A P-ADIC SUPERCONGRUENCE CONJECTURE OF VAN HAMME ERIC MORTENSON Abstract. In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the context of the Beuers-lie supercongruences of Rodriguez-Villegas. This conjecture is a p-adic analog of a formula of Ramanujan.. Introduction Recently, van Hamme [vh] made several conjectures concerning p-adic analogs of several formulas of Ramanujan. In this paper we prove one of these conjectures by maing a connection between it and one of the Beuers-lie supercongruences discovered by Rodriguez-Villegas [FRV]. We begin with numbers Apery used in his proof of the irrationality of ζ( and ζ(3: n ( n ( n + A(n :, where B(n : n ( n ( n + (a : a(a + (a +, and, ( n : ( ( n,! are the standard notations for the raising factorial and binomial coefficent respectively. For p an odd prime, Beuers made the following two conjectures concerning these numbers and the coefficients of two modular forms: ( p (. A a(p (mod p, ( p (. B b(p (mod p, where a(nq n : q Π n qn 4 q 4n 4 S 4 (Γ 0 (8, and b(nq n : q Π n q4n 6 S 3 (Γ 0 (6,4 d, q : eiz. 000 Mathematics Subject Classification. Primary: 33C0; Secondary: S80.
ERIC MORTENSON Both of these conjectures have multiple guises. Another form of (. is found in [vh]: (.3 ( 3 b(p (mod p. Beuers proved these modulo p, [Be], [Be]. For (., a partial proof was given by Ishiawa [I], and a complete proof was given by Ahlgren and Ono [AO]. For (.3, proofs have been given by Ishiawa [I], van Hamme [vh], and Ahlgren [A]. (For techniques that yield a computer free version of [A], see Mortenson [M3]. Finite field analogs of classical hypergeometric series [G] play large roles in [AO], [A], and [M3]. In [FRV], Rodriguez-Villegas discovered numerically a number of Beuers-lie supercongruences. This was motivated by his joint wor with Candelas and de la Ossa [COV], where they studied Calabi-Yau manifolds over finite fields. For proofs of some of these congruences, see [M], [M], [M3], and [K], where again the theory of finite field analogs of classical hypergeometric series plays a large role. (The supercongruence of [K] is also found in the list of conjectures in [vh]. For example, we have the following: Theorem.. [M], [M] Let p be an odd prime p 5, and denote by φ p (x the Legendre symbol modulo p. Then ( (! φ p ( (mod p. It should be noted that the proof in [M] is software pacage dependent, whereas the proof in [M] is not. The motivation for this research comes from a recent paper by McCarthy and Osburn [McO], where they prove the following conjecture of van Hamme [[vh], page 6, (A.]: Conjecture. [vh] (A. If p is an odd prime, then 5 (4 + p b(p (mod p 3. Upon examining their proof, one is immediately struc by the strong shadow of one of Beuers supercongruences, e.g. (.3. In other words, one finds, 5 (4 + p ( 3 (mod p 3. This leads one to speculate as to the potential role of other Beuers-lie supercongruences, and brings us to the goal of this paper. With this idea, Theorem., and the arsenal of transformation formulas for generalized hypergeometric series found in Bailey s tract [B], we prove the next supercongruence conjecture on van Hamme s list [[vh], page 6, (B.]:
A p-adic SUPERCONGRUENCE CONJECTURE OF VAN HAMME 3 Conjecture. [vh] (B. Let p be an odd prime, then 3 (4 + p φ p ( (mod p 3. This is the p-adic analog of the following formula of Ramanujan [[vh], page 6, (B.]: (4 + 3 The paper is organized as follows. In section 3 we recall necessary bacground information and in section 4 we prove a technical lemma. In section 5, we prove van Hamme s conjecture (B... Acnowledgements The author would lie to than George Andrews, Karl Mahlburg, Ken Ono, and the referee for their helpful comments and suggestions. 3. Bacground information We begin this section by recalling a theorem of Morley. Theorem 3.. [Mo] Let p be an odd prime. Then ( p p φ p ( (mod p 3. We recall the gamma function as defined by Weierstrass and found in [[WW], Ch. XII]. Definition 3.. Let z C, then {( Γ(z : zeγz Π n where γ is Euler s constant. + z n e z } n, From this definition it is apparent that Γ(z is analytic except at the points z 0,,,..., where it has simple poles. We also recall some gamma function properties, which are found in [[WW], CH. XII]. Proposition 3.3. Let z C. Then the following are true,. Γ(,. Γ(, 3. Γ(z + zγ(z, 4. Γ(zΓ z sin(z, 5. z Γ(zΓ(z + Γ( Γ(z.
