Related papers: Explicit open images for elliptic curves over $\ma…

We consider elliptic curves $E / \mathbb{Q}$ for which the image of the adelic Galois representation $\rho_E$ is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their…

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E for a…

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The…

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when…

Let $N$ be a positive integer. Let $\operatorname{H}$ be a group of level $N$ and let $E$ be an elliptic curve defined over the rationals with $\textit{j}_{E} \neq 0, 1728$. Then the image…

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex…

Let E be an elliptic curve over Q, and rho_l: Gal(Q) --> GL_2(Z_l) its l-adic Galois representation. Serre observed that for l>3 there is no proper closed subgroup of SL_2(Z_l) that maps surjectively onto SL_2(Z/lZ), and concluded that if…

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of…

Let $E$ be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb{Q}$ has open image, and in particular there is a minimal natural number $C_E$…

A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N \geq 1$. For a given subgroup $G \le…

We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop…

We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the…

Using an analogue of Serre's open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve $E$ defined over a number field $F$ a natural number $\mathcal{I}(E/F)$ which describes how big the…

Given an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation $\rho_E$. In particular, if $\operatorname{h}_{\mathcal{F}}(E)$…

We investigate the higher Chow groups, specifically $SK_1(E)$ for elliptic curves $E$ over number fields $F$. Focusing on the kernel $V(E)$ of the norm map $SK_1(E)\to F^{\times}$, we analyze its mod $p$ structure. We provide conditions,…

We gives an explicit genus 3 curve over Q such that the Galois action on the torsion points of its Jacobian is a large as possible. That such curves exist is a consequence of a theorem of D. Zureick-Brown and the author; however, those…

We develop an algorithm to test whether a non-CM elliptic curve $E/\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing…

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to…

We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of…

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational…