Frobenius structure and $p$-adic zeta values

TL;DR

This paper proves that for certain Calabi-Yau families, the $p$-adic Frobenius structure matrix entries explicitly involve rational combinations of $p$-adic zeta values, confirming a conjecture for these cases.

Contribution

It establishes the conjectured appearance of $p$-adic zeta values in the Frobenius matrices of specific Calabi-Yau hypersurfaces, expanding understanding of $p$-adic structures in algebraic geometry.

Findings

Frobenius matrix entries are rational linear combinations of $zeta_p(k)$

The phenomenon holds for simplicial and hyperoctahedral Calabi-Yau families

Confirmed conjecture for these specific Calabi-Yau hypersurfaces.

Abstract

For differential operators of Calabi-Yau type, Candelas, de la Ossa and van Straten conjecture the appearance of $p$-adic zeta values in the matrix entries of their $p$-adic Frobenius structure expressed in the standard basis of solutions near a MUM-point. We prove that this phenomenon holds for simplicial and hyperoctahedral families of Calabi-Yau hypersurfaces in $n$ dimensions, in which case the Frobenius matrix entries are rational linear combinations of products of $\zeta_p(k)$ with $1 < k < n$.