Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series

TL;DR

This paper explores the connection between integer cubes, Shintani zeta functions, and Weyl group multiple Dirichlet series, revealing new invariants and mappings in the context of prehomogeneous spaces.

Contribution

It establishes that the Shintani zeta function for 2x2x2 integer cubes coincides with the A3 Weyl group multiple Dirichlet series and identifies key invariants determining orbit classes.

Findings

Shintani zeta function matches A3 Weyl group multiple Dirichlet series

Three arithmetic invariants classify integer cube orbits

Finitely surjective map to a moduli space for semi-stable orbits

Abstract

We investigate the Shintani zeta functions associated to the prehomogeneous spaces, the example under consideration is the set of $2 \times 2\times 2$ integer cubes. We show that there are three relative invariants under a certain parabolic group action, they all have arithmetic nature and completely determine the equivalence classes. We show that the associated Shintani zeta function coincides with the $A_3$ Weyl group multiple Dirichlet series. Finally, we show that the set of semi-stable integer orbits maps finitely and surjectively to a certain moduli space.