Mohammed Ismail, Harris Ahmed2018-09-212018-09-212018-07http://hdl.handle.net/11299/200296University of Minnesota Ph.D. dissertation. July 2018. Major: Mathematics. Advisor: Steven Sperber. 1 computer file (PDF); xv, 219 pages.Dwork’s trace formula is a seminal result proven by Bernard Dwork [Dwo60] (Section 2: Lemma 2), and it is one of the main ingredients in his celebrated proof of the rationality of the zeta function of an (affine or projective) algebraic variety over a finite field. In this thesis, we will prove a generalization of Dwork’s trace formula that applies to exponential sums over Galois rings and the associated L-function. Using the generalized trace formula, we will prove more results on the L-function, analogous to the classical results by Dwork and Bombieri who studied L-functions associated to such exponential sums over finite fields. In particular, we will construct an analogue of the Dwork complex, and then prove the rationality of the L-function, and then obtain estimates on the degree (the number of zeros minus the number of poles) of the L-function (or its reciprocal) as in [Bom66] and the improvement by Adolphson and Sperber [AS87a]. We will conclude with a brief discussion on some interesting applications and extensions of this work that are worth investigating. To the reader who lacks sufficient mathematical background: Finding integer solutions to polynomial equations (called as Diophantine problems) have been of great interest to humanity since antiquity. These fundamental problems have been driving significant developments in modern number theory and algebraic geometry. An algebraic variety is a geometric structure determined by the common zeros (solutions) to a system of polynomial equations. The zeta function of an algebraic variety encodes the information on the number of solutions to a system of polynomial equations in certain mathematical structures generalizing commonly used systems of numbers such as the integers, the rational numbers and the real numbers. More precisely, it encodes the sequence of the number of “rational points” on the variety. L-functions associated to exponential sums are related to the zeta function of an algebraic variety. This motivates the study of L-functions associated to exponential sums. On the other hand, exponential sums themselves are important objects of interest. For example, exponential sums like Gauss sums and Kloosterman sums play a fundamental role in analytic number theory. In general, the study of L-functions associated to exponential sums is of great interest due to the similarity with and the relationships to other L-functions and zeta functions all of which share important properties with the classical Riemann zeta function, the study of which has been of great interest for centuries for to its fundamental place in number theory.enThesis or Dissertation