Marino, Alicia (Author), (Wai Kiu Chan) (Thesis advisor)
A quadratic form is a homogeneous polynomial of degree two. In the arithmetic theory of integral quadratic forms, a main question is the representation problem: given an integral quadratic form f, for which integers a does there exist a solution to f(x) = a? We call an integral quadratic form regular if the existence of solutions locally everywhere implies the existence of a solution over the rational integers. We can strengthen this notion of regularity to strict regularity by demanding that the solutions are primitive, i.e. the coordinates of the solutions are coprime. In 2014, Earnest-Kim-Meyer proved that there are finitely many equivalence classes of primitive positive definite integral strictly regular quadratic forms in four variables. The main result of this thesis extends their result in the context of a higher dimensional analogue of strict regularity. We obtain the following result: for n >= 2, there are only finitely many equivalence classes of primitive positive definite integral strictly n-regular quadratic forms of n + 4 variables., Old URL: https://wesscholar.wesleyan.edu/etd_diss/76, In Copyright – Non-Commercial Use Permitted (InC-NC)
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)
Kaylor, Lisa (Author), (Wai Kiu Chan) (Thesis advisor)
A quadratic form is a homogeneous polynomial of degree 2 in n variables. One of the most fundamental questions in the study of quadratic forms is the classification problem. For rational quadratic forms, there is a complete solution to this question due to the local - global principle of Hasse; however, for integral quadratic forms, this principle does not hold in general. This leads to the study of invariants of integral quadratic forms and how they can be used for classification. In this thesis, we use the geometric language of quadratic spaces and lattices and restrict our attention to quaternary even positive definite integral Z-lattices and their theta series. For such lattices with discriminant 389 and minimum 2, Kitaoka showed that there is a linear dependence relation among the theta series corresponding to the classes of these lattices. However, Hsia and Hung showed that the degree 2 theta series corresponding to the classes of positive definite even quaternary integral lattices of discriminant p a prime congruent to 1 mod 4 with minimum 2 are linearly independent. We consider those lattices with discriminant 4p where p > 13 is a prime congruent to 3 mod 4. There are two genera of lattices in this case, which are considered separately. We follow the strategy of Hsia and Hung to show that the degree 2 theta series of the classes with nontrivial orthogonal group are linearly independent within each genus., Old URL: https://wesscholar.wesleyan.edu/etd_diss/111, In Copyright – Non-Commercial Use Permitted (InC-NC)