Zeta functions enumerating subforms of quadratic forms

2024/09/09 by Daejun Kim, Seok Hyeong Lee, Kim, Daejun +3 · 1 citation
Mathematics · #11E12 #11E16 #11H06 #11M41 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.05625

openalex publication_date 2024/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Zeta functions of quadratic lattices of a hyperbolic plane

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