Related papers: Elliptic curves and Hilbert's tenth problem for al…

Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The…

Caraiani and Newton have proven that if $F$ is an imaginary quadratic number field such that $X_0(15)$ has rank $0$ over $F$, then every elliptic curve over $F$ is modular. This paper is concerned with the quadratic fields…

This version improves the old version entitled "On the modularity of elliptic curves with a residually irreducible representation". Let $E$ be an elliptic curve over an abelian totally real field $K$ unramified at 3,5, and 7. We prove that…

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…

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a…

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…

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…

For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is…

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism…

We prove that the existential theory of any function field $K$ of characteristic $p> 0$ is undecidable in the language of rings provided that the constant field does not contain the algebraic closure of a finite field. We also extend the…

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give…

We study real trigonal curves and elliptic surfaces of type $\I$ (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's \emph{dessins d'enfants}. We give a…

Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (\zeta_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that…

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form…

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K/\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the…

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational…