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

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

Theorem 1 (Hurwitz)If is a compact Riemann surface of genus , then there are at most automorphisms of .

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 together with Riemann surfaces with exactly automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic , 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 2If is a compact Riemann surface of genus for a prime, then has order strictly less than . (more…)