Apparaît dans la collection : Summer School 2022 - Cohomology, Geometry and Explicit number theory (COGENT)

In the problem of deciding integer or rational solutions of polynomial equations (i.e. finding integer/rational points of a variety), we often first look at the “local” solutions over all the Q_p. When does a collection of local solutions give rise to an honest global solution over Q? There are necessary conditions given by certain cohomological groups like the Brauer group, and these conditions sometimes turn to be sufficient, defining the Brauer-Manin obstruction. We can also use such tools to study the problem of weak-approximation. Going beyond number fields, we can consider such problems of certain function fields of the shape C(X,Y), C((X,Y)), Q_p(X) etc and I’ll try to present some of my results on such problems.