William Messing (University of Minnesota)
Friday 10th October, 12.05-12.55pm, Carslaw 159
p-adic Hodge theory and p-adic periods
Classical Hodge theory deals with projective and smooth algebraic varieties defined over C. If X is such a variety defined over the subfield Q, then associated to X are two cohomology theories:
- The Betti cohomology of X, denoted H*B(X). It is defined as the singular cohomology of the underlying space X(C) of the complex points of X with coefficients in Q.
- The algebraic de Rham cohomology of X, denoted H*dR(X). This is defined as the hypercohomology of the de Rham complex Omega*X/Q.
We will discuss the p-adic analogues of this classical situation. Beginning with the work of Barsotti, Tate, Grothendieck during the 1950's and 1960's, the theory of p-divisible groups developed. In his Nice ICM talk, Grothendieck raised the problem of the "mysterious functor". In 1978 Fontaine introduced a ring Bcrys and a field BdR containing it, each equipped with additional structure, in order to obtain p-adic analogues of the period isomorphism described above. In Fontaine's 1982 Annals paper, a number of conjectures relating p-adic etale cohomology to crystalline and de Rham cohomology were made. These conjectures have all been established due to the work, over the last thirty years, of various mathematicians including Fontaine, Bloch, Breuil, Colmez, Faltings, Gabber, Hyodo, Illusie, Kato, Kisin, Messing, Niziol, Olsson, Tsuji, .... Just as with classical Hodge theory, p-adic Hodge theory has had important applications in number theory and arithmetic geometry, including, in particular, the proof by Khare and Wintenberger of Serre's Modularity Conjecture.