Solvable Points on Projective Algebraic Curves

[摘要] We examine the problem of finding rational points defined oversolvable extensions on algebraic curves defined over general fields.We construct non-singular, geometrically irreducible projective curveswithout solvable points of genus $g$, when $g$ is at least $40$, overfields 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 provethat every genus $1$ curve defined over a local field ofcharacteristic zero with residue field of characteristic $p$ has adivisor of degree prime to $6p$ defined over a solvable extension.

[学科分类] 数学（综合）