Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

The arithmetic of curves defined by iteration

100%

Hindes W.

Acta Arithmetica

2015

tom 169

nr 1

1-27

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.

Points on elliptic curves parametrizing dynamical Galois groups

100%

Hindes W.

Acta Arithmetica

2013

tom 159

nr 2

149-167

We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials x²+c whose third iterate has a "small" Galois group by determining the rational points on some elliptic curves. It follows as a corollary that the only integer value with this property is c=3, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational points on quadratic twists of a hyperelliptic curve, we indicate how the ABC conjecture implies a finite index result, suggesting a geometric interpretation of this problem.

Ograniczanie wyników

2 Acta Arithmetica

2 Hindes W.

1 2015

1 2013

Wyniki wyszukiwania

The arithmetic of curves defined by iteration

Points on elliptic curves parametrizing dynamical Galois groups