Zeta functions enumerating subforms of quadratic forms

TL;DR

This paper introduces a Dirichlet series that counts subforms of quadratic forms, linking it to ideal class counting in quadratic fields, and explores its explicit formulas and analytic properties.

Contribution

It establishes a novel connection between subform enumeration and ideal class counting, providing explicit formulas and analyzing the series' properties.

Findings

Formulas linking subform Dirichlet series with ideal class series

Explicit formulas for square and hexagonal lattices

Analysis of the Dirichlet series' analytic properties

Abstract

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series.