Unit-root crystal conjecture for higher Hasse–Witt matrices

Let $f$ define a nonsingular hypersurface $X_f=\{f=0\}$, and let $H_f=H^{\cdot}_{dR}(X_f)$ be its de Rham cohomology, equipped with the Gauss–Manin connection and Frobenius maps $F_{\sigma}:H_{\sigma(f)}\otimes_R\widehat R_p\to H_f\otimes_R\widehat R_p$. Let $\bar\alpha_1$ denote the matrix whose determinant occurs in the Hasse–Witt invertibility condition, and let $g$ be the asserted rank. Unit-root crystal conjecture. When $\det\bar\alpha_1$ is invertible, there is a rank $g$ free subquotient in $H_f\otimes_R\widehat R_p$ on which both the Frobenius structure and the connection descend, with matrices given in a suitable basis by the limiting matrices

$$and$$

, respectively. Equivalently, when the Hasse–Witt matrix is invertible, these matrices describe the Frobenius operator and Gauss–Manin connection on the unit-root crystal attached to $f$. This conjecture interprets the limiting Frobenius and connection matrices as the structures on the unit-root part of the crystal associated with the hypersurface. The paper presents it as a general interpretation, with the subsequent discussion giving elementary consequences and evidence through a proposed lift of a zeta-function factor.