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

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, first we give the classification of positive definite ternary quadratic forms of level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 upper N"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 upper N"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 upper N"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As applications, we derive an explicit base of Eisenstein series space of modular forms of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 upper N"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and give new proofs of some interesting identities involving representation number of ternary quadratic forms.