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…)