On trace-zero norm forms of orders of rational quaternion algebras with applications

Abstract

In this paper, by normalizing the trace-zero norm forms according to the parity of orders of definite rational quaternion algebras, we give an explicit formulation to the even Clifford correspondence followed by the reciprocity of ternary forms. We obtain an exact identity between ternary representation numbers and the number of elements of an order with prescribed reduced norm and trace, which applies to all orders without squarefreeness constraint on the reduced discriminant. Applications include a criterion for realizing ternary forms as trace-zero norm forms, a formula for traces of Brandt matrices in terms of representation numbers, and explicit theta series relations associated with local reciprocity. We also describe reciprocal orders and order chains, and interpret the ternary forms occurring in the congruent number problem through their associated quaternion orders.