Talk: Congruences connecting modular forms and truncated hypergeometric series (AG17)

Date: 2017/07/31
Occasion: SIAM Conference on Applied Algebraic Geometry, Minisymposium on Symbolic Combinatorics
Place: Georgia Tech

Abstract

We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated $_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $\zeta(3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Apéry numbers and another Apéry-like sequence. We will highlight the role of computer algebra in this work, and give explicit examples indicating the need for algorithmic approaches to certifying $A \equiv B$.

This is joint work with Robert Osburn and Wadim Zudilin.

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