Twisted period relations for Lauricella's hypergeometric functions $F_{A}$

Abstract

We study Lauricella's hypergeometric function $F_{A}$ of $m$ variables and the system $E_{A}$ of differential equations annihilating $F_{A}$, by using twisted (co)homology groups. We construct twisted cycles with respect to an integral representation of Euler type of $F_{A}$. These cycles correspond to $2^{m}$ linearly independent solutions to $E_{A}$, which are expressed by hypergeometric series $F_{A}$. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's $F_{A}$.