Introductions to log-growth and Frobenius slope filtrations.

I plan to give an overview of the literature on this topic, which originates from the variation of $p$-adic De Rham cohomology in a family of varieties over a field of positive characteristic $p$. So, I would like to start discussing the two polygons that vary over the moduli space: the geometric Hodge polygon and the Newton polygon of $p$-adic size of Frobenius eigenvalues. The evidence emerging from Dwork's theory, lead to the "Katz conjecture", stating that the Frobenius polygon stays above the Hodge one. This was proven by Mazur, who conveniently replaced the geometric Hodge polygon by a $p$-adic one, which will be the same under favorable circumstances. The filtration of the solution space by the $p$-adic order of Frobenius eigenvalues, leads to a factorization of the connection over relatively large analytic domains. This leads to good analytic formulas for the unit-root part of the zeta function. More generally, the analytic continuation of the "unit-root $F$-subcrystal", should in principle generate lots of $p$-adic formulas of Gross-Koblitz type. Moreover, in a number of interesting examples where a Frobenius structure exists, the filtration by the eigenvalues of Frobenius, tends to coincide with a more general type of filtration, the one by the "(log-)growth of solutions at the boundary of their disk of convergence". This is not at all a general phenomenon, but it suggests the independent study of $p$-adic differential equations, equipped with the log-growth filtration. Does this lead to a filtration of the differential equations themselves over large analytic domains? I will report on the progress on these topics made by André and Chiarellotto-Tsuzuki. I will try to give a couple of examples.