Siegel's mass formula and averages of Dirichlet L-functions over function fields

TL;DR

This paper provides elementary proofs of Siegel's formulas for representation numbers and mass of ternary lattices over function fields, revealing a connection with averages of Dirichlet L-functions.

Contribution

It offers new, self-contained proofs of classical formulas and uncovers a link between lattice mass and Dirichlet L-function averages in the function field setting.

Findings

Elementary proofs of Siegel's formulas for representation numbers and mass.

Establishment of a relation between lattice mass and Dirichlet L-function averages.

Enhanced understanding of lattice theory over function fields.

Abstract

Let D be a square-free polynomial in F_q[t], where q is odd, and let G be a genus of definite ternary lattices over F_q[t] of determinant D. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the weighted sum of primitive representations numbers over the classes of G and for the mass of G. Our proof of the mass formula shows an interesting relation with certain averages of Dirichlet L-functions.