Related papers: Elliptic curves with bounded ranks in function fie…

Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log…

By reformulating and extending results of Elkies, we prove some results on $\mathbb Q$-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees~$\ell$ which an elliptic curve without CM may…

We discuss the distribution of Mordell--Weil ranks of the family of elliptic curves $y^2=(x+\alpha f^2)(x+\beta b g^2)(x+\gamma h^2)$ where $f,g,h$ are coprime polynomials that parametrize the projective smooth conic $a^2+b^2=c^2$ and…

Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves…

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over…

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an…

The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree $d$, $d=4,5,6$, is not completely determined. However, the list of groups $\Phi^{\infty}(d)$, $d=4,5,6$, that can be realized as…

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any…

Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $\nu_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that…

Let E be an elliptic curve having Complex Multiplication by the full ring O_K of integers of K=Q(\sqrt{-D}), let H=K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an O_K-module, and its structure denpends on its…

Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i…

We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D…

For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the…

Let $e_1,e_2,e_3$ be nonzero integers satisfying $e_1+e_2+e_3=0$. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1a^2+e_2b^2+e_3c^2=0$. Denote by $E: y^2=x(x-e_1)(x+e_2)$ and $\mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2)$.…

We describe how the Mordell-Weil group of rational points on a certain family of elliptic curves give rise to solutions to a conjecture of Erd\"{o}s on $3$-powerful numbers, and state a related conjecture which can be viewed as an elliptic…

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.

We study the structure of the Mordell--Weil groups of semiabelian varieties over large algebraic extensions of a finitely generated field of characteristic zero. We consider two types of algebraic extensions in this paper; one is of…

We consider all \emph{odd} fundamental discriminants $D \equiv 2 \bmod 3$ and their mirror discriminants $D' = -3D$, and we study the family of elliptic curves $E_{D'}: y^{2} = x^{3} + 16D'$. We denote by $r_{3}(D)$ and $r_{3}(D')$ the rank…

For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky,…

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks…