Invertible Orbifolds over Finite Fields
Marco Aldi, Andrija Perunicic
TL;DR
The paper develops an arithmetic framework for Berglund-Hübsch mirror pairs over finite fields by deforming Borisov's $p$-adic complex and employing a modified, inverse Frobenius twisted by age. Under the condition $ ext{det}(A)igm| (p-1)$, the Frobenius acts diagonally on cohomology and its eigenvalues are expressed via the $p$-adic gamma function, enabling a conjecture that crepant-resolution point counts are given by the supertrace ${ m ST}_p(A,G)$ of $p^{ ext{age}+n-1}{ m Fr}_A$ on the deformed complex’s cohomology. The authors prove the conjecture in key cases (Berglund-Hübsch elliptic curves and weighted diagonal K3 surfaces) and relate the arithmetic to congruence mirror symmetry, showing compatibility in diagonal cases but not in general non-symmetric BH pairs. They connect these counts to Gauss sums and Gross-Koblitz, thereby linking $p$-adic cohomology, orbifold Chen-Ruan data, and arithmetic mirror symmetry with explicit, testable formulas. This work advances understanding of LG/CY-type correspondences in finite-field settings and provides concrete point-count formulas for a wide class of invertible Calabi-Yau orbifolds.
Abstract
In the context of Berglund-Huebsch mirror symmetry, we compute the eigenvalues of the Frobenius endomorphism acting on a p-adic version of Borisov's complex. As a result, we conjecture an explicit formula for the number of points of crepant resolutions of invertible Calabi-Yau orbifolds defined over a finite field.