×

zbMATH — the first resource for mathematics

Arboreal Galois representations. (English) Zbl 1206.11069
Summary: Let \(G_{\mathbb{Q}}\) be the absolute Galois group of \(\mathbb{Q}\), and let \(T\) be the complete rooted \(d\)-ary tree, where \(d \geq 2\). In this article, we study “arboreal” representations of \(G_{\mathbb{Q}}\) into the automorphism group of \(T\), particularly in the case \(d = 2\). In doing so, we propose a parallel to the well-developed and powerful theory of linear \(p\)-adic representations of \(G_\mathbb{Q}\). We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of \(G_{\mathbb{Q}}\). Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of \(\operatorname{Aut}(T)\) can occur as the image of an arboreal representation of \(G_{\mathbb{Q}}\).

MSC:
11F80 Galois representations
11R32 Galois theory
20E08 Groups acting on trees
20E18 Limits, profinite groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aitken W., Hajir F., Maire C., (2005) Finitely ramified iterated extensions. Int. Math. Res. Not. 14, 855–880 · Zbl 1160.11356 · doi:10.1155/IMRN.2005.855
[2] Boston, N.: Galois groups of tamely ramified p-extensions. In: Proceedings of Journées Arithmetiques 2005, special issue of Journal de Théorie des Nombres de Bordeaux. To appear (2006) · Zbl 1123.11038
[3] Boston N., Jones R. Densely settled groups and arboreal Galois representations. preprint (2006)
[4] Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14(4), 843–939 (2001) (electronic) · Zbl 0982.11033
[5] Fontaine, J.-M., Mazur, B.: Geometric Galois representations. In: Coates, J., Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I Internat. Press, Cambridge, MA, pp. 41–78.
[6] Jones, R.: The density of prime divisors in the arithmetic dynamics of quadratic polynomials. Available at http://www.arxiv.org . · Zbl 1193.37144
[7] Labute J. Mild pro p-groups and Galois groups of p-extensions of Q. J. Reine Angew. Math. Yau, S.-T.(eds.) 596, (2006), 155–182. · Zbl 1122.11076
[8] Markšaitis G.N., (1963) On p-extensions with one critical number.Izv. Akad. Nauk SSSR Ser. Mat. 27, 463–466
[9] Nekrashevych V. (2005) Self-similar groups, Vol. 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI · Zbl 1087.20032
[10] Odoni R.W.K. The Galois theory of iterates and composites of polynomials. Proc. London Math. Soc. (3) 51(3), (1985), 385–414. · Zbl 0622.12011
[11] Serre J.-P. (1972), Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4): 259–331 · Zbl 0235.14012 · doi:10.1007/BF01405086
[12] Serre J.-P. (1987), Sur les représentations modulaires de degré 2 de \({\mathrm Gal}(\overline{Q}/Q)\) . Duke Math. J. 54(1): 179–230 · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5
[13] Shimura G., (1966) A reciprocity law in non-solvable extensions. J. Reine Angew. Math. 221, 209–220 · Zbl 0144.04204 · doi:10.1515/crll.1966.221.209
[14] Stoll M., (1992) Galois groups over Q of some iterated polynomials. Arch. Math. (Basel) 59(3): 239–244 · Zbl 0758.11045
[15] Wiles A. (1995), Modular elliptic curves and Fermat’s last theorem’. Ann. of Math. (2) 141(3): 443–551 · Zbl 0823.11029 · doi:10.2307/2118559
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.