Solutions to many problems in number theory can be described using the theory of algebraic stacks. In this talk, I will describe a Diophantine equation, the so-called “generalized Fermat equation”, whose integer solutions correspond to points on an appropriate stacky curve: a curve with extra automorphisms at prescribed points. Using étale descent over such a curve, we characterize local and global solutions to a family of such equations and give asymptotics for the local-global principle in the corresponding family of stacky curves. This is joint work with Juanita Duque-Rosero, Chris Keyes, Manami Roy, Soumya Sankar and Yidi Wang. 

The Jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group from number theory. There are multiple recent results controlling the sizes of these groups in Galois towers of graphs that mimic the classical results in Iwasawa theory, though the connection to the values of the Ihara zeta function often requires some adjustment. In this talk we will give a new way to view the Jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant Iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from.

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.