The arithmetic of curves defined by iteration

Wade Hindes

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Rational points on twists of X₀(63)

Nils Bruin, Julio Fernández, Josep González, Joan-C. Lario (2007)

Acta Arithmetica

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Trivial points on towers of curves

Xavier Xarles (2013)

Journal de Théorie des Nombres de Bordeaux

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In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

Points on elliptic curves parametrizing dynamical Galois groups

Wade Hindes (2013)

Acta Arithmetica

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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...

Pencils of irreducible rational curves and plane Jacobian conjecture

Nguyen Van Chau (2011)

Annales Polonici Mathematici

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In certain cases the invertibility of a polynomial map F = (P,Q): ℂ²→ ℂ² can be characterized by the irreducibility and the rationality of the curves aP+bQ = 0, (a:b) ∈ ℙ¹.

Quadratic forms, fibre products and some plane curves with many points

Motoko Qiu Kawakita, Shinji Miura (2002)

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Divisors on general curves and cuspidal rational curves.

D. Eisenbud, J. Harris (1983)

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Non-existence of points rational over number fields on Shimura curves

Keisuke Arai (2016)

Acta Arithmetica

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Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we extend their results to the case of number fields of higher degree. We also give counterexamples to the Hasse principle on Shimura curves.