## Geometry Seminar - Abstracts

### Talk

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*

### Abstract

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.

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