Alladi Ramakrishnan Hall

Cone theta functions, volumes of spherical polytopes, and their relations to classical theta functions

Sinai Robins

Nanyang Technological 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 rational.
�Addressing this question, we use number theoretic methods to study a new class of
polyhedral functions called conic theta functions, which are closely related to
classical theta functions. �We show that if K is a Weyl chamber for any finite
reflection group, then its cone theta function lies in a graded ring of classical
theta functions and in this sense it is `almost modular'. �It is then natural to
ask whether or not the conic theta functions are themselves modular, and we prove
that (generally) they are not. �In other words, we uncover some implications
between the class of integer polyhedral cones that have a rational solid angle, and
the class of cone theta functions that are almost modular.