Doctor of Philosophy (PhD)
This 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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Bureau, Jean Edouard, "Representation properties of definite lattices in function fields" (2006). LSU Doctoral Dissertations. 3433.