First coniveau notch of the Dwork family and its mirror

Abstract

If $X_{\lambda}$ is a smooth member of the Dwork family over a perfect field $k$, and $Y_{\lambda}$ is its mirror variety, then the motives of $X_{\lambda}$ and $Y_{\lambda}$ are equal up to motives that are in coniveau $\geq 1$. If $k$ is a finite field, this provides a motivic explanation for Wan's congruence between the zeta functions of $X_{\lambda}$ and $Y_{\lambda}$.