The Riemann Hypothesis for Elliptic Curves

and extended analytically to the whole complex plane by a functional equation (see [8, p. 14]). The original Riemann hypothesis asserts that the nonreal zeros of the Riemann zeta function ζ(s) all lie on the line Re(s) = 1/2. In his monumental paper [11] of 1859, Riemann made this assertion in order to derive an expression for the deviation of the exact number of primes ≤ x, which is denoted by π(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