Suppose that F is a function field of a single variable over a finite field. The automorphism group Aut(F) acts on the places of F and permutes all places of a given degree. I have a few questions:


1) if the action an automorphism sigma on the rational places is trivial, ie sigma fixes every rational place, does it follow that sigma is the identity? (I'm assuming that F has more than one rational place).


2) is it possible for an automorphism to switch two rational places and keep the remaining rational places fixed? (I'm assuming that F has at least 3 rational places).


In the case of the Hermitian function fields, the answer is yes to 1) and no to 2).


Thanks.
Hiren



automorphism group of a function field
Consider the automorphism of the curve $y^2=x^p-x$ over $\mathbb F\_p$ thar sends $y$ to $-y$. This fixes every $\mathbb F\_p$-point but is not trivial. Taking function fields, we get a counterexample to 1.


The same automorphism of $y^2=x^p-x^{p-1}-x+1$ fixes all the rational points but $(0,1)$ and $(0,-1)$ which it switches.


