On $L$-derivatives and biextensions of Calabi-Yau motives

TL;DR

This paper investigates differential operators related to Calabi-Yau motives, providing hypergeometric expressions for period matrices and comparing their values to derivatives of associated L-functions.

Contribution

It establishes that certain differential operators are of Picard-Fuchs type and offers explicit hypergeometric formulas for biextension period minors of Calabi-Yau motives.

Findings

Differential operators of the form DLD are of Picard-Fuchs type.

Explicit hypergeometric formulas for period minors are derived.

Numerical comparisons link period matrix values to L-function derivatives.

Abstract

We prove that certain differential operators of the form $DLD$ with $L$ hypergeometric and $D=z \frac{d}{dz}$ 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 $s=2$.