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 $N$, 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.