Relative Homotopy type and obstructions to the existence of rational points
Relative Homotopy type and obstructions to the existence of rational points
In 1969 Artin and Mazur defined the etale homotopy type $Et(X)$ of scheme $X$, as a way to homotopically realize the etale topos of a $X$. In the talk I shall present for a map of schemes $X\rightarrow S$ a relative version of this notion. We denoted this construction by $Et(X/S)$ and call it the homotopy type of $X$ over $S$. It turns out that the relative Homotopy type, can be especially useful in studying the sections of the map $X\rightarrow S$. In the special case where $S=Spec K$ is the spectrum of a field, the set of sections are just the set of rational points $X(K)$ and then the relative homotopy type $Et(X/Spec K)$ can be used to define obstructions to the existence of a rational point on $X$. When $K$ in a number fields it turns out that most known obstructions for the existence of rational points (such as Grothendieck's section obstruction , the regular and etale Brauer-Manin obstructions, etc.. ) can be obtained in this way and this point a view can be used to show new properties of these obstructions. In the case where $K$ is a general field or ring this method allows one to get new obstructions that generalized the obstructions above. This is a joint work in progress with Y. Harpaz many of the results appear in our joint paper http://arxiv.org/abs/1002.1423