Subconvexity of Shintani's zeta function

Abstract

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani $\zeta$ function enumerating class numbers of binary cubic forms.

The original paper is split into two, with the subconvexity estimate for the cusp form twisted version in a separate paper