4 ERIC MORTENSON Lastly, we recall some generalized hypergeometric series transformations as found in Bailey s tract [B]. The first may be viewed as a specialization of Whipple s famous 7 F 6 transformation [B, p.8]. The specialization is (3. where ( a, + a, b, c, d, e 6F 5 a, + a b, + a c, + a d, + a e ( Γ( + a dγ( + a e + a b c, d, e Γ( + aγ( + a d e 3 F + a b, + a c, ( a0, a,..., a n n+f n t b,..., b n (a 0 (a (a n t!(b (b n. is the classical hypergeometric series. We recall a transformation formula for a terminating 3 F [B, p., eq. ]: (3. Γ(e + mγ(f + m Γ(sΓ(eΓ(f ( a, b, m 3 F e, f ( m Γ aγ b Γ aγ(s mγ b m 3F ( e b, f b, m b m, s m where s : e + f a b + m, and m is a positive integer. Here we are careful, because we will eventually loo at the limit as a goes to one. 4. A technical lemma Here we show a technical lemma, which appears in the proof of Conjecture (B.. Lemma 4.. Let n t + be a postive odd integer, and ω + i 3. Then Proof. We use Proposition 3.3 (4 to write Γ( + ω n Γ( + ω n Γ( ω n Γ( ω n n n. Π ( + 3n Γ( + ω n Γ( + ω n Γ( ω n Γ( ω n sin( ω n sin( ω n Γ( ω n Γ( ω n Γ( ω n Γ( ω n. Writing sine in terms of the exponential function and simplifying, we obtain sin( ω n sin( ω n Γ( ω n Γ( ω n Γ( ω n Γ( ω n 4 e 3 n + e 3 n Γ( ω n Γ( ω n Γ( ω n Γ( ω n.
A p-adic SUPERCONGRUENCE CONJECTURE OF VAN HAMME 5 Using the duplication formula of Proposition 3.3 (5, yields 4 e 3 n + e 3 n Γ( ω n Γ( ω n Γ( ω n Γ( ω n 4 e 3 n + e 3 n e 3 n + e 3 n ωn ωn Γ( Γ( ωn Γ( Γ( ω n n Γ( ωnγ( ω n, where in the last step we have have used Γ(. Recalling n t+ where t is a nonnegative integer, we find n e 3 n + e 3 n Γ( ωnγ( ω n n e 3 n + e 3 n Γ(t + n i 3Γ(t + + n i 3 n e 3 n + e 3 n n Π ( + 3 4 n Γ( n i 3Γ( + n i 3 n sin( e 3 n + e 3 n n i 3 n n Π ( + 3 n, 4 n Π ( + 3 4 n where the penultimate line follows from repeated application of Proposition 3.3 (3, and the ultimate line follows from Proposition 3.3 (4. The conclusion of the lemma easily follows. 5. Proof of Conjecture (B. Using an idea of McCarthy and Osburn [McO], we mae the following choice of variables in equation (3.. (An idea ain to this is also found in [vh]. We let a, b ω p, c ω p,d p,e where ω + i 3, to produce ( 6F, 5 4, ω p, ω p p, 5 4, + ω p, + ω p, + p, Γ( + p Γ( Γ( 3 Γ(p 3F ( p, p, + ω p, + ω p This one can consider (mod p 3, and by using appropriate gamma function properties, one obtains (5. 3 (4 + p ( p ( p ( + ω p ( + ω p. (mod p 3. We now focus on the coefficient of p on the right hand side of equation (5.. We recall equation (3. and let a + ǫ, b p,m, e + ω p,f + ω p, to obtain
6 ERIC MORTENSON Γ( ω p Γ( ω p ( + ǫ, Γ( p ǫγ( + ω p Γ( + ω p 3F ( Γ( ǫγ( + p Γ( ǫγ( ǫγ( 3F p, p + ω p, + ω p ( ω p, ω p, p, ǫ. Isolating the hypergeometric series on the left, taing the limit as ǫ goes to 0, and then examining everything (mod p 3 yields, ( p ( p ( + ω p ( + ω p ( Γ(p+ Γ(p Γ( + ω p Γ( + ω p Γ( Γ( ω p Γ( ω p ( (! (mod p 3. However, we need only be concerned with the above equation (mod p. Employing Proposition 3.3 (3 and (5 produces ( p ( p ( + ω p ( + ω p ( p (p! Γ( + ω p Γ( + ω p Γ( ω p Γ( ω p ( (! (mod p. From Lemma 4., we have
