New points on curves
Volume 186 / 2018
Abstract
Let $K$ 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 $X(L)$.
