Displaying similar documents to “A Diophantine problem on algebraic curves over function fields of positive characteristic”

Descent via (3,3)-isogeny on Jacobians of genus 2 curves

Nils Bruin, E. Victor Flynn, Damiano Testa (2014)

Acta Arithmetica

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We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate-Shafarevich group.

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.