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Let C be an algebraic curve over a number field K. Given an extension of K, we say that C gains a point over this extension if it is generated by adjoining the coordinates of some point on C to K. In a recent paper, Mazur and Rubin posed the following question: to what extent does the set of number fields over which a curve gains a point determine the identity of the curve? Motivated by this ... read morequestion, we study the set of fields over which C gains a point, in the case that C is a hyperelliptic curve over the rationals with genus at least 2. For curves with odd degree and sufficiently large n, we give an unconditional lower bound on the number of degree n number fields up to a given discriminant for which C gains a point. We present progress toward a similar bound for hyperelliptic curves of even degree, and contrast these to a conditional result of Granville in the quadratic field case.read less
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