Tuesday 16 April 2024, 16:00 - 17:00 in HG02.032
Ishai Dan-Cohen (Ben Gurion University)
Towards Mordellic obstruction devices coming from rational motivic homotopy
theory
Let \(Z\) be an open subscheme of \(\mathrm{Spec}\ \mathbb{Z}\). A smooth scheme \(X\) over \(Z\) gives rise to a commutative algebra object \(C^*X\) in the derived category of motives over \(Z\). Some intuition for this rather abstract object comes from the fact that it remembers, for instance, the smooth de Rham complex of the complex analytic space associated to \(X\), regarded as a cdga. On the other hand, if one forgets the algebra structure, then \(C^*X\) lives in a category whose RHom's are governed by algebraic \(K\)-theory. One can define a natural filtration of \(C^*X\) by algebras \(C ^i\) similar in spirit to a Postnikov tower. Following work of I. Iwanari, in simple examples, it's possible to make \(C^i\) for small i quite explicit. We can then show that the space of augmentations of \(C^i\) has the structure of a finite type affine \(\mathbb Q\)-scheme. I'll explain how this leads to new proofs of very special cases of Siegel's theorem for curves, and I'll indicate how these techniques may be relevant for studying integral (or rational) points of higher dimensional varieties.