Galois Image Research Articles

2-Adic Galois Images of Non-CM Isogeny-Torsion Graphs Defined over ℚ

Let E be a Q -isogeny class of elliptic curves defined over Q without CM. The isogeny graph associated to E is a graph which has a vertex for each elliptic curve in E and an edge for each Q -isogeny of prime degree that maps one elliptic curve in E to another elliptic curve in E , with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over Q of the corresponding elliptic curve. Then, the main statement of the article is a classification of the 2-adic Galois image that occurs at each vertex of all isogeny-torsion graphs consisting of elliptic curves defined over Q without CM.

Products of hyperelliptic Jacobians with maximal Galois image

In this note we study the associated adelic representation of a product of hyperelliptic Jacobians. We give a simple criterion that ensures that this representation has maximal Galois image in a certain sense. As an application, we provide a method of constructing products of Jacobians with division fields as disjoint as they can be.

Remarks on the inverse Galois problem over function fields

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of ℓ-adic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible systems.

2-Adic Galois images of isogeny-torsion graphs over ℚ with CM

Let [Formula: see text] be a [Formula: see text]-isogeny class of elliptic curves defined over [Formula: see text]. The isogeny graph associated to [Formula: see text] is a graph which has a vertex for each elliptic curve over [Formula: see text] of [Formula: see text] and an edge for each [Formula: see text]-isogeny of prime degree that maps one elliptic curve in [Formula: see text] to another elliptic curve in [Formula: see text], with the degree of the isogeny recorded as a label of the edge. The isogeny-torsion graph associated to [Formula: see text] is the isogeny graph associated to [Formula: see text] where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over [Formula: see text] of the corresponding elliptic curve. The main result of the article is a classification of the [Formula: see text]-adic Galois image at each vertex of the isogeny-torsion graphs whose associated [Formula: see text]-isogeny class consists of elliptic curves over [Formula: see text] with complex multiplication.

Families of Abelian Varieties and Large Galois Images

Abstract Associated with an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho _A\colon \operatorname {Gal}({\overline {K}}/K)\to \operatorname {GL}_{2g}(\widehat {{\mathbb {Z}}})$. The representation $\rho _A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a non-empty open subvariety $U\subseteq {\mathbb {P}}^n_K$. We show that away from a set of density $0$, the image of $\rho _A$ will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results that assumed that the family of abelian varieties has “big monodromy”. We also give a version for families of abelian varieties with a more general base.

Some uniform bounds for elliptic curves over ℚ

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of $E$, and the Kummer extensions generated by points of infinite order in $E(\mathbb Q)$.

A bound for the image conductor of a principally polarized abelian variety with open Galois image

Let A A be a principally polarized abelian variety of dimension g g over a number field K K . Assume that the image of the adelic Galois representation of A A is an open subgroup of G S p 2 g ( Z ^ ) GSp_{2g}(\hat {\mathbb {Z}}) . Then there exists a positive integer m m so that the Galois image of A A is the full preimage of its reduction modulo m m . The least m m with this property, denoted m A m_A , is called the image conductor of A A . Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for m A m_A , in terms of standard invariants of A A , in the case that A A is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.

A classification of ECM-friendly families of elliptic curves using modular curves

In this work, we establish a link between the classification of ECM-friendly elliptic curves and Mazur’s program B, which consists in parameterizing all the families of elliptic curves with exceptional Galois image. Motivated by Barbulescu et al. [ANTS X–proceedings of the tenth algorithmic number theory symposium, Berkeley, CA, 2013], we say an elliptic curve is ECM-friendly if it does not have complex multiplication and if its Galois image is exceptional for some level. Building upon two recent works which treated the case of congruence subgroups of prime-power level which occur for infinitely many j j -invariants, we prove that there are exactly 1525 families of rational elliptic curves with distinct Galois images which are cartesian products of subgroups of prime-power level. This makes a complete list of rational families of ECM-friendly elliptic curves with cartesian Galois images, out of which less than 23 were known in the literature. We furthermore refine a heuristic of Montgomery to compare these families and conclude that the best 4 families which can be put in a = − 1 a=-1 twisted Edwards’ form are new.

Boundedness results for 2‐adic Galois images associated to hyperelliptic Jacobians

AbstractLet K be a number field, and let C be a hyperelliptic curve over K with Jacobian J. Suppose that C is defined by an equation of the form for some irreducible monic polynomial of discriminant Δ and some element . Our first main result says that if there is a prime of K dividing but not (2Δ), then the image of the natural 2‐adic Galois representation is open in and contains a certain congruence subgroup of depending on the maximal power of dividing . We also present and prove a variant of this result that applies when C is defined by an equation of the form for distinct elements . We then show that the hypothesis in the former statement holds for almost all and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.

