On L-derivatives and biextensions of Calabi–Yau motives

Abstract

We prove that certain differential operators of the form $ DLD $ with $ L $ hypergeometric and $ D=z\frac{\partial }{dz} $ are of Picard–Fuchs type. We give closed hypergeometric expressions for minors of the biextension period matrices that arise from certain rank 4 weight 3 Calabi–Yau motives presumed to be of analytic rank 1. We compare their values numerically to the first derivative of the $ L $ -functions of the respective motives at $ s=2 $ .

Keywords

• Birch-Swinnerton-Dyer-type conjecture
• Calabi–Yau motives
• hypergeometric differential equations