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Abstract Let F₆ be the family of monic odd-degree hyperelliptic curves of genus g over Q. Poonen and Stoll have shown that for every g 3, a positive proportion of curves in F₆ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each g 4, a positive proportion of curves in F₆ have no points defined over quadratic extensions except those that arise by pulling back rational points from P^1.
Laga et al. (Tue,) studied this question.