Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,α)$, where $f$ is a polynomial over a number field $K$ and $α\in K$, such that the dynamical Galois group of the pair $(f,α)$ is abelian. In this paper we focus on the case of unicritical polynomials $f$, and more general dynamical systems attached to sequences of unicritical polynomials.
After having reduced the conjecture to the post-critically finite case, we establish it for all polynomials with periodic critical orbit, over any number field. We next establish the conjecture in full for all monic unicritical polynomials over any quadratic number field. Finally we show that for any given degree $d$ there exists a finite, explicit set of unicritical polynomials that depends only on $d$, such that if $f=ux^d+1$ is a unicritical polynomial over a number field $K$ that lies outside such exceptional set, then there are at most finitely many basepoints $α$ such that the dynamical Galois group of $(f,α)$ is abelian.
To obtain these results, we exploit in multiple ways the group theory of the generic dynamical Galois group to force diophantine relations in dynamical quantities attached to $f$. These relations force in all cases, outside of the ones conjectured by Andrews--Petsche, a contradiction either with lower bounds on the heights in abelian extensions, in the style of Amoroso--Zannier, or with the computation of rational points on explicit curves, carried out with techniques from Balakrishnan--Tuitman and Siksek.
In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton-Patel, and Vivaldi-Hatjispyros in the early 1990s con…
This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.
We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory for curves over a $p$-adic field.
We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.
Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $ρ_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $ρ_A$ 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 nonempty open subvariety $U\subseteq \mathbb{P}^n_K$. We show that away from a set of density $0$, the image of $ρ_A$ will be very large; more precisely, it will have uniformly bounded in…
We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.
In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.
Let $K$ be a number field, $f\in K[x]$ and $α\in K$. A recent conjecture of Andrews and Petsche predicts that the dynamical Galois group of the pair $(f,α)$ is abelian if and only if the pair $(f,α)$ is $K^{\text{ab}}$-conjugated to $(g,β)$, where $g$ is a power or a Chebyshev map and $β$ is $ζ$ or $ζ+ζ^{-1}$, respectively, and $ζ$ is a root of unity. We review three completely different approaches that allow to prove several cases of the conjecture.
Let $A$ be a principally polarized abelian variety of dimension $g$ over a number field $K$. Assume that the image of the adelic Galois representation of $A$ is an open subgroup of $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. Then there exists a positive integer $m$ so that the Galois image of $A$ is the full preimage of its reduction modulo $m$. The least $m$ with this property, denoted $m_A$, is called the image conductor (also called the level) of $A$. Jones recently established an upper bound for $m_A$, in terms of standard invariants of $A$, in the case that $A$ is an elliptic curve witho…
Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $> 3$. The result stated in the title is a special case.
We prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers of K to be free over its associated order in the rational group algebra Q[G].
We develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.
For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.
