Relative Homotopy type and obstructions to the existence of rational points

T. Schlank, Hebrew University
Fine Hall 214

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