Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent space) is of the form
\begin{bmatrix} r\cos(\theta) & -r\sin(\theta) \\ r\sin(\theta) & r\cos(\theta) \end{bmatrix}
The local Lefschetz number is always $1$, unless the tangent map is identity, which will impose $f$ to be the identity map. This fact can be easily seen when $f$ is lifted to the universal covering of $X$.
Can we carry on this idea to say something more about the group of automorphism of a compact Riemann surface of genus $g>1$? Recall that on such a Riemann surface, we have exactly $g(g-1)(g+1)$ Weierstrass points, counted with weight, and a biholomorphism must take Weierstrass points to Weierstrass points.
Thank you!