On Vlasenko's formal group laws

TL;DR

The paper studies $F_f$ attached to Laurent polynomials over rings flat over $\mathbb{Z}$. It proves integrality by embedding the construction into toric geometry and identifying $F_f$ as a coordinate system for an Artin–Mazur type formal group functor attached to the ideal sheaf of a toric hypersurface $X$. In the second part, it analyzes higher Hasse–Witt matrices and shows that, under invertibility of the first matrix, the $p$-adic limit $\alpha = \lim_{s\to\infty} \alpha_{s+1}(\alpha_s^{\sigma})^{-1}$ yields the Frobenius on the Cartier–Dieudonné module of the mod $p$ reduction $\Gamma_f$. A universal $\delta$-ring framework and base change then identify the Frobenius action on the Dieudonné crystal $\mathbf{D}^{*}(\Gamma_f)$ and extract unit-root information from the (possibly singular) rigid cohomology via base change. These results connect the arithmetic of $F_f$ to $p$-adic cohomology theories (Dwork crystal, unit roots) and provide a combinatorial route to unit-root Frobenius in families of toric hypersurfaces.

Abstract

Given a Laurent polynomial over a ring flat over \(\mathbb{Z}\), Vlasenko defines a formal group law. We identify this formal group law with a coordinate system of a formal group functor, prove its integrality. When the Hasse--Witt matrix of the Laurent polynomial is invertible, Vlasenko defines a matrix by taking a certain \(p\)-adic limit. We show that this matrix is the Frobenius of the Dieudonné module of this formal group modulo \(p\).

Theorems & Definitions (34)

• Theorem 1
• Remark 2: We do not need the flatness of the hypersurface defined by $f$
• Theorem 3: = Theorem \ref{['theorem:dieudonne-module-frob']}
• Theorem 1.4
• Remark 1.5: We can assume $R$ is noetherian
• Lemma 1.9
• proof
• Lemma 1.11
• proof
• Lemma 1.13