1,720,978 research outputs found
Exceptional scatteredness in prime degree
Full text linkLet q be an odd prime power and n an integer. Let l∈Fq(n) be a q-linearized t-scattered polynomial of linearized degree r. Let d=max{t,r} be an odd prime number. In this paper we show that under these assumptions it follows that l=x. Our technique involves a Galois theoretical characterization of t-scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field Fq
The set of stable primes for polynomial sequences with large Galois group
Full text linkLet K be a number field with ring of integers OK, and let {fk}k∈N be a sequence of monic polynomials in OK[x] such that for every n ∈ N, the composition f(n) = f1 ◦ f2 ◦ ... ◦ fn is irreducible. In this paper we show that if the size of the Galois group of f(n) is large enough (in a precise sense) as a function of n, then the set of primes p ⊆ OK such that every f(n) is irreducible modulo p has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f(n) is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences
On the existence of infinite, non-trivial F-sets
Full text linkIn this paper we prove a conjecture of J. Andrade, S.J. Miller, K. Pratt and M. Trinh, showing the existence of a non-trivial infinite F-set over Fq[x] for every fixed q. We also provide the proof of a refinement of the conjecture, involving the notion of width of an F-set, which is a natural number encoding the complexity of the set
The inverse problem for arboreal Galois representations of index two
Full text linkThis paper introduces a systematic approach towards the inverse problem for
arboreal Galois representations of finite index attached to quadratic
polynomials. Let be a field of characteristic , be
monic and quadratic and be the arboreal Galois representation
associated to , taking values in the group of
automorphisms of the infinite binary tree. We give a complete description of
the maximal closed subgroups of each closed subgroup of index at most two of
in terms of linear relations modulo squares among certain
universal functions evaluated in elements of the critical orbit of . We use
such description in order to derive necessary and sufficient criteria for the
image of to be a given subgroup of index two of . These
depend exclusively on the arithmetic of the critical orbit of . Afterwards,
we prove that if , then there exist exactly five
distinct subgroups of index two of that can appear as images
of for infinitely many , where
is the specialized polynomial. We show that two of them appear
infinitely often, and if Vojta's conjecture over holds true, then
so do the remaining ones. Finally, we give an explicit description of the
derived series of each subgroup of index two. Using this, we introduce a
sequence of combinatorial invariants for subgroups of index two of
. With a delicate use of these invariants we are able to
establish that such subgroups are pairwise non-isomorphic as topological
groups, a result of independent interest. This implies, in particular, that the
five aforementioned groups are pairwise distinct topological groups, and
therefore yield five genuinely different instances of the infinite inverse
Galois problem over .Comment: Comments are welcome
Constraining images of quadratic arboreal representations
Full text linkIn this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-critically finite, and we combine our results with local class field theory to classify quadratic pairs over with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other hand we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence towards Jones' finite index conjecture
An equivariant isomorphism theorem for mod reductions of arboreal Galois representations
Full text linkLet be a quadratic, monic polynomial with coefficients in , where is a localization of a number ring . In this paper, we first prove that if is non-square andnon-isotrivial, then there exists an absolute, effective constant with the following property: for all primes such that the reduced polynomial \phi_\mathfrak p\in (\mathcalO_{F,D}/\mathfrak p)[t][x] is non-square and non-isotrivial, the squarefree Zsigmondy set of is bounded by . Using this result, we prove that if is non-isotrivial and geometrically stable thenoutside a finite, effective set of primes of the geometric part of the arboreal representation of is isomorphic to that of . As an application of our results we prove R. Jones' conjecture on the arboreal Galois representation attached to the polynomial .<br
Strongly modular models of Q-curves
Full text linkLet E be a Q-curve without complex multiplication. We address the problem of deciding whether E is geometrically isomorphic to a strongly modular Q-curve. We show that the question has a positive answer if and only if E has a model that is completely defined over an abelian number field. Next, if E is completely defined over a quadratic or biquadratic number field L, we classify all strongly modular twists of E over L in terms of the arithmetic of L. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of L
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