First coniveau notch of the Dwork family and its mirror

TL;DR

This paper investigates the relationship between the motives of Dwork family varieties and their mirrors, revealing a coniveau filtration connection that explains certain zeta function congruences over finite fields.

Contribution

It establishes a motivic equivalence up to coniveau 1 between Dwork family members and their mirrors, providing a new perspective on mirror symmetry and zeta function congruences.

Findings

Motives of Dwork family members and their mirrors are equal up to coniveau ≥ 1.

Provides a motivic explanation for Wan's zeta function congruence.

Connects mirror symmetry with coniveau filtration in motives.

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 $Y_{\lambda}$.