Related papers: Elliptic curves with non-abelian entanglements

In this work, we study elliptic curves of the form $E_L: y^2 = x(x-1)(x-L)$, where $L^2-L+1 \in \left(\mathbb{Q}^{\times}\right)^2$ and $L \in \mathbb{Q} \setminus \{0,1\}$. We will show that for almost all quadratic extensions…

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K^{ab} of K. We give: (i) an explicit family of diagonal plane cubic curves with…

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a…

We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any…

In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective representations $\rho$ of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic…

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

In this article, we consider a family of elliptic curves defined by $E_{m}: y^2= x^3 -m^2 x + (pqr)^2 $ where $m $ is a positive integer and $p, q, ~\text{and}~ r$ are distinct odd primes and study the torsion as well the rank of…

Let $2^+S_4$ denote the double cover of $S_4$ corresponding to the element in $H^2(S_4,\Z/2\Z)$ where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 (denoted $\tilde S_4$ in…

Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$…

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We…

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed…

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$…

A complex elliptic curve $E$ can be defined as the quotient of the analytic space $\mathbb{C}^*$ by a discrete action of the cyclic group $q^{\mathbb{Z}}$ for $\vert q\vert \neq 1$. We study the boundary case when $\vert q\vert =1$, which…

In this paper, we will study modular Abelian varieties with odd congruence numbers by examining the cuspidal subgroup of $J_0(N)$. We will show that the conductor of such Abelian varieties must be of a special type. For example, if $N$ is…

For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

Let $E$ be an elliptic curve over $\mathbb{Q}$ described by $y^2= x^3+ Kx+ L$ where $K, L \in \mathbb{Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb{Q})$ for $i=1, 2, \cdots, k$, is said to be a sequence of consecutive cubes on $E$…

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve $E$ in $\mathbb P^2$ and blow up nine general points on $E$. Then the complement $M$ of the…

Let k be a number field, let E/k be an elliptic curve, and let S be a finite set of places of k contianing the archimedean places. Let F be an algebraic closure of k. We prove that if a point P in E(F) is nontorsion, then there are only…

We generalize the notion of Elkies primes for elliptic curves to the setting of abelian varieties with real multiplication (RM), and prove the following. Let $A$ be an abelian variety with RM over a number field whose attached Galois…