Rank Of Elliptic Curves Research Articles

On the rank of elliptic curves with a reational point of order 3

On the rank of elliptic curves with a reational point of order 3

Ranks of Elliptic Curves in Families of Quadratic Twists

We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series.

A rank 3 family of elliptic curves of arbitrary j -invariant

Using Shioda's theory of Mordell–Weil lattices, we produce a family of elliptic surfaces of rank 3 over ℚQ(i)(t) each having a unique fiber with j-invariant j. Two of the generators are rational over ℚ, reproducing Mestre's result [M] that for any j-invariant there are infinitely many elliptic curves of rank 2 or greater.

Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights

Let E/Q be an elliptic curve of rank 1. We describe an algorithm which uses the value of L'(E, 1) and the theory of canonical heghts to efficiently search for points in E(Q) and E(Z S ). For rank 1 elliptic curves E/Q of moderately large conductor (say on the order of 10 7 to 10 10 ) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set E(Z S ) contains non-torsion points.

Heron triangles and elliptic curves

In this paper we present a proof that there exist infinitely many rational sided triangles with two rational medians and rational area. These triangles correspond to rational points on an elliptic curve of rank one. We also display three triangles (one previously unpublished) which do not belong to any of the known infinite families.

On certain twisted families of elliptic curves of rank 8

On certain twisted families of elliptic curves of rank 8

ℚ(T)-rank of elliptic curves and certain limit coming from the local points

ℚ(T)-rank of elliptic curves and certain limit coming from the local points

The Order of the $p$-Selmer Groups and the Rank of Elliptic Curves

The Order of the $p$-Selmer Groups and the Rank of Elliptic Curves

The Order of the $p$-Selmer Groups and the Rank of Elliptic Curves

The Order of the $p$-Selmer Groups and the Rank of Elliptic Curves

On the rank of elliptic curves with three rational points of order $2$

On the rank of elliptic curves with three rational points of order $2$

On the rank of elliptic curves with three rational points of order $2$, II

On the rank of elliptic curves with three rational points of order $2$, II

An Algorithm for the Rank of Elliptic Curves

For elliptic curves the rank is one of the most essential numerical charactenstics. By means of finding the independent generators of the rational points on such a curve one may determine the lower bound of the rank. In this note we obtain a formula for the rank of elliptic curves defined over the rational numbers. Moreover, if we admit a conjecture suggested by S. Lang, then the proof of our formula provides an algorithm for calculating the rank.

The Square-Free Sieve and the Rank of Elliptic Curves

The Square-Free Sieve and the Rank of Elliptic Curves

The square-free sieve and the rank of elliptic curves

Let E be an elliptic curve over Q. A celebrated theorem of Mordell asserts that E(Q), the (abelian) group of rational points of E, is finitely generated. By the rank of E we mean the rank of E(Q). Thus the rank of E is positive if and only if E possesses an infinity of rational points. Relatively few general qualitative assertions can be made about the rank as E varies. How large can the rank get? Although we expect that there are elliptic curves over Q with arbitrarily high ranks, this is presently unknown. What is the average size of the rank? Recently Brumer and McGuinness have reported on their study of 310,716 elliptic curves of prime conductor less than 108 where they have found that 20.06% of those curves have even rank > 2. (Cf. [BM]; also see forthcoming work of Brumer where, subject to a number of standard conjectures, he shows that 2.3 is an upper bound for the average rank for all elliptic curves over Q ordered in terms of their Faltings height.) What is the behavior of the rank over the family of twists of a given elliptic curve? Here, one has three natural kinds of families of twists: (1) Quadratic twists. One can take any elliptic curve and systematically twist it by all quadratic characters (this is the type of family of elliptic curves we are concerned with in this paper). Specifically, if E is an elliptic curve over Q given by the Weierstrass equation Y2 = X + A * X + B, and D is any squarefree integer, the (quadratic) twist of E by D, ED, is given by the equation

On some elliptic curves defined over Q of free rank ?9

By Q (respectively Z ) we denote the field of rational numbers (respectively the ring of rational integers). The famous theorem of Mordell states that the set of all Qrational points on an elliptic curve defined over Q is a finitely generated abelian group. The rank of such a curve is defined to be the number of free generators of this group. Wiman [9] had examples of rank ~ 4 elliptic curves. D. Penny and C. Pomerance [5], [6] found examples of elliptic curves of rank ~ 7 by using computer. F. G. Grunewald and R. Zimmert [i] constructed examples of elliptic curves of rank ~ 8. Although N~ron [3], [4] proved that elliptic curves of rank ~ ii exist, there are no examples. It seems to be a practical benefit that we have examples of elliptic curves with large rank. In this paper, we shall explicitly construct an infinity of elliptic curves of rank ~ 9, modifying the method in [i].

The rank of elliptic curves

The rank of elliptic curves

Bounded ranks and Diophantine error terms

We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very weak version of Lang's error term conjecture would be enough to deduce boundedness of ranks for quadratic twists of elliptic curves over number fields. This can be seen as evidence for boundedness of ranks not relying on probabilistic heuristics on elliptic curves.