On Possible Genus 0 adelic Galois Images of non CM Elliptic Curves over Q\mathbb{Q}

Let $E$ be an elliptic curve defined over $\mathbb{Q}$/ Associated to $E$, there is an adelic Galois representation $\rho_E \colon {\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_2(\hat{\mathbb{Z}})$. In this article, we give possibilities for groups generated by $\rho_E({\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}))$ and $-I$ as $E$ varies over non CM elliptic curves defined over $\mathbb{Q}$ such that the group $\pm \rho_E({\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}))$ has genus 0. However, our list is not minimal.Comment: Comments are welcom

Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.Comment: 24 page

Serre curves relative to obstructions modulo 2

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 $\ell$-adic images, compute all examples of conductor at most 500,000, precisely describe the image of $\rho_E$, and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves

Drinfeld modules with maximal Galois action on their torsion points

To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and R\"utsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explicit examples of his openness theorem for elliptic curves. Using our specific examples, we will numerically test analogues of some well-known elliptic curve conjectures

Cyclic reduction densities for elliptic curves

For an elliptic curve E defined over a number field K, the heuristic density of the set of primes of K for which E has cyclic reduction is given by an inclusion–exclusion sum δE/K involving the degrees of the m-division fields Km of E over K. This density can be proved to be correct under assumption of GRH. For E without complex multiplication (CM), we show that δE/K is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of E and a universal infinite Artin-type product. For E admitting CM over K by a quadratic order O , we show that δE/K admits a similar ‘factorization’ in which the Artin type product also depends on O . For E admitting CM over K¯ by an order O⊄ K , which occurs for K= Q , the entanglement of division fields over K is non-finite. In this case we write δE/K as the sum of two contributions coming from the primes of K that are split and inert in O . The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities

Towards a Classification of Isolated jj-invariants

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 the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$.Comment: With an appendix by Maarten Derickx and Mark van Hoei