Related papers: On some special classes of complex elliptic curves

We classify the log del Pezzo surface (S,B) of rank 1 with no 1-,2-,3-,4-, or 6-complements with the additional condition that B has one irreducible component C which is an elliptic curve, and that C has the coefficient b in B with…

We split the program of explicit descent of elliptic curves into two parts. For $n=3$ and $n=5,$ we first display a model for the universal elliptic curve $E$ with full level $n$ structure and describe the map of rational points of $E$ to…

We consider the universal family $E_n^d$ of superelliptic curves: each curve $\Sigma_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $\Sigma_n^d\hookrightarrow E_n^d\to…

We say that two elliptic curves E_1, E_2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character \chi of K, there is an isomorphism between the n-Selmer groups Sel_n(E_1^\chi/K) and…

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of…

In this paper, we give some explicit Diophantine parameters of the cyclic torsion subgroups of odd order of elliptic curves over $\mathbb{Q}$.

Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N…

Let $\mathcal{E}$ be a CM elliptic curve defined over a number field $K$, with Weiestrass form $y^3=x^3+bx$ or $y^2=x^3+c$. For every positive integer $m$, we denote by ${\mathcal{E}}[m]$ the $m$-torsion subgroup of ${\mathcal{E}}$ and by…

Let $C$ be a curve of genus 2 and $\psi_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $\phi_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering.…

Let $E$ be a $\mathbb Q$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb Q$-curve. We show that the question has a positive answer if and only…

Let $E$ be an elliptic curve over $\mathbb{Q}$. In this paper we study two certain modular curves which parameterize families of elliptic curves which are directly (resp. reverse) 6-congruent to $E$ together with the explicit…

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}_1.$ We determine the odd-order torsion groups that can arise as $E(L)_{\text{tor}}$ where $L$ is a…

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod l^n…

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class…

In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq…

Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \mathbb{N}}$ on this curve. If one wishes to study $\mathbb{Z}$--linear combination of…

Let $K$ be a global field of finite characteristic $p\geq2$, and let $E/K$ be a non-isotrivial elliptic curve. We give an asympotoic formula of the number of places $\nu$ for which the reduction of $E$ at $\nu$ is a cyclic group. Moreover…

In this paper, we find a power series expansion of the invariant differential $\omega_E$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we introduce…

Let $M \mid N$ be positive integers, and let $\Delta$ be the discriminant of an order in an imaginary quadratic field $K$. When $\Delta_K < -4$, the first author determined the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the…

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least…