A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve

Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.

Non-openness of v-adic Galois representation for A-motives

Drinfeld modules and A-motives s are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and [Formula: see text]-motives, one can construct their [Formula: see text]-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for [Formula: see text]-motives. Here, we clarify the rank-one case for A-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.

On polycyclic codes over a finite chain ring

Galois images of polycyclic codes over a finite chain ring $ S $ and their annihilator dual are investigated. The case when a polycyclic code is Galois-disjoint over the ring $ S, $ is characterized and, the trace codes and restrictions of free polycyclic codes over $ S $ are also determined giving an analogue of Delsarte's theorem relating the trace code and the annihilator dual code.

Large 2-adic Galois image and non-existence of certain abelian surfaces over ${\mathbb Q}$

Large 2-adic Galois image and non-existence of certain abelian surfaces over ${\mathbb Q}$

The Galois variance of constacyclic codes

This paper studies the Galois images of constacyclic codes over Fqm of length relatively prime to q, and determines when those images are equal and when they intersect only at the zero codeword. The subfield subcodes and trace codes of constacyclic codes are also determined.

Genus of abstract modular curves with level-ℓ structures

Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect 𝔽 ℓ {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with ℓ {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in 𝔽 ℓ {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.

COMPUTING IMAGES OF GALOIS REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES

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 present two probabilistic algorithms to simultaneously determine$G_{E}(\ell )$up to local conjugacy for all primes$\ell$by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine$G_{E}(\ell )$up to one of at most two isomorphic conjugacy classes of subgroups of$\mathbf{GL}_{2}(\mathbf{Z}/\ell \mathbf{Z})$that have the same semisimplification, each of which occurs for an elliptic curve isogenous to$E$. Under the GRH, their running times are polynomial in the bit-size$n$of an integral Weierstrass equation for $E$, and for our Monte Carlo algorithm, quasilinear in$n$. We have applied our algorithms to the non-CM elliptic curves in Cremona’s tables and the Stein–Watkins database, some 140 million curves of conductor up to $10^{10}$, thereby obtaining a conjecturally complete list of 63 exceptional Galois images$G_{E}(\ell )$that arise for$E/\mathbf{Q}$without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images$G_{E}(\ell )$that arise for non-CM elliptic curves over quadratic fields with rational$j$-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the$j$-invariant is irrational.

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

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, \frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\equiv 0\pmod 3. We relate this sequence $\{b_n\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match that of $E$.

Large Galois images for Jacobian varieties of genus 3 curves

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation \rho_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).

Images of 2-adic representations associated to hyperelliptic Jacobians

TextLet k be a subfield of C which contains all 2-power roots of unity, and let K=k(α1,α2,…,α2g+1), where the αi's are independent and transcendental over k, and g is a positive integer. We investigate the image of the 2-adic Galois action associated to the Jacobian J of the hyperelliptic curve over K given by y2=∏i=12g+1(x−αi). Our main result states that the image of Galois in Sp(T2(J)) coincides with the principal congruence subgroup Γ(2)◁Sp(T2(J)). As an application, we find generators for the algebraic extension K(J[4])/K generated by coordinates of the 4-torsion points of J. VideoFor a video summary of this paper, please visit http://youtu.be/VXEGYxA6N8w.

Controlling the Galois images in one-dimensional families of ℓ-adic representations

Let k be a finitely generated field of characteristic 0 and ℓ a prime. Let S be a smooth, separated and geometrically connected scheme over k and let ρ:π1(S)→GLr(Zℓ) be an ℓ-adic representation of the étale fundamental group of S. Let G and G¯ denote the images of π1(S) and π1(Sk¯) respectively. Given a closed point s∈S, we write Gs for the image of the absolute Galois group Γk(s) of the residue field k(s) viewed as a decomposition group at s in π1(S). By previous works of the authors, it is known that, when S is a curve, for all d≥1 and all but finitely many s∈S with [k(s):k]≤d, the codimension of Gs in G is at most 2. In this note, we improve this rigidity result as follows. Write g, g¯, gs for the Lie algebras of G, G¯, Gs respectively. Then for all but finitely many s∈S with [k(s):k]≤d, one of the following holds: (i) the codimension of Gs in G is at most 1 and gs contains D(g¯):=[g¯,g¯]; or (ii) the codimension of Gs in G is 2 and gs contains D2(g¯):=D(D(g¯)). We also obtain an arithmetic variant of this result, which involves the derived series of g.