Supercongruences for truncated n+1Fn hypergeometric series with applications to certain weight three newforms

By Eric T. Mortenson. 2004

120 citations recorded by the source corpus.

We prove general results on supercongruences between values of truncated n+1 F n hypergeometric functions and their character analogs. As a consequence of the main results of this paper, we prove Beukers-type supercongruences for certain weight three newforms.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1a80882f24245b7418d0bee19d89781599d47b98

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Citations in the corpus

The trace of Frobenius of elliptic curves and the p-adic gamma function (2012, background): “Hypergeometric functions over finite fields have been applied to many areas but most interestingly perhaps has been their relationships to modular forms [Ahlgren and Ono 2000; Evans 2010; Fuselier 2010; Frechette et al. 2004; McCarthy 2012b; Mortenson 2005; Ono 1998; Papanikolas 2006] and their use ”
On sums involving products of three binomial coefficients (2010, methodology): “mod p) and hence (1.22) follows. (iii) Finally we suppose p ≡ 5 (mod 12) and write p = x2 + y2 with x odd and y even. Once again it suﬃces to show the mod p form of (1.23) in view of Mortenson’s work [M]. As Z.-H. Sun observed, (p− 5)/6+ k 2k 2k k ≡ k −5/6 2k 2k k = 3k k 6k 3k (−432)k (mod p) ”
Open Conjectures on Congruences (2009, unknown): “∤m. A conjecture of Rodriguez-Villegas [RV] has the following equivalent form: pX−1 k=0 2k k 2 3k k 108k ≡ ˆ 4x −2p (mod p2) if (p 3) = 1 & p = x2 +3y2 (x,y ∈ Z), 0 (mod p2) if p ≡ 2 (mod 3). See [M3] for related result. Mathematicayields that X∞ k=0 k 2 k k 3 k 54k = √ π 9Γ( 4 3)Γ( 7 6) and X∞ ”
On congruences related to central binomial coefficients (2009, background): “other conjecture of Roderiguez-Villeags [RV] has the following equivalent form: pX−1 k=0 2k k 2 3k k 108k ≡ ˆ 4x 2−2p (mod p ) if (p 3) = 1 & p = x2 +3y2 (x,y ∈ Z), 0 (mod p2) if ≡ 2 (mod 3). See [M3] for related result. Conjecture 3.12. Let p > 3 be a prime. If (p 7 ) = 1 and p = x2 + 7y2 wi”
Super congruences and Euler numbers (2011, background): “ 13 Conjecture1.10. Foranyprimep ≡ 1 (mod 4) wehave pX−1 k=0 36k +5 123k 6k 3k 3k k 2k k ≡ 0 (mod p2). Remark. A related conjecture of Rodriguez-Villega [RV] partially conﬁrmed by Mortenson [M05] states that if p > 3 is a prime then Xp−1 k=0 6k 3k 3k k 2k k 123k = pX−1 k=0 (6k)! (3k)!(k!)3 17”
On sums of Ap\'ery polynomials and related congruences (2011, unknown): “ ) = −1, i.e.,p ≡ 5,7 (mod 8). Also, pX−1 k=0 (−1)kA k ≡ −1 p pX−1 k=0 (−1)kA k 1 16 ≡ ˆ 4x2 −2p (mod p) ifp ≡ 1 (mod 3) andp = x2 + 3y2 (x,y ∈ Z), 0 (mod p2) ifp ≡ 2 (mod 3). (1.10) Proof. By [M05] and [Su4], we have pX−1 k=0 4k k,k,k,k 256k ≡ c(p) (mod p2) as conjectured in [RV]. (Here we only ”
A p-adic analogue of a formula of Ramanujan (2007, unknown): “2 in [16] and Corollary 5 in [21], we have that p(3)3F2(1) = { − p Γp( 34) 4 (mod p 3) if p ≡ 1 (mod 4) 0 (mod p3) if p ≡ 3 (mod 4) .”