TITLE: Upper bound for the order of the group of automorphisms of Riemann surfaces of genus 2
QUESTION [7 upvotes]: By the Hurwitz's automorphisms theorem there is an upper bound $|\text{Aut}(C)|\leq 84(g-1)$ for all Riemann surfaces $C$ with $g(C)\geq 2$, but it is not sharp if $g=2$. What is the sharp upper bound for genus $2$ Riemann surfaces? Also, I'm asking for the simple proof of the fact that there is no genus $2$ Riemann surfaces with $|Aut(C)|=84$.

REPLY [7 votes]: It follows immediately from Hurwitz' theorem on the genus that a curve of genus 2 cannot have an automorphism of order 7, by considering the quotient map by this action.  Hurwitz himself proved that if 84(g-1) does not occur, the next largest possible bound is 48(g-1).  I.e. his method gives a list of possible bounds, not just one maximal bound.
Mathematische Annalen
1892, Volume 41, Issue 3, pp 403-442
This is offered as an argument for reading the original work.
see also:
Hurwitz's automorphisms theorem with deformations