有理点的一致性:更新和修正

IF 0.8 Q2 MATHEMATICS Tunisian Journal of Mathematics Pub Date : 2022-03-30 DOI:10.2140/tunis.2022.4.183
L. Caporaso, J. Harris, B. Mazur
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引用次数: 2

摘要

在[2]中,我们断言,假设强朗猜想(下面的猜想1)成立,一个非常强的有界性形式成立:对于每一个g≥2,存在一个有限界N(g) -不依赖于K!-使得对于任意数域K,只有有限多个在K上定义的g属曲线的同构类具有大于N(g)个K-有理点。问题是,在这个表述中我们是指K上,还是代数闭包K上,有有限多个同构类吗?本文以较强的形式证明了K上同构的命题,但证明只建立了较弱的命题,即有有限多条大于N(g)的曲线指向K上同构。
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Uniformity of rational points: an up-date and corrections
In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below), a very strong form of boundedness holds: for every g ≥ 2 there is a finite bound N(g)—not depending on K!—such that for any number field K there are only finitely many isomorphism classes of curves of genus g defined over K with more than N(g) K-rational points. The issue is, in that statement do we mean finitely many isomorphism classes over K, or over the algebraic closure K? The paper asserts the statement in the stronger form—up to isomorphism over K—but the proof establishes only the weaker statement that there are finitely many curves with more than N(g) points up to isomorphism over K.
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来源期刊
Tunisian Journal of Mathematics
Tunisian Journal of Mathematics Mathematics-Mathematics (all)
CiteScore
1.70
自引率
0.00%
发文量
12
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