Poincaré Series of Divisors on Graphs and Chains of Loops

Madhusudan Manjunath

Published 2020-11-24Version 1

We study Poincar\'e series associated to a finite collection of divisors on i. a finite graph and ii. a certain family of metric graphs called chain of loops. Our main results are proofs of rationality of the Poincar\'e series in both these cases. For a finite graph, our main technique involves studying a certain homomorphism from a free Abelian group of finite rank to the direct sum of the Jacobian of the graph and the integers. For chains of loops, our main tool is an analogue of Lang's conjecture for Brill-Noether loci on a chain of loops and adapts the proof of rationality of the Poincar\'e series of divisors on an algebraic curve (over an algebraically closed field of characteristic zero). In both these cases, we express the Poincar\'e series as a finite integer combination of lattice point enumerating functions of rational polyhedra.