If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then $E\left({\mathbb{Q}}^{ab}\right)$ has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then $E\left(K{\mathbb{Q}}^{ab}\right)$ has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over $K{\mathbb{Q}}^{ab}$.

Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E:y^2=x^3-4p^{2}x$. Further, let $N\left(p\right)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\ge 17$, then $N\left(p\right)\le 4$ or $1$ depending on whether $p\equiv 1(mod8)$ or $p\equiv -1(mod8)$.

We show a $p$-parity result in a $D_{2p^n}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W(E/K,1\oplus \eta \oplus \tau )={(-1)}^{〈1\oplus \eta \oplus \tau ,X_p(E/L)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}(L/K)=D_{2p^n}$, and $X_p(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_0$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture (using...

For $n\in {𝔽}_q\left[T\right]$ let $G$ be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve $X_0\left(𝔫\right)$. We determine all $𝔫$ and $G$ for which the quotient curve $G\setminus X_0\left(𝔫\right)$ is rational or elliptic.

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $K$ be an imaginary quadratic field, and let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$. Given a set $\Sigma$ of primes of $K$, containing the primes above $p$, and the primes of bad reduction for $E$, write $K_{\Sigma }$ for the maximal algebraic extension of $K$ which is unramified outside $\Sigma$. This paper is devoted to the study of the structure of the cohomology groups $H^i(K_{\Sigma }/K_{\infty },E_{p^{\infty }})$ for $i=1,2,$ and of the $p$-primary Selmer group Sel${}_{p^{\infty }}(E/K_{\infty })$, viewed as discrete modules over the Iwasawa algebra of $K_{\infty }/K.$