Solvable Points on Projective Algebraic Curves
Solvable Points on Projective Algebraic Curves
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射影代数曲线上的可解点
DOI：
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发表时间：
2004
期刊：
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通讯作者：
Ambrus Pál
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作者：
Ambrus Pál
摘要研究了在一般域上定义的代数曲线上的可解扩张上求有理点的问题。我们构造了非奇异的，几何上不可约的射影曲线没有可解点的亏格$g$，当$g$是至少40，在域的任意特征。证明了定义在任意域上的亏格为0、2、3或4的光滑的几何不可约射影曲线都有可解点。最后证明了定义在特征为零的局部域上的亏格为1的曲线在可解扩张上都有一个与6p互质的度因子。
Abstract We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus $g$ , when $g$ is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic $p$ has a divisor of degree prime to $6p$ defined over a solvable extension.