Supercongruences for rigid hypergeometric CalabiYau threefolds
Abstract
We establish the supercongruences for the fourteen rigid hypergeometric CalabiYau threefolds over $\mathbb Q$ conjectured by RodriguezVillegas in 2003. Our first method is based on Dwork's theory of $p$adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in executing the both approaches are the modularity of the underlying CalabiYau threefolds and a $p$adic perturbation method applied to hypergeometric functions.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.01663
 Bibcode:
 2017arXiv170501663L
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory;
 11F33;
 11T24;
 14G10;
 14J32;
 14J33;
 33C20
 EPrint:
 42 pages