Lectures & Talks

[PMI Number Theory Seminar] - Supercongruences for rigid hypergeometric Calabi--Yau threefold

Date: 2017 Jun 20 16:00 - 00:00
Speaker: Ling Long (Louisiana State University)
Place: Math. Bldg. 404
첨부파일: poster_Prof. Ling Long.pdf
PMI Number Theory Seminar

Speaker: Ling Long(Louisiana State University)

Date&Time: 2017.06.20(Tue), 4pm
Place: Math. Bldg. 404

Title: Supercongruences for rigid hypergeometric Calabi--Yau threefold

Abstract:
We establish the supercongruences for the fourteen rigid
hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by
Rodriguez-Villegas in 2003. Two different approaches are implemented, and
they both successfully apply to all the fourteen supercongruences. Our
first method is based on Dwork's theory of $p$-adic unit roots, and it
allows us to establish the supercongruences 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
Calabi--Yau threefold and a $p$-adic perturbation method applied to
hypergeometric functions. This is a joint project with Fang-Ting Tu,
Noriko Yui, and Wadim Zudilin.