The Riemann Hypothesis for Elliptic Curves

Abstract

tion of the exact number of primes < x, which is denoted by n(x), from the estimate x/ log x that had been conjectured by Gauss, Legendre, and others. Riemann alluded to returning to this matter later by saying that he was setting it aside for the time being. Apparently Riemann did not live long enough to do that. To this day, no one has been able to prove the Riemann hypothesis despite overwhelming numerical evidence in its favor. However, many generalizations and analogs of the Riemann zeta function have been formulated by, among others, Dirichlet, Dedekind, E. Artin, F. K. Schmidt, and Weil, and the Riemann hypothesis has been shown to be true in some of these cases. One such case is the Riemann hypothesis for elliptic curves, originally conjectured by E. Artin (see [1, pp. 1-94]) and proved by Hasse, and therefore also known as Hasse's theorem. We begin by laying out the statement of this result in Section 2 below. We then turn to the two main topics of this article: (i) a brief explanation of the fact that these two Riemann hypotheses are not only closely analogous, but indeed two examples of a single more general framework; and (ii) an elementary proof of the Riemann hypothesis for elliptic curves over finite fields. This is carried out in Sections 3 and 4 respectively, and these may be read independently of one another. Our proof is based on an idea of Manin. The presentation is self-contained except for an appeal to the which is a technical lemma stated in (19) be low. The proof of the Basic Identity, although somewhat complicated, is completely elementary (see [5]) and is the least illuminating part of our proof of this Riemann hypothesis.