Isospectral Definite Ternary F_q[t]-Lattices

TL;DR

This paper proves that for large enough finite fields, the representation numbers of a ternary definite quadratic form over F_q[t] uniquely determine its integral equivalence class, linking theta series to form classification.

Contribution

It establishes that representation numbers determine the form's equivalence class over F_q[t] for sufficiently large q, connecting theta series with form classification.

Findings

Representation numbers determine the quadratic form's class for large q.

Quadratic forms are classified by their theta series in this setting.

The result applies to definite ternary quadratic forms over F_q[t].

Abstract

We prove that the representations numbers of a ternary definite integral quadratic form defined over F_q[t], where F_q is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to its successive minima. Equivalently, such a quadratic form is determined up to integral isometry by its theta series.