# automorphisms of an étale cover of a curve

The base field is algebraically closed and of chatacteristic zero. If $$X$$ is a smooth projective curve and $$Y\to X$$ is an étale covering of $$X$$ of degree $$d$$, then what can we say about the automorphism group $$\mathrm{Aut}(Y/X)$$ of $$Y$$ over $$X$$? Is it always nontrivial?

• By etale covering do you mean finite etale map? If so, this is just Galois theory. The map corresponds to a finite index index $d$ subgroup $H\le\pi_1(X)$, and the automorphism group is just the automorphism group of the corresponding $\pi_1(X)$-set. IIRC this should just be the quotient of the normalizer of $H$ by $H$. Sometimes it'll be trivial. For example you can fix a surjection $f : \pi_1(X)\rightarrow S_3$ (symmetric group on 3 things), and let $H$ be the preimage of an order 2 subgroup of $S_3$. The resulting cover will have trivial aut group for the same reason that $Q(\sqrt{3})$ does Jul 30, 2020 at 4:17
• I mean finite étale maps. I am confused that there may exist a nontrivial finite separable extension $F$ of $K(X)$ with trivial automorphism group $\mathrm{Aut}(F/K(X))$ and $F$ corresponds to a curve $Y$ by $Y=F(X)$. Jul 30, 2020 at 4:24
• What is $F(X)$? Jul 30, 2020 at 4:30
• I just checked your additional comments and try to make the picture clear. $K$ means the base field, and my comments comes from the correspondences between 1. smooth algebraic curves and its function field $X\leftrightarrow K(X)$, 2. finite etale morphism $Y\to X$ and finite separable field extension $K(Y)/K(X)$, and 3. the automorphism groups $\mathrm{Aut}(Y/X)\leftrightarrow \mathrm{Aut}(K(Y)/K(X))$. Jul 30, 2020 at 4:37
• Could explain what is the corresponding $\pi_1(X)$-set? How can say that the automorphism is trivial in your example? Jul 30, 2020 at 4:38