New Points on Curves

TL;DR

This paper explores the existence of new points on algebraic curves over finite extensions, establishing conditions under which infinitely many hyperelliptic and elliptic curves possess such points, depending on the genus and extension degree.

Contribution

It provides new results on the existence of curves with new points over finite extensions, particularly for hyperelliptic and elliptic curves, based on genus and extension degree.

Findings

Existence of infinitely many hyperelliptic curves with new points for genus g ≥ [d/4] in characteristic ≠ 2.

Existence of infinitely many elliptic curves with new points for degrees 1 to 10.

Construction of non-isomorphic curves with pairwise distinct j-invariants and new points.

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 the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and a finite separable extension L/K of degree d. We investigate in this article 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 of characteristic different from 2 and g is bigger or equal to [d/4], then there exist infinitely many hyperelliptic curves X/K of genus g, pairwise non-isomorphic over the algebraic closure of K, and with a new point in X(L). When d is between 1 and 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).