Explicit Local Solubility and Applications (24rit020)

The Banff International Research Station will host the "Explicit Local Solubility and Applications" Research in Teams workshop in Banff from December 8 - 15, 2024.

The question of whether or not a collection of equations has a solution in the integers is notoriously challenging, with the answer depending heavily on the geometry of the constituent equations. An important reduction is localization - looking at the equations modulo a prime or over the real numbers. Having solutions locally is a key necessary condition for the equations to have an integral solution and much more tractable to determine explicitly.

The proposed research involves determining explicit and exact expressions for how often certain families of equations have local solutions, which can sometimes yield explicit results for how often an equation has an integral solution. In particular, we propose an approach to determine how often a degree 3 polynomial in n+1 variables has an integral zero, by reducing to the local probabilities (via recent work of Browning, Le Boudec, and Sawin) and determining them by a careful recursive argument. We also propose to find how often certain families of superelliptic curves fail to have integral solutions by determining how often they possess an \'etale descent obstruction. Both of these directions promise to shed light on producing a framework for studying the solubility properties of more general families of varieties.