Let C be a smooth plane curve of degree d ≥ 4 defined over a global field k of characteristic p = 0 or p > (d − 1)(d − 2)/2 (up to an extra condition on Jac(C)). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions k(D) when k is a number field, in which we may have more points of C than these over k. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets ℰ and 𝒟 of isomorphism classes of smooth projective plane quartic curves over k with a prescribed automorphism group, such that all members of ℰ (respectively 𝒟) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field k. We verify the conjecture over k = ℚ for G = ℤ/6ℤ and GAP(16, 13). The analog of the conjecture over global fields with p > 0 is also considered.

中文翻译：

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双椭圆光滑平面曲线和二次点

让C是平滑的度数平面曲线d ≥ 4在全局字段上定义ķ有特色的p = 0要么p > (d - 1)(d - 2)/2（最多附加条件江淮(C)）。除非曲线是四次双椭圆曲线，否则我们观察到它总是包含有限多个二次点。我们进一步表明，只有有限多个二次扩展ķ(D)什么时候ķ是一个数字字段，其中我们可能有更多的点C比这些ķ. 特别是，我们有这种渐近现象适用于费马和克莱因方程。其次，我们推测有两个无限集ℰ和𝒟光滑射影平面四次曲线的同构类ķ具有规定的自同构群，使得ℰ（分别𝒟) 是双椭圆的，并且在一个数域上具有有限（分别为无限）许多二次点ķ. 我们验证猜想ķ = ℚ为了G = ℤ/6ℤ和差距(16, 13). 全球场猜想的类比p > 0也被考虑。