# Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

@article{Cerchia2019UniformBO, title={Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves}, author={Michael Cerchia and Jeremy A. Rouse}, journal={arXiv: Number Theory}, year={2019} }

Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha)$ is the $\ell$-adic arboreal Galois representation $\omega_{E,\alpha,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to \mathbb{Z}_{\ell}^{2} \rtimes {\rm GL}_{2}(\mathbb{Z}_{\ell})$ describing the action of ${\rm Gal}(\overline{F}/F)$ on points $\beta_{n}$ so that $\ell^{n} \beta_{n} = \alpha$. We give an explicit bound on…

## 5 Citations

$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

- Mathematics
- 2021

We discuss the l-adic case of Mazur’s “Program B” over Q, the problem of classifying the possible images of l-adic Galois representations attached to elliptic curves E over Q, equivalently,…

Effective Kummer Theory for Elliptic Curves

- MathematicsInternational Mathematics Research Notices
- 2021

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha $ be the set of $N$-division points of $\alpha $ in…

N T ] 1 9 Ju l 2 02 1 lADIC IMAGES OF GALOIS FOR ELLIPTIC CURVES OVER Q

- 2021

We discuss the l-adic case of Mazur’s “Program B” over Q, the problem of classifying the possible images of l-adic Galois representations attached to elliptic curves E over Q, equivalently,…

SOME UNIFORM BOUNDS FOR ELLIPTIC CURVES OVER Q

- 2021

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image…

Some uniform bounds for elliptic curves over $\mathbb Q$

- Mathematics
- 2021

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E/Q. We consider in particular the subgroup of scalars in the image…

## References

SHOWING 1-10 OF 20 REFERENCES

Explicit Kummer Theory for Elliptic Curves

- Mathematics
- 2019

Let $E$ be an elliptic curve defined over a number field $K$, let $\alpha \in E(K)$ be a point of infinite order, and let $N^{-1}\alpha$ be the set of $N$-division points of $\alpha$ in $E(\bar{K})$.…

The density of primes dividing a particular non-linear recurrence sequence

- Mathematics
- 2015

Define the sequence $\{b_n\}$ by $b_0=1,b_1=1, b_2=2,b_3=1$, and $$b_n=\begin{cases} \frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\not\equiv 0\pmod 3,…

Pursuing polynomial bounds on torsion

- MathematicsIsrael Journal of Mathematics
- 2018

We show that for all ϵ > 0, there is a constant C(ϵ) > 0 such that for all elliptic curves E defined over a number field F with j(E) ∈ Q we have $$\# E\left( F \right)\left[ {tors} \right] \leq…

Elliptic curves over ℚ
$\mathbb {Q}$
and 2-adic images of Galois

- Mathematics
- 2014

AbstractWe give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over ℚ
$\mathbb {Q}$
. To this end, we compute the ‘arithmetically maximal’…

Galois theory of iterated endomorphisms

- Mathematics
- 2010

Given an abelian algebraic group A over a global field F, � 2 A(F), and a prime `, the set of all preimages ofunder some iterate of (`) generates an extension of F that contains all `-power torsion…

A UNIFORM OPEN IMAGE THEOREM FOR `-ADIC REPRESENTATIONS I

- 2008

Let k be a field finitely generated over Q and let X be a smooth, separated and geometrically connected curve over k. Fix a prime `. A representation ρ : π1(X) → GLm(Z`) is said to be geometrically…

Curves with infinitely many points of fixed degree

- Mathematics
- 1994

Thed-th symmetric productC(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang [Hi] proved by Faltings…

Elliptic curves over Q and 2-adic images of Galois

- 2015

We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over Q. To this end, we compute the ‘arithmetically maximal’ tower of 2-power level…

Density of odd order reductions for elliptic curves with a rational point of order 2

- MathematicsInternational Journal of Number Theory
- 2019

Suppose that [Formula: see text] is an elliptic curve with a rational point [Formula: see text] of order [Formula: see text] and [Formula: see text] is a point of infinite order. We consider the…

Fundamentals Of Diophantine Geometry

- Computer Science
- 2016

The fundamentals of diophantine geometry is universally compatible with any devices to read, and is available in the digital library an online access to it is set as public so you can get it instantly.