This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over ℚ . The focus is on practical aspects of this problem in the case that the genus of C is at least 2 , and therefore the set of rational points is finite.
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Michael Stoll (2011) studied this question.
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