The Mordell-Weil rank of elliptic curves

Complete characterization of the Mordell-Weil group of some families of elliptic curves

 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...

Elliptic Curves Group Law and Mordell-weil

This paper assumes no background on elliptic curves and culminates with a proof of the Mordell-Weil theorem. The Riemann-Roch and Dirichlet unit theorem are recalled but used without proof, but everything else is self-contained. After some elementary properties of elliptic curves are given, the group structure is explored in detail.

Hodge Theory and the Mordell-weil Rank of Elliptic Curves over Extensions of Function Fields

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension a...

Solution to the Inverse Mordell-weil Problem for Elliptic Curves

In [Ro76], M. Rosen showed that for any countable commutative group G, there is a field K, an elliptic curve E/K and a surjective group homomorphism E(K) → G. From this he deduced that any countable commutative group whatsoever is the ideal class group of an elliptic Dedekind domain – the ring of all functions on an elliptic curve which are regular away from some (fixed, possibly infinite) set ...

Construction of Rational Elliptic Surfaces with Mordell-Weil Rank 4