11 Citations
Hypergeometric Supercongruences
- Mathematics2017 MATRIX Annals
- 2019
We discuss two related principles for hypergeometric supercongrences, one related to accelerated convergence and the other to the vanishing of Hodge numbers. This is an extended abstract of a talk…
(q-)Supercongruences hit again
- Mathematics
- 2020
Using an intrinsic $q$-hypergeometric strategy, we generalise Dwork-type congruences $H(p^{s+1})/H(p^s)\equiv H(p^s)/H(p^{s-1})\pmod{p^3}$ for $s=1,2,\dots$ and $p$ a prime, when $H(N)$ are truncated…
A Whipple $$_7F_6$$ Formula Revisited
- MathematicsLa Matematica
- 2022
A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$,…
Some supercongruences of arbitrary length
- Mathematics
- 2018
We prove supercongruences modulo $p^2$ for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are $d$ copies of $1/2$ and $d$ copies of $1$…
Proof of some conjectural supercongruences of Guo and Schlosser
- Mathematics
- 2020
In this paper, we deduce certain supercongruence relations concerning truncated hypergeometric series. As particular cases, we confirm some recent conjectural supercongruences of Guo and Schlosser on…
Dwork-type supercongruences through a creative q-microscope
- MathematicsJ. Comb. Theory, Ser. A
- 2021
$p$-adic analogues of hypergeometric identities
- Mathematics
- 2017
α(α+ 1) · · · (α+ k − 1), if k ≥ 1, 1, if k = 0. It is easy to see that (1.1) absolutely converges whenever |z| < 1, or |z| = 1 and R(β1 + · · ·+ βm) > R(α0+ · · ·+αm). If (1.1) is convergent, it is…
A local-global theorem for $p$-adic supercongruences
- Mathematics
- 2019
Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of…
References
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GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
- Mathematics
- 1995
We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise…
On the supercongruence conjectures of van Hamme
- Mathematics
- 2015
In 1997, van Hamme developed $$p$$p–adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs…
Computing the Level of a Modular Rigid Calabi-Yau Threefold
- MathematicsExp. Math.
- 2004
This work recalls a result of Serre giving a bound for the conductor of “integral” two-dimensional compatible families of Galois representations and applies this result to give an algorithm that determines the level of a modular rigid Calabi-Yau threefold.
On the modularity of rigid Calabi−Yau threefolds: epilogue
- Mathematics
- 2009
In a recent paper by F. Gouvea and N. Yui, a detailed account is given of a patching argument due to Serre that proves that the modularity of all rigid Calabi–Yau threefolds defined over $$…
The 14th case VHS via K3 fibrations
- Mathematics
- 2016
We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these…
On a supercongruence conjecture of Rodriguez-Villegas
- Mathematics
- 2012
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,…
Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds
- Mathematics
- 2006
Abstract In this paper we are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the…
Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
- Mathematics
- 1993
We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric…
Zeta functions of alternate mirror Calabi–Yau families
- MathematicsIsrael Journal of Mathematics
- 2018
We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common…