Correspondences, integral structures, and compatibilities in <italic>p</italic>-adic cohomology.

Abstract

Fix a discrete valuation ring <italic>R</italic> with fraction field <italic> K</italic> of characteristic zero and residue field <italic>k</italic>. In this thesis, we study the Hodge filtration of the first de Rham cohomology of a smooth, proper, and geometrically connected curve <italic>X_K</italic> over <italic>K</italic> and the endomorphisms of it induced by correspondences on the curve. Our investigations proceed along two courses: in one direction, we equip the Hodge filtration with a canonical integral structure, i.e. a short exact sequence of free <italic>R</italic>-modules that is functorial in finite <italic>K</italic>-morphisms of <italic>X_K</italic> and recovers the Hodge filtration of H1dR (<italic>X_K/K</italic>) after extending scalars to <italic> K</italic>. In the other direction, we suppose that <italic>k</italic> has characteristic <italic>p</italic> > 0 and employ techniques from rigid and formal geometry to study the Hodge filtration of H1dR (<italic>X_K/K</italic>)---together with its canonical integral structure---via <italic>p</italic>-adic cohomology. We give two different constructions of a canonical integral structure on the Hodge filtration of H1dR (<italic>X_K/K</italic>). One construction uses certain models of <italic>X_K</italic> over <italic>R</italic> and Grothendieck duality theory, while the other uses the Neron model of the Jacobian of <italic>X_K</italic> and the theory of rigidified extensions of Grothendieck and Mazur-Messing. We will prove under mild hypotheses that these different constructions yield the same integral structure. When <italic>k</italic> has characteristic <italic>p</italic> > 0, we can attach several different <italic>p</italic>-adic cohomology theories to <italic> X_K</italic>. We will study the comparison maps between these theories, and will be especially concerned with their interaction with endomorphisms induced by correspondences on <italic>X_K</italic> and Frobenius. A significant impetus for writing this thesis comes from Gross' beautiful paper [22] on Galois representations. In that paper, Gross' main theorem is conditional on certain compatibilities between <italic>p</italic>-adic cohomology theories. As a consequence of our work, we resolve these compatibility issues. Our work concerning canonical integral structures moreover provides a reference for several results that play a key role in [22] but that we have been unable to find in the literature.