Hidden structures on curves

Let V be the ring of integers of a finite extension of Q(p) and let X be a proper curve over V with semistable special fiber and smooth generic fiber. In this article we explicitly describe the Frobenius and monodromy operators on the log crystalline cohomology of X with values in a regular log F-isocrystal in terms of p-adic integration. We have a version for open curves and as an application we prove that two differently defined L-invariants, attached to a split multiplicative at p new elliptic eigenform, are equal.