Award Date

5-1-2012

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Committee Member

Arthur Baragar

Second Committee Member

Peter Shiue

Third Committee Member

Gennady Bachman

Fourth Committee Member

Evangelos Yfantis

Number of Pages

36

Abstract

The purpose of this paper is to derive the Hasse-Weil zeta function of a special class of Algebraic varieties based on a generalization of the Markoff equation. We count the number of solutions to generalized Markoff equations over finite fields first by using the group structure of the set of automorphisms that generate solutions and in other cases by applying a slicing method from the two-dimensional cases. This enables us to derive a generating function for the number of solutions over the degree k extensions of a fixed finite field giving us the local zeta function. We then create an Euler product of our local zeta functions for fields of prime order for all prime numbers similar to the derivation of the Riemann-zeta function.

Keywords

Algebraic varieties; Finite fields; Functions, Zeta; Hasse-weil; Local zeta function; Markoff equation; Markov processes; Mariscal, Juan; Zeta function

Disciplines

Algebraic Geometry | Mathematics

Language

English


Share

COinS