Abstract:

For proper stacks, unlike schemes, there is a distinction between rational and integral points; this is addressed by the notions of tuning stacks & stacky height functions which were introduced by Ellenberg, Satriano, and Zureick-Brown. In this talk, I will present a geometric approach to heights on cyclotomic stacks and the construction of moduli spaces of rational points of fixed height in the function field case. As an application, I will explain how the distinction exactly accounts for the main term and lower order terms appearing in counts of elliptic curves over global function fields ordered by discriminant and also a natural generalization of Tate's algorithm. This is based on joint work with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).