Cone theta functions and rational volumes of spherical polytopes

01/27/2015 - 12:15pm
01/27/2015 - 1:10pm
Sinai Robins (Nanyang Technological University - Singapore and Brown University)

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 associated cone theta functions, which we define from scratch. 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. One of our results shows 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 relations 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.

Mudd Science Library 126, Pomona College