On a supercongruence conjecture of Rodriguez-Villegas
By Dermot McCarthy. 2012
53 citations recorded by the source corpus.
In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b46822ed370e90241b4f4b235145d97332bf2a1f
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- The trace of Frobenius of elliptic curves and the p-adic gamma function (2012, methodology): “The motivation for developing nGn[· · · ] is that it can often allow these results to be extended to a wider class of primes [13, 14], as we exhibit in our main result below.”
- Super congruences and Euler numbers (2011, unknown): “−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 papers [AO], [K], [MO], [M03b], [M05], [M08], [Mc] and [OS] for such advanced approach to super congruences. Recently the author’s twin brother Z.”