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The classification and representations of positive definite ternary quadratic forms of level 4N

Yifan Luo, Haigang Zhou

Abstract

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive definite ternary quadratic forms of level $4N$ explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level $4N$, which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level $4N$. As applications, we derive an explicit base of Eisenstein series space of modular forms of weight $3/2$ and level $4N$, and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

The classification and representations of positive definite ternary quadratic forms of level 4N

Abstract

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer , first we give the classification of positive definite ternary quadratic forms of level explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight and level , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.
Paper Structure (21 sections, 29 theorems, 180 equations, 1 table)

This paper contains 21 sections, 29 theorems, 180 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ternary quadratic forms
  3. Level and discriminant
  4. Lehman's bijection
  5. Watson transformation
  6. Isotropic
  7. Quaternion algebras
  8. Maps between Eichler orders and ternary quadratic forms
  9. Two trace zero lattices attached to Eichler orders
  10. Even Clifford algebras
  11. Proof of Theorem 1.1
  12. Proof of Theorem 1.2 and Theorem 1.3
  13. Proof of Theorem 1.3
  14. Proof of Theorem 1.2
  15. Some applications
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $N$ be a product of $s$ distinct odd primes. Then for all primitive positive definite ternary quadratic forms of level $4N$, there are $2^{2s+1}$ genera. These are $G_{4N,N^2/N_r,N^{\text{odd}}}$, $G_{4N, 4N^2/N_r,N^{\text{odd}}}$, $G_{4N,4N^2/N_r,2N^{\text{even}}}$ and $G_{4N,16N^2/N_r,N^{\text

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Remark 2.4
  • Proposition 2.5
  • Theorem 2.6
  • ...and 32 more