245 research outputs found
Algebraic curves with many automorphisms
Let be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus defined over an algebraically closed field of odd
characteristic . Let be the group of all automorphisms of which
fix element-wise. It is known that if then the -rank
(equivalently, the Hasse-Witt invariant) of is zero. This raises the
problem of determining the (minimum-value) function such that whenever
then has zero -rank. For {\em{even}} we prove
that . The {\em{odd}} genus case appears to be much more
difficult although, for any genus , if has a solvable
subgroup such that then has zero -rank and fixes a
point of . Our proofs use the Hurwitz genus formula and the Deuring
Shafarevich formula together with a few deep results from finite group theory
characterizing finite simple groups whose Sylow -subgroups have a cyclic
subgroup of index . We also point out some connections with the Abhyankar
conjecture and the Katz-Gabber covers
Curves with more than one inner Galois point
Let be an irreducible plane curve of
where is an algebraically closed field of characteristic . A point is an inner Galois point for if
the projection from is Galois. Assume that has two
different inner Galois points and , both simple. Let and
be the respective Galois groups. Under the assumption that fixes ,
for , we provide a complete classification of and we exhibit a curve for each such . Our proof relies on deeper
results from group theory
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