An Algorithm for computing Isomorphisms and Automorphism Groups of Function
Fields of Algebraic Curves
Florian Hess
TU, Berlin
Abstract:
Let $F_1/k$ and $F_2/k$ denote two algebraic function fields of some not
necessarily non-singular irreducible algebraic curves over a perfect field
$k$. An isomorphism $\phi$ of $F_1/k$ and $F_2/k$ is an isomorphism of fields
$\phi : F_1 \rightarrow F_2$ whose restriction to $k$ is the identity map.
The talk presents an algorithm available in Magma to compute the set of
isomorphisms $\phi$ of $F_1/k$ and $F_2/k$ if these algebraic function fields
have genus greater than or equal to two. The isomorphisms $\phi$ are described
by their action on the corresponding coordinates or field generators. For the
special case $F_1 = F_2 = F$ the algorithm computes the elements of the
automorphism group $\text{{\rm Aut}}_k(F)$ of $F/k$. We restrict to genus
greater than or equal to two since otherwise the number of isomorphisms and
the automorphism groups may be infinite and the task would involve quite
different techniques. Permutation and matrix representations of the
automorphism groups may then be computed using for example the action on
Weierstrass places or on the space of holomorphic differentials.