Gamma factors for Asai representations of ${\mathrm{GL}}_2$

Abstract

Let $E$ be a quadratic semisimple extension of a local field $F$ of characteristic zero. We determine explicit relation between gamma factors for Asai representations of $R_{E/F}{\mathrm{GL}}_{2/E}$ defined by the Weil-Deligne representations and local zeta integrals. When $E=F\times F$, the results were due to Henniart and Jacquet. We completed the theory in this article based on explicit calculation.