Asymptotically good towers of global function fields and bounds for the Ihara function

This is a thesis in Algebraic Number Theory, concerned with the study of bounds for the Ihara function, an asymptotic measure comparing the number of rational places of a global function field with the genus of that field. The exact behavior of this function is unknown; however, some bounds on its values are known. There is a sharp upper bound, proven by Drinfeld and Vladut, and this bound is achieved when the size of the finite field is square. When the size of the finite field is not a square, all that is known are lower bounds on the values of the function. In this thesis, we present some improvements on the known explicit lower bounds for the Ihara function when the size of the finite field is a small prime