Canad. J. Math. 56(2004), 612-637
Printed: Jun 2004
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.
14H25 - Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
11D88 - $p$-adic and power series fields