Abstract:Let C be a curve of genus at least three defined over a number field, and let r be the rank of the rational points of its Jacobian. Under mild hypotheses on r, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on C by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on r: when r is small, there should be fewer points on C. In a 2013 paper, Stoll established such a "rank-favorable" bound for hyperelliptic curves… Show more
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