Type of Document Dissertation Author Bureau, Jean Edouard Author's Email Address email@example.com URN etd-11142006-132217 Title Representation Properties of Definite Lattices in Function Fields Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee
Advisor Name Title Jorge Morales Committee Chair Brendan Owens Committee Member Jerome W. Hoffman Committee Member Padmanabhan Sundar Committee Member Robert Perlis Committee Member Shengmin Guo Dean's Representative Keywords
- quadratic forms
- theta series
Date of Defense 2006-11-10 Availability unrestricted AbstractThis work is made of two different parts. The first contains results concerning isospectral quadratic forms, and the second is about regular quadratic forms.
Two quadratic forms are said to be isospectral if they have the same representation numbers. In this work, we consider binary and ternary definite integral quadratic form defined over the polynomial ring F[t], where F is a finite field of odd characteristic. We prove that the class of such a form is determined by its representation numbers. Equivalently, we prove that there is no nonequivalent definite F[t]-lattices of rank 2 or 3 having the same theta series.
A quadratic form is said to be regular (resp. spinor-regular) if it represents any element represented by its genus (resp. by its spinor genus). A form is said to be universal if it represents any integral element. We prove that regular and spinor-regular definite F[t]-lattices must have class number one and we give a characterization of definite universal F[t]-lattices.
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