Symplectic Eichler criterion

TL;DR

This paper establishes new criteria for classifying primitive vectors in symplectic lattices using discriminant groups, simplifying the understanding of their equivalence and splitting properties under certain symplectic group actions.

Contribution

It introduces an Eichler-type criterion and a related splitting criterion for primitive vectors in symplectic lattices, based on discriminant groups.

Findings

Provides a simple method to determine vector equivalence under symplectic congruence groups.

Offers criteria to identify when a primitive vector is splitting.

Enhances understanding of symplectic lattice structures and their automorphisms.

Abstract

We prove an Eichler-type criterion for symplectic lattices which determines in a simple way when two primitive vectors are equivalent under a canonical congruence subgroup of the symplectic group. This is supplemented by another, related criterion which determines when a given primitive vector is splitting. Both criteria use the discriminant groups.