The following is a folklore result : Let $X$ be a compact Riemann surface of genus at least $2$ and let $f : X \rightarrow X$ be a biholomorphism. Then $f$ acts nontrivially on $H_1(X;\mathbb{Z})$.

I have two questions about this.

- Who proved this first?
- One proof I have been told derives this from the Lefschetz fixed point theorem. Namely, all the fixed points of $f$ have degree $1$, so the sum of the degrees of the fixed points is nonnegative. However, if $f$ acts as the identity on $H_1(X;\mathbb{Z})$, then the Lefschetz fixed point theorem says that the sum of the degrees of the fixed points is $1-2g+1$, which is negative. Where is the original source for this proof?

I found a couple of sources that give the reference

J.-P. Serre. Rigidit´e de foncteur d’Jacobi d’´echelon n ≥ 3. Sem. H. Cartan, 1960/1961, Appendix to Exp. 17, 1961.

for both questions 1 and 2. However, I see nothing in this reference about the Lefschetz fixed point theorem, and also no reference to Riemann surfaces (it is all about self-maps of abelian varieties).