the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...

1. What is an elliptic curve? 2 2. Mordell-Weil Groups 5 2.1. The Group Law on a Smooth, Plane Cubic Curve 5 2.2. Reminders on Commutative Groups 8 2.3. Some Elementary Results on Mordell-Weil Groups 9 2.4. The Mordell-Weil Theorem 11 2.5. K-Analytic Lie Groups 13 3. Background on Algebraic Varieties 15 3.1. Affine Varieties 15 3.2. Projective Varieties 18 3.3. Homogeneous Nullstellensätze 20 3...

We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is Mordell-Weil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one copy of Λ. We prove that every rigid lattice is Mordell-Weil. In particular, we show that the Leech lattice can...

Let K be a number field and A an abelian variety over K. The K-rational points of A are known to constitute a finitely generated abelian group (Mordell-Weil theorem). The problem studied in this paper is to find an explicit upper bound for the rank r of its free part in terms of other invariants of A/K. This is achieved by a close inspection of the classical proof of the so-called ‘weak Mordell...

Three lectures on elliptic surfaces and curves of high rank Noam D. Elkies Over the past two years we have improved several of the (Mordell–Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P i ∈ E(Q) (the previous record was 24, by R. Martin and W. McMillen 2000), and the fi...

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

In this paper we examine diierences between the two standard methods for computing the 2-Selmer group of an elliptic curve. In particular we focus on practical diierences in the timings of the two methods. In addition we discuss how to proceed if one fails to determine the rank of the curve from the 2-Selmer group. Finally we mention brieey ongoing research i n to generalizing such methods to t...