence we get a congruence relation 1 weaker for one ower of. Formulas 1.4) and 1.5) follow from Lemma 4.2 with the choice of arameters as above. Proof of Corollary 1.4. We aly Lemma 4.2 to cus forms f n τ) and h n τ), where we choose tq) to be the inverse of lq) under comosition. The claim follows from a n ) = b n ) = 0.

9 MODULAR FORMS, HYPERGEOMETRIC FUNCTIONS AND CONGRUENCES 9 References [1] F. Beukers, Another congruence for the Aéry numbers, J. Number Theory, ), [2] G. Bol, Invarianten linearer differentialgleichungen, Abh. Math. Sem. Univ. Hamburg, ), [3] F. Jarvis and H. A. Verrill, Suercongruences for the Catalan-Larcombe-French numbers, Ramanujan J., ), [4] M. Kazalicki and A. J. Scholl, Modular forms, de Rham cohomology and congruences, rerint. [5] M. Kontsevich and D. Zagier, Periods, in Mathematics unlimited 2001 and beyond, Sringer, Berlin, 2001, [6] D. McCarthy, R. Osburn, and B. Sahu, Arithmetic roerties for Aéry-like numbers, rerint. [7] R. Osburn and B. Sahu, Congruences via modular forms, Proc. Amer. Math. Soc., ), [8] G. Shimura, Introduction to the arithmetic theory of automorhic functions, Publications of the Mathematical Society of Jaan, No. 11. Iwanami Shoten, Publishers, Tokyo, Kanô Memorial Lectures, No. 1. [9] J. Stienstra and F. Beukers, On the Picard-Fuchs equation and the formal Brauer grou of certain ellitic K3-surfaces, Math. Ann., ), [10] H. A. Verrill, Congruences related to modular forms, Int. J. Number Theory, ), [11] D. Zagier, Integral solutions of Aéry-like recurrence equations, in Grous and symmetries, vol. 47 of CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 2009, Deartment of Mathematics, University of Zagreb, Croatia, Zagreb, Bijenička cesta 30 address: mkazal@math.hr