In recent work with Rachel Newton, we study the p-torsion in the Brauer group of a variety over a p-adic field. Specifically, we relate the function obtained by evaluating an element of the Brauer group to Kato’s refined Swan conductor. I will describe our results and some applications to rational points on varieties.
We count monic cubic and quartic polynomials with prescribed Galois group. We'll see that the problem reduces to counting rational points on varieties. We obtain the order of magnitude for $D_4$ quartics, and show that if $d \in \{3,4\}$ then irreducible non-$S_d$ polynomials of degree $d$ are less prevalent than reducible polynomials of degree $d$. The latter confirms the cubic and quartic cases of a 1936 conjecture of van der Waerden. Joint with Rainer Dietmann.
In this talk, I'll describe a Tannakian approach to patching problems in various contexts, including field patching, and formal patching. In particular, we show that patching is effective for objects in any Noetherian algebraic stack once it is effective for coherent sheaves. As a consequence, we show that various patching results extend to torsors under algebraic groups, families of torsors over proper algebraic spaces of finite presentation, families of stable varieties over proper algebraic spaces of finite presentation, and other natural moduli problems. This gives us not only new contexts for patching, but also a natural framework for relating some of these together, which has been used to obtain new local-global principles for torsors over semiglobal fields. This is joint work with Bastian Haase and Max Lieblich.
Say you're trying to find rational points on an algebraic variety that are really close to some fixed rational point. It turns out that the best approximations line up, like airplanes in a flight corridor, along rational curves on the variety. Or at least, that's been humanity's experience with things so far. In my talk, I'm going to discuss various reasons this might be the case, both conjectural and not.
Patching, under different forms, has been used in the past for purposes such as the inverse Galois problem and the local-global principle. Recently, it has become one of the main tools in a multitude of works.
In this talk I will present an extension of the patching technique to the framework of Berkovich spaces. As an application, we will then obtain a local-global principle for function fields of curves. Finally, I will also present a first step towards such results in higher dimensions.
In this talk we discuss joint work with Marta Pieropan on the distribution of Campana points on toric varieties. We discuss how this problem leads us to studying a generalised version of the hyperbola method, which had first been developed by Blomer and Bruedern. We show how duality in linear programming is used to interpret the counting result in the context of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alvarado.