# Posts Tagged ‘rational points’

## London workshop on arithmetic geometry and homotopy theory

Posted by Andreas Holmstrom on May 8, 2012

At Imperial College London, from May 30th to June 1st. Many interesting talks, but the main theme appears to be the recent work of Harpaz and Schlank in which they use etale homotopy theory to study rational points and in particular obstructions to the local-global principle. Their big preprint is available here.

Related to this theme there is also a longer summer course in Lausanne in July/August with lectures by Harpaz and Schlank, as part of the of the EPFL program on rational points and algebraic cycles.

Towards the end of the paper, he speculates about a possible connection between the homotopy types of a variety and rational points on the variety. The study of rational points is one of the main themes of arithmetic geometry, as they correspond to integer or rational solutions of (systems of) polynomial equations. The famous section conjecture of Grothendieck, explained in these notes of Kim, is supposed to give a conceptual proof of Faltings’ theorem, aka the Mordell conjecture. Faltings’ theorem says that a curve of genus at least 2, defined over $\mathbb{Q}$, only has a finite number of rational points. Toen suggests a generalization of the section conjecture to higher-dimensional varieties, using his notion of homotopy types.
Another main theme of arithmetic geometry is L-functions of various kinds. To any variety over $\mathbb{Q}$, one can attach an L-function, which encodes lots of information about the arithmetic properties of the variety. Many outstanding conjectures in number theory are formulated in terms of these functions, for example the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture, as well as many other more accessible conjectures. The building blocks of an L-function are precisely the various Weil cohomology groups, and one could speculate about the significance of Toen’s conceptual approach to Weil cohomologies. Could it give us some new tools for approaching questions about L-functions? Or could it be that L-functions are not the right thing to consider, but that the notion of homotopy types could lead us to some better objects of study?