A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function

By Eric T. Mortenson. 2003

121 citations recorded by the source corpus.

Fernando Rodriguez-Villegas has been studying hypergeometric families of Calabi–Yau manifolds, and from his investigations he has found (numerically) many possible supercongruences. For example, he conjectures for every odd prime p that Image Here, we use the theory of Gaussian hypergeometric series, the properties of the p-adic Γ-function, and a strange combinatorial identity to prove this conjecture.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/48e1252650cb26698d3d5c2c5906a2dcae32daae

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

An extension of the Apéry number supercongruence (2006, unknown): “These have been proved by Mortenson ([10]–[12]).”
A new series for $\pi^3$ and related congruences (2010, unknown): “ 4kH k−1 k2 2k k = 7ζ(3), X∞ k=1 4kH 2k−1 k2 2k k = 21 2 ζ(3), A NEW SERIES FOR π3 AND RELATED CONGRUENCES 7 X∞ k=1 2k k k4k H(2) k = 3 2 ζ(3), X∞ k=1 4kH(2) −1 k2 2k k = π4 24 . In 2003 E. Mortenson [M1] conﬁrmed the following conjecture of F. Rodriguez-Villegas [RV] via character sums: (p−X1)/2 k=”
Open Conjectures on Congruences (2009, unknown): “We also have (pX−1)/2 k=1 1 k2 2k k ≡ (−1)(p−1)/2 4 3 Ep−3 (mod p), X p/2<k<p 2k k 2 16k ≡ −2p2E p−3 (mod p 3), and X p/2<k<p k 2k k 2 16k ≡ p2 2 Ep−3 (mod p 3). Remark. In 2003 Mortenson [M1] conﬁrmed the following conjecture of Rodriguez-Villegas [RV]: P (p−1)/2 k=0 2k k 2 /16k ≡ (−1)(”
Super congruences and Euler numbers (2011, methodology): “n hypergeometric series and the CalabiYau manifolds. The most elegant one of the 22 super congruences is as follows: pX−1 k=0 2k k 2 16k ≡ (−1)(p−1)/2 (mod p2), which was later proved by E. Mortenson [M03a] via the p-adic Γ-function and the Gross-Koblitz formula. See also K. Ono’s book [O] and the p”
On sums of Ap\'ery polynomials and related congruences (2011, unknown): “e following identities (cf. [G, (3.32) and (3.66)]): X2n k=0 (−1)k 2n k 2 = (−1)n 2n n and Xn k=0 n k 2 = 2n n . Corollary 1.4. Letp beanoddprime. (i) (Conjectured in [RV] and proved in [M03]) Wehave pX−1 k=0 2k k 2 16k ≡ −1 p (mod p2). (ii) (Conjectured by the author [Su1] and conﬁrmed in [”
Gaussian hypergeometric functions and traces of Hecke operators (2004, background): “In addition to their relationship with coefficients of modular forms, Gaussian hypergeometric functions have also played important roles in the proofs of “supercongruence” conjectures of Beukers [5], [6] and Rodriguez-Villegas [20] (see [3], [15], [16]).”
Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions (2015, methodology): “[30] Eric Mortenson.”
A p-ADIC SUPERCONGRUENCE CONJECTURE OF VAN HAMME (2008, background): “Theorem 1.1. [ M1 ], [M2] Let p be an odd prime p � 5, and denote by φp(x) the Legendre symbol modulo p. Then”
Congruences concerning Legendre polynomials (2010, methodology): “E. Mortenson, Supercongruences between truncated 2F1 by hypergeometric functions and their Gaussian analogs, Trans.”
Super congruences and Euler numbers (2010, methodology): “n hypergeometric series and the CalabiYau manifolds. The most elegant one of the 22 super congruences is as follows: pX−1 k=0 2k k 2 16k ≡ (−1)(p−1)/2 (mod p2), which was later proved by E. Mortenson [M03a] via the p-adic Γ-function and the Gross-Koblitz formula. See also K. Ono’s book [O] and the p”
On congruences related to central binomial coefficients (2009, methodology): “ for related results on p-adic valuations. Let p be an odd prime. In 2003 Roderiguez-Villeags [RV] conjectured that pX−1 k=0 2k k 2 16k ≡ (−1)(p−1)/2 (mod p2). This was later conﬁrmed by E. Mortenson [M1] via the Gross-Koblitz formula. Mortenson [M2] also used the p-adic Γ-function to conﬁrm the fol”
On a supercongruence conjecture of Rodriguez-Villegas (2012, methodology): “These cases have been proven by Mortenson [19], [20].”