Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero) {X}. One of the first facts one observes in their theory is that the group {\mathrm{Aut}(X)} of automorphisms of {X} is quite large when {X} has genus zero or one. When {X} has genus zero, it is the projective line, and its automorphism group is {\mathrm{PGL}_2(\mathbb{C})}, a fact which generalizes naturally to higher projective spaces. When {g = 1}, the curve {X} acquires the structure of an elliptic curve from any distinguished point. Thus, translations by any element of {X} act on {X}. So {\mathrm{Aut}(X)} has points corresponding to each element of {X} (and a few more, such as inversion in the group law).

But when the genus of {X} is at least two, things change dramatically. It is a famous theorem that the number of automorphisms is bounded:

Theorem 1 (Hurwitz) If {X} is a compact Riemann surface of genus {g \geq 2}, then there are at most {84(g-1)} automorphisms of {X}.

I won’t write out a proof here; a discussion is in lecture 9 of the notes I’m taking in an algebraic curves class. In fact, the hard part is to show that there are finitely many automorphisms, after which it is a combinatorial argument.

This bound is often sharp. There are infinitely many genera {g} together with Riemann surfaces with exactly {84(g-1)} automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic p, because the Riemann-Hurwitz formula is no longer necessarily true (because of the existence of non-separable morphisms), and counterexamples are given in the notes.

What I want to describe today is that this bound is often not sharp.

Theorem 2 If {X} is a compact Riemann surface of genus {g = p+1} for {p>84} a prime, then {\mathrm{Aut}(X)} has order strictly less than {84(g-1)}. (more…)