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Wai Kiu Chan




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.



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