In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for
$p$
-adic differential equations
$Dx=0$
on the
$p$
-adic open unit disc
$|t|<1$
, which measure the asymptotic behavior of solutions
$x$
as
$|t|\to 1^{-}$
. Then, Dwork calculated the log-growth filtration for
$p$
-adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for
$(\varphi ,\nabla )$
-modules over
$K[\![t]\!]_0$
, which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to
$(\varphi ,\nabla )$
-modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.
Jacobi sum, differential equation modulo prime and logarithmic growth
