Work

The classification and representations of ternary quadratic forms with level $ 8N $

Year: 2025

Type: article

Abstract: In this paper, we investigated positive definite ternary quadratic forms of level $ 8N $, where $ N $ is an odd positive squarefree integer. Our study makes two main contributions. First, we provided an explicit classification of positive definite ternary quadratic forms of level $ 8N $. Second, we derived exact formulas for the weighted sum of representations over each class within every genus of ternary quadratic forms of level $ 8N $, which involved modified Hurwitz class numbers. The proof of our main results leverages the relations between ternary quadratic forms, quaternion algebras, and weight $ 3/2 $ modular forms of level $ 8N $. As applications, we obtained exact formulas for the class number of positive ternary quadratic forms of level $ 8N $.

Source: AIMS Mathematics

DOI: 10.3934/math.2025664

Authors: Yifan Luo, Haigang Zhou

Institutions: Tongji University, North China University of Water Resources and Electric Power, Henan Academy of Sciences

Language: English

Topic: Advanced Mathematical Identities

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $1,200

Funder: National Natural Science Foundation of China