Periodic Reporting for period 1 - LocalGlobal0Cycles (Local-global principles and zero-cycles)
Reporting period: 2019-10-01 to 2021-09-30
The aim of the project was to use the theory of obstruction sets to investigate, through local-global considerations, both the quantitative and qualitative arithmetic behaviours of ``zero-cycles'', which can be viewed as generalised rational solutions to systems of polynomial equations. Whereas the theory of rational solutions to systems of polynomial equations is accessible from many perspectives (including analytic methods), the theory of zero-cycles seems to be much less accessible and requires a deeper geometric component. As a result, the arithmetic of zero-cycles is not nearly as studied as the arithmetic of rational points. Part of the motivation for the project was indeed to open up the area to a wider range of tools and to lay down the foundations for new research directions. Specifically, the researcher focused on the following four objectives, some of which have an open-ended, exploratory nature, due to the novelty of the research directions outlined: classification problems for zero-cycles; constructing new obstruction sets for zero-cycles; integral zero-cycles; arithmetic statistics of zero-cycles.
Due to the close relationship between rational points over general number fields and zero-cycles, it is natural to study the arithmetic of rational points over general number fields first, and then try to transfer the acquired knowledge to the realm of zero-cycles. In this direction, the Researcher has obtained several results: she has shown that the étale-Brauer obstruction is the only one for strong approximation for homogeneous spaces of connected linear algebraic groups (with the results published in the Proceedings of the American Mathematical Society); in a collaboration with Nick Rome, she has shown that, by replacing Schinzel hypothesis with a Bateman-Horn conjecture on average, one can prove interesting arithmetic statistical results; in a collaboration with Alexis Johnson and Rachel Newton, she has proven uniform bound results for Brauer groups of Kummer varieties, with applications to effectivity results for the Brauer-Manin set of Kummer surfaces; in a collaboration with Jennifer Park and Alexandra Shlapentokh, she has shown that the inverse Galois problem can be reduced to Hilbert's tenth problem.
All the obtained results are novel, and some of them introduce new ideas that potentially open the field to new techniques and researc directions. All the results have proven to be of great interest to the arithmetic geometry community.