A conjecture due to Honda predicts that given any abelian variety over a number field $K$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda’s conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound...

We explicitly determine the elliptic $K3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$, for each of the two possible maximal fibre types, $I_{19}$ and $I_{14}^{*}$, the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I_{13}^{*}$, and there exist two (resp. one) one-parameter families of such surfaces.

Let $p$ be a prime and let $K$ be a number field. Let ${\rho }_{E,p}:{\mathrm{G}}_K\to \mathrm{Aut}\left(T_{p}E\right)\cong {\mathrm{GL}}_2\left({\mathbb{Z}}_p\right)$ be the Galois representation given by the Galois action on the $p$-adic Tate module of an elliptic curve $E$ over $K$. Serre showed that the image of ${\rho }_{E,p}$ is open if $E$ has no complex multiplication. For an elliptic curve $E$ over $K$ whose $j$-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of ${\rho }_{E,p}$.

Stein and Watkins conjectured that for a certain family of elliptic curves E, the X₀(N)-optimal curve and the X₁(N)-optimal curve of the isogeny class 𝓒 containing E of conductor N differ by a 3-isogeny. In this paper, we show that this conjecture is true.

For i = 0,1, let $E_i$ be the $X_i\left(N\right)$-optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational point of order...