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

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.