On algebraic and analytic properties of Jacobian varieties of Riemann surfaces

The main purpose of this dissertation is to study some basic properties of Riemann surfaces. The Jacobian of a Riemann surface is one of the most important algebraic and analytic characteristics for the surface. Related to Jacobian of a Riemann surface are Riemann Period Matrix, Jacobian Lattice, and Jacobian Variety. There are algebraic and analytic aspects of study of Jacobians.; In Chapter 2, we will establish a number of algebraic properties of complex lattices and tori that are fundamental to the study of Jacobians of Riemann surfaces. In Chapter 3, we will consider an extremal problem of Riemann surface. The problem was first studied by Buser and Sarnak (BS), who introduced the concept of maximal minimal norms of the Jacobian lattices of Riemann surfaces, and obtained a number of properties for a Riemann surface of large genus. We will answer to their conjecture that Klein's surface would be an absolute extremal Riemann surface in the case of genus 3, and prove that their conjecture is not really true. To complete our proof, we will first give out a sufficient, possibly necessary, condition for a Riemann surface to have a maximal minimal norm of its Jacobian lattice, and then prove that, based on the results of Quine (Q2), there exists a local extremal Riemann surface in the case of genus 3 that has a bigger minimal norm than Klein's surface does. It seems to be extremely difficult to find out all the extremal Riemann surfaces no matter how one nontrivially defines the extremality.; Among all the essential work to this paper are the results obtained by Rauch and Lewittes (RL), Quine (Q1) (Q2), and the classic discussions of perfect and eutactic forms introduced by Voronoi (VO) and studied extensively by Barnes (BA3) and many other mathematicians (CS2).; Buser and Sarnak in their paper (BS) have obtained a number of interesting characteristics for surfaces of large genus. Our discussion is very computational and probably not applicable to higher genus case. We will also provide some information on Riemann surfaces of genus 3 such as Klein's surface and Fermat's surface, to hopefully help further study in this area.