Points on elliptic curves parametrizing dynamical Galois groups

• Autorzy: Wade Hindes
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 14G05: Rational points
• 37P55: Arithmetic dynamics on general algebraic varieties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 2
• Strony: 149-167

Daty

wydano

2013

Twórcy

autor

Wade Hindes

• Department of Mathematics, Brown University, Providence, RI 02912, U.S.A.

Identyfikatory

DOI

10.4064/aa159-2-5