Ternary positive definite quadratic forms are determined by their theta series
We consider the set \(D\) of pairs of ternary positive definite real forms which are reduced in the sense of Seeber-Eisenstein and have the same theta series. \(D\) is embedded into \(\mathbb{R}^{12}\) by the coefficients of the Gram matrices. The reduction conditions define a unique representative of every class (with respect to integral equivalence). By restricting to subsets of \(D\) with finitely many prescribed minimal vectors, the condition on the theta series yields linear equalities. We describe a series of finite coverings (one refining the other) of \(D\) with rational polyhedral cones arising from the prescribed minimal vectors and the equality of the first theta coefficients. Explicit computations (done by computer) show that the equality of the theta series of two reduced forms \(f\) and \(g\) up to a bound \(b(f)\) implies \(f=g\). We may choose \(b(f)=7/2 \cdot s_3(f)\), where \(s_3(f)\) denotes the third successive minimum of \(f\).
ternary positive definite quadratic forms
scientific article; zbMATH DE number 1057252