Jasbir S. Chahal

The Riemann Hypothesis for Elliptic Curves

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 Hasses 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 Basic Identity, 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.

Congruent Numbers and Elliptic Curves

1. CONGRUENT NUMBER PROBLEM. It has been known since antiquity that six is the smallest natural number that is the area of a right triangle the lengths of whose sides are whole numbers and that the right triangle with sides three, four, and five is the only such triangle with area six. In 1225, Fibonacci discovered the right triangle with sides 3/2, 20/3, and 41/6 whose area is five, also a natural number and smaller than six. This raised the question: Can every natural number n be represented as the area of a right triangle whose side-lengths are rational numbers? About 350 years ago, Ferm?t proved that the answer is no for n ? 1, 2, or 3 (hence, for n = 4). Thus, Fibonacci had already found the smallest congruent number, meaning a natural number that is the area of a rational right triangle. The congruent number problem is to determine which natural numbers are con gruent numbers. To realize the challenge, the reader is asked to prove that seven is a congruent number and to exhibit a rational right triangle with area seven. No one has yet found an unconditional algorithm that would decide in a finite number of steps whether a given natural number n is congruent or not. Of course, one way to show that a given n is a congruent number is to produce a rational right triangle with area n. But finding such a triangle is probably a more, not less, difficult task. It took a mathemati cian like Zagier to find a right triangle with all sides of rational length whose area is the prime number 157 (see [4, p. 5]). Recently the congruent number problem came into the limelight again, with the discovery of a deep connection between this problem and the arithmetic of elliptic curves, a subject that has become very popular during the last few decades. If two triangles are similar, their sides are proportional. If c is the constant of propor tionality, the ratio of their areas is c2. Hence, to treat the general case one need consider only the congruent number problem for natural numbers n having no square factor larger than one. For example, since 1 and 2 are not congruent numbers, 4 = 22 1, 9 = 32 1, and 8 = 22 2 cannot be congruent numbers either. From now on, we as sume that the natural numbers under consideration as potential congruent numbers are square free (i.e., have no repeated prime factors). Among the six square-free natural numbers under ten, three (namely five, six, and seven) are congruent numbers, while the remaining three are not. Perhaps due to this observation, the following well-known conjecture has become folklore:

Foundations of basic geometry

The basic notions of length, area and volume were not alien to the prehistoric civilizations. The pyramids, palaces and great baths built more than 4000 years ago provide ample evidence. We begin our investigation of geometry with a discussion of areas of simple geometric objects.

Solution of the Cubic

Cardanos formula for solving a cubic is the crowning achievement of renaissance mathematics. Yet, it does not receive the same recognition in our curricula as does the quadratic formula, which was discovered long before it. It is rather surprising that there have not been attempts to simplify further the messy formulas of Cardano (see ([1],pp.606-616), or ([2], pp.187-189)) to a form that would be easier for the students to remember. Apparently the messy nature of the formulas for solving the cubic is a reason for the lack of their popularity. Another reason could be the Galois theory, which modern authors use in their exposition of Cardanos formula. We show that a simple trick, namely a rescaling of the discriminant, reduces not only the formula to a simpler form , but also its verification to a trivial calculation, with no reference to Galois theory. Although Galois theory is an indispensable tool in algebra and number theory, it is not necessary to wait until one learns it, for Cardanos formula. Cardanos formula can be introduced in a first course on complex numbers.

A remark on the torsion subgroups of elliptic curves

Abstract A uniform bound is given for the order of the torsion subgroup of E ( K ), the group of K -rational points on an elliptic curve E defined over a number field k , with K quadratic over k .

Albime triangles over quadratic fields

This note uses a diophantine problem arising in elementary geometry as a prerequisite to illustrate some theory of elliptic curves. As a typical example, Proposition~ref {5.3} and Theorem~ref {main} determine the exact set of ration-al numbers for which the specialization homomorphism from the torsion free rank~2 group of rational points on some elliptic curve over Q(t), is well defined and injective.

A supplement to Manin's proof of the Hasse inequality

In 1956 Yu.I. Manin published an elementary proof of Helmut Hasse’s 1933 result stating that the Riemann hypothesis holds in the case of an elliptic function field over a finite field. We briefly explain how Manin’s proof relates to more modern proofs of the same result. This enables us to present an analogous elementary proof for the case of finite fields of characteristic two, which was excluded in the original argument.

Carlitz–Wan conjecture for permutation polynomials and Weill bound for curves over finite fields

The authors requested a withdrawal of this article shortly after it first appeared in press in 2014. Unfortunately, the withdrawal was implemented as a retraction instead by the publisher and an earlier version of this notice contained allegations in relation to the article that, following further editorial consideration, the journal has found to be incorrect. The Publisher apologizes to the authors and has now re-published the article with minor changes at https://doi.org/10.1016/j.ffa.2018.07.006.

Foundation of basic arithmetic

This is the second of a series of articles, in which we demonstrate the utility of the place value system of representing numbers.