Li, Freda (Freda Li) (author), (Wai Kiu Chan) (Thesis advisor), Wesleyan University Mathematics (Degree grantor)
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 n- regular if L globally represents all Hermitian lattices of rank n that are locally represented by L. In 2005 Rokicki 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 sub- lattice, 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., In Copyright – Non-Commercial Use Permitted (InC-NC)