A p-adic SUPERCONGRUENCE CONJECTURE OF VAN HAMME 7 ( p ( p ( + ω p ( + ω p ( p (p! ( (p! ( p Π Π ( p ( + 3p ( p ( (! (mod p, ( (! (mod p ( (! (mod p where the last line follows from Morley s result Theorem 3.. We now return to equation (5. and see 3 (4 + p p ( p ( p ( + ω p ( + ω p where the last congruence follows from Theorem.. ( (! (mod p (mod p 3 ( (! (mod p 3 p φ p ( (mod p 3 References [A] S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics. Dev. Math., 4, Kluwer, Dodrecht, (00. [AO] S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apery number supercongruences, J. Reine Angew. Math., 58 (000, pages 87-. [B] W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, no. 3, Cambridge University Press, (935. [Be] F. Beuers, Some congruences for Apery numbers, J. Number Theory,, (985, pages 4-55. [Be] F. Beuers, Another congruence for the Apery numbers, J. Number Theory, 5, (987, pages 0-0. [COV] P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas, Calabi-Yau manifolds over finite fields I, http://xxx.lanl.gov/abs/hep-th/0033. [G] J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 30, (987, pages 77-0. [I] T. Ishiawa, On Beuers congrunce, Kobe J. Math. 6, (989, pages 49-5. [K] T. Kilbourn, An extension of the Apery number supercongruences, Acta Arith. 3 (006, no. 4, pages 335-348.
8 ERIC MORTENSON [McO] [M] [M] [M3] D. McCarthy and R. Osburn, A p-adic analogue of a formula of Ramanujan, preprint. E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, Journal of Number Theory, 99 (003, pages 39-47. E. Mortenson, Supercongruences Between Truncated F Hypergeometric Functions and their Gaussian Analogs, Trans. Amer. Math. Soc., 335 (003, pages 987-007. E. Mortenson, Supercongruences for the truncated n+f n hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc., 33, (005, pages3-330. [Mo] F. Morley, Note on the congruence 4n ( n (n!/(n!, where n + is prime, Annals of Math., 9 (895, pages 68-70. [O] K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc., 350 (998, pages 05-3. [O] K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series, CBMS Regional Conference Series in Mathematics, 0, Providence, RI, 004. [FRV] F. Rodriguez-Villegas, Hypergeometric Families of Calabi-Yau Manifolds. Calabi-Yau varieties and mirror symmerty, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 003. [WW] E.T. Whittaer and G.N. Watson, Modern Analysis, fourth ed., Cambridge University Press, 97. [vh] L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, p-adic functional analysis (Nijmegen, 996, pages 3-36, Lecture Notes in Pure and Appl. Math., 9, Deer, 997. Department of Mathematics, Penn State University, University Par, Pennsylvania 680 E-mail address: mort@math.psu.edu