Riemann Hypothesis Finite Fields Elliptic Curves Number Theory
Abstract:
The Riemann Hypothesis has been a result eluding mathematicians for nearly 200 years. Analogs of this result have been found for elliptic curves over finite fields, which is the subject of this thesis. We begin by establishing algebraic foundations that will be used to prove larger results regarding the Riemann Hypothesis. Next, we explore an elementary number theoretic approach to this problem, and deal with a very particular type of elliptic curve. The crux of this paper is found in the next chapter, where we state and prove the Riemann Hypothesis for elliptic curves over finite fields. Finally, we investigate some examples of specific elliptic curves to see the applications of the theorems proved earlier.
