Rational solutions of an elliptic curve

Abstract

The study of elliptic curves is long standing. A fundamental problem for an elliptic curve is to find all rational solutions. The problem has played and continues to play a fundamental role in the development of many areas of number theory. The theory of elliptic curves was crucial in the proof of Fermat’s Last Theorem. In this paper, we introduce rational solutions of lines and conics first. A natural problem of rational solutions of plane cubic arises, which is the problem of rational solutions of an elliptic curve in most cases. There is a natural addition on the rational solutions of an elliptic curve. So the set of rational solutions of an elliptic curve forms an abelian group. In 1922, Mordell proved that the set of rational solutions of an elliptic curve was a finitely generated abelian group, which confirmed a conjecture of Poincare in 1901. By the fundamental structure theorem of finitely generated abelian group, the group of rational solutions of an elliptic curve is completely determined by its rank and torsion subgroup. The rank of the rational solutions of an elliptic curve is called the rank of the elliptic curve, which is an import invariant of an elliptic curve. Therefore, an elliptic curve having infinitely many rational solutions is equivalent to the rank of elliptic curve being positive. After many efforts, people are clear about the torsion subgroup of the rational solutions of an elliptic curve. However, people know a little about the rank of an elliptic curve. There are many conjectures of the rank of elliptic curves. One of them says most elliptic curves has rank 0 or 1. Birch and Swinnerton-Dyer reduce an elliptic curve over rational numbers to the elliptic curve over a finite field. The advantage of an elliptic curve over a finite field is that its solutions are finite. Then Birch and Swinnerton-Dyer define Hasse-Weil L -function L ( E , s ) regarding of solutions of the elliptic curve over the finite field. The Hasse-Weil L -function L ( E , s ) has good analytic properties and is somehow computable. Birch and Swinnerton-Dyer conjecture that the rank of an elliptic curve E and the order of s =1 of the Hasse-Weil L -function L ( E , s ) are equal. The BSD conjecture offers an approach to determine whether an elliptic curve has infinitely many rational solutions. However, the conjecture is extremely difficult, and it was chosen as one of the seven Millennuim Prize Problems listed by Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof. We will introduce current progress of the conjecture. In the end, we introduce Fermat’s Last Theorem and congruent numbers. These two problems are closely related to the theory of elliptic curves. Fermat’s Last Theorem was complete solved by Wiles in 1995. The problem of congruent numbers is not clear so far. We will introduce the recent important progress of the problem by Chinese mathematician Ye Tian. Besides, important and useful applications of elliptic curves have been found in many fields, such as cryptography, factoring large integers, and primality proving.