Existence and enumeration of integral points on spherical stacky curves (25frg010)

The Banff International Research Station will host the “Existence and enumeration of integral points on spherical stacky curves” workshop in Banff from October 26 - November 2, 2025.

Stacky curves are spaces that typically arise when parametrizing objects of arithmetic or geometric interest. While inherently geometric objects, the study of the arithmetic of stacky curves has seen recent advances. Like algebraic curves, stacky curves have a notion of \textit{genus}, which can be a rational number $g$ rather than an integer. The proposed research is to investigate the Hasse principle for integral points in a specific family of stacky curves arising from generalized Fermat equations.

A generalized Fermat equation is a Diophantine equation of the form $Ax^p + By^q = Cz^r$ for integers $A,B,C,p,q,r$ with $p,q,r > 1$. Integer solutions to such equations, which are of classical interest, correspond to integer points on a stacky curve, embedded in a weighted projective stack. When the resulting stacky curve admits a cover by the projective line, it is said to be \textit{spherical}. This makes available explicit methods of descent, which are proposed to be used to identify conditions for these stacky curves to satisfy or fail the Hasse principle for their integral points.

The proposed approach has applications to the statistics of how often stacky curves in these families are everywhere locally soluble, or satisfy the Hasse principle. These efforts have the potential to to deepen our understanding of the arithmetic of stacky curves of low genus as well as that of the Diophantine equations from which they arise.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines, and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta’s Advanced Education and Technology, and Mexico’s Consejo Nacional de Ciencia y Tecnología (CONACYT).