Theta series of ternary quadratic lattice cosets

TL;DR

This paper analyzes the decomposition of theta series for ternary lattice cosets, linking them to invariants of genus, spinor genus, and class, and extends $p$-neighbor concepts with an algorithm for class representatives.

Contribution

It extends known lattice theta series decompositions to lattice cosets, introduces new invariants, and develops an algorithm for class representatives using $p$-neighbors.

Findings

Decomposition of theta series into Eisenstein, unary theta, and cuspidal forms.

Identification of invariants for genus, spinor genus, and class.

Algorithm for computing class representatives via $p$-neighborhoods.

Abstract

In this paper, we consider the decomposition of theta series for lattice cosets of ternary lattices. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary theta functions correspond to the theta series for the genus, the deficiency of the theta series for the spinor genus from that of the genus, and the deficiency of the theta series for the class from that of the spinor genus, respectively. These three pieces are hence invariants of the genus, spinor genus, and class, respectively, extending known results for lattices and verifying a conjecture of the first author and Haensch. We furthermore extend the definition of $p$-neighbors to include lattice cosets and construct an algorithm to compute respresentatives for the classes in the genus or spinor genus via the $p$-neighborhoods.