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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