Related papers: Group structures of elliptic curves over finite fi…

In this paper, we show that the maximal divisible subgroup of groups $K_1$ and $K_2$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$-groups modulo this uniquely divisible subgroup are explicitly…

In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many…

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G = E(\mathbb{Q})_{\rm tors}$ we study what groups (up to isomorphism) can occur as the torsion subgroup of $E$ base-extended to $K$, a degree 6 extension of $\mathbb{Q}$. We…

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K)_{tors}$, where $K$ is the compositum of all $\mathbb{Z}_{p}$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve…

Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct…

This master thesis describes how Selmer groups can be used to determine the Mordell-Weil group of elliptic curves over a number field K. The Mordell-Weil Theorem states that $E(K) = E(K)_{tors} \times Z^r$, where $r$ is the rank of $E$, and…

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to…

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a…

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$. In a previous article the author conjectured that the fine $p^{\infty}$-Selmer…

We combine the exact counting of all elliptic curves over $K = \mathbb{F}_q(t)$ with $\mathrm{char}(K) > 3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields…

We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is…

For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

We prove that for every number field $K$, there exist infinitely many elliptic curves $E$ over $K$ with rank exactly equal to 1.

Let $E$ be an elliptic curve over a quartic field $K$. By the Mordell-Weil theorem, $E(K)$ is a finitely generated group. We determine all the possibilities for the torsion group $E(K)_{tor}$ where $K$ ranges over all quartic fields $K$ and…

For any quadratic extension $L/K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E(K)$ and $E(L)$ both have rank $1$. In particular, there are infinitely many elliptic…

We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p>3, let S denote a finite set of places of K and let E/K denote a…

Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an…

Let $E$ and $E'$ be 2-isogenous elliptic curves over $\Q$. Following \cite{ck}, we call a good prime $p$ \emph{anomalous} if $E(\F_p) \simeq E'(\F_p)$ but $E(\F_{p^2}) \not \simeq E'(\F_{p^2})$. Our main result is an explicit formula for…

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in…