Explicit formulas for Masses of Ternary Quadratic Lattices of varying determinant over Number Fields

TL;DR

This paper derives explicit formulas for the total mass Dirichlet series of integer-valued ternary quadratic lattices over number fields with specific local conditions, enabling detailed analysis of their distribution and applications to class groups.

Contribution

It provides explicit formulas for the mass Dirichlet series of ternary quadratic lattices over number fields, using local invariants and mass formulas, with computational verification.

Findings

Formulas match with known tables of positive definite ternary quadratic forms.

Software implementation verifies results for determinants up to 20,000.

Applications to the study of 2-part class groups of cubic fields.

Abstract

This paper gives explicit formulas for the formal total mass Dirichlet series for integer-valued ternary quadratic lattices of varying determinant and fixed signature over number fields F where p = 2 splits completely. We prove this by using local genus invariants and local mass formulas to compute the local factors of the theory developed in [11]. When the signature is positive definite these formulas be checked against tables of positive definite ternary quadratic forms over Z, and we have written specialized software which checks these results when the Hessian determinant is \leq 2 \times 10^4. This work can also be applied to study the 2-parts of class groups of cubic fields (e.g. see [2]).