Joint IAS/Princeton University Number Theory Seminar

We consider an ordinary linear $p$-adic differential equation \[Dy=d^ny/dx^n+a_{n-1}d^{n-1}y/dx^{n-1}+\dots+a_0y=0, a_i\in\mathbb{Z}_p[[x]][p^{-1}]\] whose formal solutions in $\mathbb{Q}_p[x]$ converge in the open unit disc $|x|<1$. In 1973, Dwork proved that $y$ has a log-growth $n-1$, that is, $|y|_{\rho}=O((\log{1/\rho})^{1-n})$ as $\rho\uparrow 1$, where $|y|_{\rho}$ denotes the $\rho$-Gaussian norm of $y$. Moreover, Dwork defined the log-growth filtration of the solution space of $Dy=0$ by measuring the log-growth of $y$. Then, Dwork conjectured that the log-growth filtration can be compared with the Frobenius slope filtration when $Dy=0$ admits a Frobenius structure. Recently, some partial results on Dwork's conjecture have been obtained by André, Chiarellotto-Tsuzuki, and Kedlaya. In this talk, we discuss the rationality of breaks of the log-growth filtration.