Numerical Calculation of Periods on Schoen's Class of Calabi-Yau Threefolds

Abstract

Through classical modularity conjectures, the period integrals of a holomorphic $3$-form on a rigid Calabi-Yau threefold are interesting from the perspective of number theory. Although the (approximate) values of these integrals would be very useful for studying such relations, they are difficult to calculate and generally not known outside of the rare cases in which we can express them exactly. In this paper, we present an efficient numerical method to compute such periods on a wide class of Calabi-Yau threefolds constructed by small resolutions of fiber products of elliptic surfaces over $\mathbf{P}^1$, introduced by C. Schoen in his 1988 paper. Many example results are given, which can easily be calculated with arbitrary precision. We provide tables in which each result is written to a precision of 30 decimal places and then compared to integrals of the appropriate modular form, to confirm accuracy.