Number Theory Seminar: Abinhandan

Number Theory Seminar: Abinhandan (IMJ-PRG)

(phi, Gamma)-modules and syntomic complexes

In the first part of the talk, we will look at the classification of p-adic crystalline representations of the absolute Galois group G_{Q_p} of the field of p-adic numbers Q_p, in terms of certain (phi, Gamma)-modules known as Wach modules. Then, we will define syntomic complexes with coefficients in a Wach module, and relate its cohomology to the Galois cohomology of the associated crystalline representation. In the second part, we will look at a generalisation of these results to the case of “small” affine algebras, i.e. we will classify p-adic crystalline representations of a certain etale fundamental group in terms of relative Wach modules, and compute the Galois cohomology of such representations using syntomic complexes. If time permits, we will also look at a globalisation of these statements to smooth proper p-adic formal schemes.