In this paper, we consider a family of elliptic curves over Q with 2-torsion part Z2. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0. Mathematics Subject Classifications (2000). 11G05, 11L40, 14H52.

Watkins conjectured that for an elliptic curve $E$ over $\mathbb {Q}$ of Mordell-Weil rank $r$, the modular degree is divisible by $2^r$. If has non-trivial rational $2$-torsion, we prove conjecture all quadratic twists squarefree integers with sufficiently many prime factors.

For modular elliptic curves over number fields of narrow class one, and with multiplicative reduction at a collection p-adic primes, we define new invariants. Inspired by Nekovář Scholl's plectic conjectures, believe these invariants control the Mordell–Weil group higher rank support our expectations numerical experiments.

We use the hyperelliptic curve C : y + xy = x + tx + x + t over the field F2[t]/(t 113 + t + 1) which was generated by Wouter Castryck, Katholieke Universiteit Leuven, Belgium. The choice of the finite field takes into account three aspects: Firstly, it allows for an order of the divisor class group of appropriate size for the desired security level. Secondly, the extension degree of F2 was cho...

Chromatic Selmer groups are modified with local information for supersingular primes p. We sketch their role in establishing the p-primary part of Birch–Swinnerton-Dyer formula Sections 2–5, and then study growth Mordell–Weil rank along ℤ p 2 -extension a quadratic imaginary number field which splits Section 6.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...

We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field Fp2 of p 2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2 -valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves ...