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Displaying similar documents to “The genus of curves over finite fields with many rational points.”

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.

Trivial points on towers of curves

Xavier Xarles (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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.