On Arithmetic Over Function Fields

Abstract

There are two parts in this dissertation. The first part (Chapter 1) is trace formula of a Brandt matrix. Given a division quaternion algebra over a global function field which is definite with respect a fixed chosen infinity place. A family of Brandt matrices is then introduced to encode information from the arithmetic of the division quaternion algebra. Adapting Eichler’s method from the rational field case, we build up a fine formula expressing the trace of these Brandt matrices in terms of class numbers of specific orders inside imaginary quadratic extensions of the global function field embeddable into the division quaternion. The proof is base on a detailed study of the so-called optimal embeddings of quadratic orders into the division quaternion. The second part (Chapter 2 and 3) is multiple Dirichlet series over a global function field. Fisher and Friedberg [c.f. FF and FF2] constructed and studied a family of multiple (2 and 3 variables) Dirichlet series over general function fields. They proved that these multiple Dirichlet series satisfy finite, non-abelian groups of functional equations and are rational functions with specific denominator. We then are interested in explicitly finding these multiple Dirichlet series over a curve. In the 2-variable case, we work on in particular the elliptic curve C : y^2 = x^3 + 2x over F_5. For the 3-variable case, we compute the curve P^1 over F_q with q odd which the function field is rational function field over F_q.