My thesis is concerned with the cohomological study of smooth curves over a p-adic field K of characteristic zero. The central object of study is the Hodge filtration of the first de Rham cohomology group (over K) of such a curve. Specifically, I investigate integral structures on the Hodge filtration and endomorphisms of them induced by correspondences on the curve. I study several types of p-adic cohomology theories (de Rham, crystalline, rigid, Monsky-Washnitzer) that can be attached to a smooth curve over K and its Jacobian, and the relationships between them. I am particularly interested in compatibilities between these different theories, especially with regard to endomorphisms induced by correspondences,integral structures, and Frobenius.

The single greatest impetus for this work comes from Gross's beautiful paper on Galois representations and companion forms [Gro90] in which he employs the different p-adic cohomology theories above in the case of modular curves to prove Serre's "modular criterion" for the splitting of the local Galois representation attached to a mod p modular form. In the introduction to [Gro90], Gross writes:

The proof of Serre's conjecture on companion forms uses p-adic techniques, and specifically the different $p$-adic cohomology theories (de Rham, crystalline, Washnitzer-Monsky) of modular curves and their Jacobians. Here we confess that we have occasionally used rather artificial methods for defining the action of Hecke operators on these cohomology groups, and have not always checked that the actions are compatible with isomorphisms between the theories. In particular, the assertions preceding (15.4), (15.7), and (16.7) depend on an unchecked compatibility.

As a consequence of my work, I resolve this "unchecked compatibility," thereby completing Gross's work. In addition, I provide a reference for several results regarding integral structures on $p$-adic cohomology groups that play a key role in Gross's work that I have been unable to find in the literature.

I rely heavily on work of Berthelot, Bosch-Lutkebohmert, Grothendieck, Mazur-Messing, Monsky-Washnitzer, Raynaud, and Raynaud-Gruson. Throughout, the hypotheses and methods are kept as general as possible to ensure that the results apply equally to modular curves as they do to other curves of arithmetic interest (e.g. Shimura curves).