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New points on curves

Volume 186 / 2018

Qing Liu, Dino Lorenzini Acta Arithmetica 186 (2018), 101-141 MSC: 11G05, 11G20, 11G30, 14G05, 14G25. DOI: 10.4064/aa170322-23-8 Published online: 26 October 2018


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 $X(L)$.


  • Qing LiuUniversité de Bordeaux
    Institut de Mathématiques de Bordeaux
    33405 Talence, France
    School of Mathematical Sciences
    Xiamen University
    361005 Xiamen, China
  • Dino LorenziniDepartment of Mathematics
    University of Georgia
    Athens, GA 30602, U.S.A.

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