We explain how to equip the de Rham cohomology of curves and of abelian varieties over a p-adic field K with a canonical integral structure, i.e. a functorial (in K-morphisms) O_K-lattice. For curves, the construction uses a certain class of proper flat models and relative dualizing sheaves, while for abelian varieties it uses the canonical extension of Mazur-Messing. For curves that have a model with generically smooth closed fiber, the canonical isomorphism between the de Rham cohomology of the curve and that of its Jacobian identifies the two constructions; we also explain the proof of this fact.

The curves portion of this talk is contained contained (and generalized) in Canonical Integral Structures on the de Rham Cohomology of Curves