Hypergeometric motives for rigid hypergeometric Calabi-Yau threefolds

Abstract: 

In 2003, Fernando Rodriguez-Villegas conjectured fourteen congruences modulo p3p3 that relate hypergeometric sums truncated at p−1p−1 to the Fourier coefficients a(p)a(p) of weight 4 modular forms. Such "supercongruences" are now understood as particular instances of hypergeometric motives (HGMs). In my talk I will review some ingredients of the theory of HGMs and illustrate its features on the fourteen examples of the underlying rigid Calabi-Yau threefolds. I will further outline some ideas in the proofs of Villegas's conjectures given recently in my joint work with Ling Long, Fang-Ting Tu and Noriko Yui.