Finiteness Of Strictly N-regular And Almost N-regular Hermitian Lattices

Abstract

Let E be an imaginary quadratic field. A Hermitian lattice L is said to be regular if L globally represents all elements that are locally represented by L. It is nregular if L globally represents all Hermitian lattices of rank n that are locally represented by L. In 2005 Rokicki [23] proved that, for a fixed imaginary quadratic field, there exist only finitely many isometry classes of nondegenerate normalized positive definite n-regular Hermitian lattices of rank n + 1. The notion of n-regularity can be strengthened to strict n-regularity. A sublattice, N , of a Hermitian lattice, L, is primitive in L if N is a direct summand of L. We say that L is strictly n-regular if L primitively represents all Hermitian lattices of rank n that are locally primitively represented by L. In this thesis, we prove an analogous result to Rokicki's for strictly n-regular Hermitian lattices, for n ≥ 2. Another notion of regularity is almost n-regularity. A Hermitian lattice L is almost n-regular if L globally represents all but finitely many Hermitian lattices of rank n that are locally represented by L. In this thesis, we also consider almost n-regular lattices, and prove a finiteness result for almost n-regular lattices of rank n + 1.