# The arithmetic of curves defined by iteration

Acta Arithmetica (2015)

- Volume: 169, Issue: 1, page 1-27
- ISSN: 0065-1036

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topWade Hindes. "The arithmetic of curves defined by iteration." Acta Arithmetica 169.1 (2015): 1-27. <http://eudml.org/doc/286431>.

@article{WadeHindes2015,

abstract = {We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of rational quadratic polynomials x² + c, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when f(x) = x² + 3. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.},

author = {Wade Hindes},

journal = {Acta Arithmetica},

keywords = {quadratic polynomial; dynamical arboreal representation; Hall-Lang conjecture; rational points on curves},

language = {eng},

number = {1},

pages = {1-27},

title = {The arithmetic of curves defined by iteration},

url = {http://eudml.org/doc/286431},

volume = {169},

year = {2015},

}

TY - JOUR

AU - Wade Hindes

TI - The arithmetic of curves defined by iteration

JO - Acta Arithmetica

PY - 2015

VL - 169

IS - 1

SP - 1

EP - 27

AB - We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of rational quadratic polynomials x² + c, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when f(x) = x² + 3. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.

LA - eng

KW - quadratic polynomial; dynamical arboreal representation; Hall-Lang conjecture; rational points on curves

UR - http://eudml.org/doc/286431

ER -

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