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Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) in K[z]$ be a separable polynomial of the form $z^ell-c.$ Given $a in K$, we examine the Galois groups and ramification groups of the extensions of $K$ generated by the solutions to $f^n(z)=a$. The behavior depends upon $v(c)$, and we find that it shifts dramatically as $v(c)$ crosses a certain value: $0$ in the case $p mid ell$, and $-p/(p-1)$ in the case $p=ell$.
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We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. This allows us to study the behavior of the monodromy groups under specialization of the maps, and to derive applications to dynamical sequences.
The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0in K$, analogous to the action of Galois on the $ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=mathbb{Q}$, our condition holds for infinitely many choices of $x_0$.
We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / QQ_p$ we will construct a bijection [ CL_g : CA^0_g(G_2,K) rightarrow CG^0(G_2,K) ] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $rho : W_K to G_2(CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.
We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^{dag}(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaines formalism as presented in [Tsu98a]. Our method of proof will allow us to give a new interpretation of the Refined Swan Conductor.
Let $F/F^+$ be a CM field and let $widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $overline{r}: mathrm{Gal}(overline{mathbb{Q}}/F)rightarrow mathrm{GL}_n(overline{mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $mathrm{GL}_n(F_{widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $widetilde{v}|_{F^+}$ determines $overline{r}|_{mathrm{Gal}(overline{mathbb{Q}}_p/F_{widetilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.
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