It is natural to ask when the spherical volume defined by the intersection of a sphere at the apex of an integer polyhedral cone is a rational number. This work sets up a dictionary between the combinatorial geometry of polyhedral cones with `rational volume’ and the analytic behavior of certain associate cone theta functions.
We use number theoretic methods to study a new class of polyhedral functions called cone theta functions, which are closely related to classical theta functions. We show that if $K$ is a Weyl chamber for any finite crystallographic reflection group, then its cone theta function lies in a graded ring of classical theta functions (of different weights/dimensions) and in this sense is ‘almost’ modular. It is then natural to ask whether or not the conic theta functions are themselves modular, and we prove that in general they are not. In other words, we uncover some connections between the class of integer polyhedral cones that have a rational solid angle at their apex, and the class of cone theta functions that are almost modular. This is joint work with Amanda Folsom and Winfried Kohnen.