The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve deened over a eld K by 2-descent. The results lead to some simpliications to the method rst presented in (Birch and Swinnerton-Dyer, 1963), and can be applied to give a more eecient algorithm for determining Mordell-Weil groups over Q, as well as being...

We give a structure theorem for the $m$-torsion of Jacobian general superelliptic curve $y^m=F(x)$. study existence torsion on curves form $y^q=x^p-x+a$ over finite fields characteristic $p$. apply those results to bound from below Mordell-Weil ranks Jacobians certain $\mathbb Q$.

Let $A$ be an abelian variety defined over a number field $k$, let $p$ odd prime and $F/k$ cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give interpretation the $p$-component relevant case equivariant Tamagawa conjecture in terms integral congruence relations involving evaluation on appropriate points ${\rm Gal}(F/k)$-valued height pairing Mazur Tate. We then discu...

Abstract We prove that $164\, 634\, 913$ is the smallest positive integer a sum of two rational sixth powers, but not powers. If $C_{k}$ curve $x^{6} + y^{6} = k$ , we use existence morphisms from to elliptic curves, together with Mordell–Weil sieve, rule out points on for various k .

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 ...