New points on curves
Volume 186 / 2018
Abstract
Let be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to \bigcup_F X(F), where F runs over all proper subfields K \subseteq F \subset L. Fix now an integer g \gt 0 and a finite separable extension L/K of degree d. We investigate whether there exists a smooth proper geometrically connected curve of genus g with a new point in X(L). We show for instance that if K is infinite with {\rm char}(K)\neq 2 and g \geq \lfloor d/4\rfloor, then there exist infinitely many hyperelliptic curves X/K of genus g, pairwise non-isomorphic over \overline{K}, and with a new point in X(L). When 1 \leq d \leq 10, we show that there exist infinitely many elliptic curves X/K with pairwise distinct j-invariants and with a new point in X(L).
