A large arboreal Galois representation for a cubic postcritically finite polynomial

被引：12

作者：

Benedetto R.L. ^{[1
]}

Faber X. ^{[2
]}

Hutz B. ^{[3
]}

Juul J. ^{[1
]}

Yasufuku Y. ^{[4
]}

机构：

[1] Amherst College, Amherst, MA

[2] Center for Computing Sciences, Institute for Defense Analyses, Bowie, MD

[3] Saint Louis University, Saint Louis, MO

[4] College of Science and Technology, Nihon University, Tokyo

来源：

Research in Number Theory | / 3卷 / 1期

基金：

美国国家科学基金会; 日本学术振兴会;

关键词：

D O I：

10.1007/s40993-017-0092-8

中图分类号：

学科分类号：

摘要：

We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not conjugate to a power map, Chebyshev polynomial, or Lattès map. The associated Galois action on an infinite ternary rooted tree has Hausdorff dimension bounded strictly between that of the infinite wreath product of cyclic groups and that of the infinite wreath product of symmetric groups. We deduce a zero-density result for prime divisors in an orbit under this polynomial. We also obtain a zero-density result for the set of places of convergence of Newton’s method for a certain cubic polynomial, thus resolving the first nontrivial case of a conjecture of Faber and Voloch. © 2017, The Author(s).

引用

共 15 条

[1]

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[2]

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[3]

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[4]

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[5]

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[6]

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[7]

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[8]

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[9]

Juul J., Iterates of generic polynomials and generic rational functions. arXiv:1410.3814v4 [math.NT], preprint, (2016)

[10]

Looper N.R., Dynamical Galois groups of trinomials and Odoni’s conjecture. arXiv:1609.03398v1 [math.NT], preprint, (2016)

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