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.

We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of Q, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.

We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant j-invariant. Moreover, we classify all such elliptic threefolds if the j-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in P(2, 3, 1, 1, 1) that admit no variation of Hodge structure. (Remark 9.3.)

Computing a lower bound for the canonical height is a crucial step in determining a Mordell–Weil basis of an elliptic curve. This paper presents a new algorithm for computing such lower bound, which can be applied to any elliptic curves over totally real number fields. The algorithm is illustrated via some examples.

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.