Explicit Class number formulas for Siegel--Weil averages of ternary quadratic forms

TL;DR

This paper derives explicit formulas connecting class numbers and lattice point counts in ternary quadratic forms, extending previous results and providing asymptotic estimates with error terms for these mathematical objects.

Contribution

It generalizes Jones's result by establishing explicit class number formulas for Siegel--Weil averages of ternary quadratic forms, including asymptotic formulas with error terms.

Findings

Derived explicit class number formulas for ternary quadratic forms.

Established asymptotic estimates for lattice point counts with error bounds.

Extended previous theoretical results to broader classes of quadratic forms.

Abstract

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes $t_j\mathbb{Z}^2$) relating the number of lattices points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice.