EQUATIONS OF RIEMANN SURFACES OF GENUS 4 , 5 , AND 6 WITH LARGE AUTOMORPHISM GROUPS

We say a Riemann surface has a large automorphism group if #Aut(C) > 4(g − 1). We find equations of all genus 4 and 5 Riemann surfaces C with large automorphism groups, and all but two of the genus 6 Riemann surfaces with large automorphisms groups. The strategy used is based on the Eichler trace formula and the projection formula from the character theory of finite groups. It is not completely algorithmic, but surprisingly yields enough information that we can finish each example by hand. In the last section we discuss how an algorithm for computing flattening stratifications could make the last step algorithmic as well. Every finite group G arises as a group of automorphisms of some smooth compact Riemann surface S (or complex algebraic curve C) or as the full automorphism group of a punctured Riemann surface [3, 7]. However, the property of #G being large compared to the genus g of S is special. A general curve has no nontrivial automorphisms at all. There is the well-known bound due to Hurwitz: #G ≤ 84(g − 1), but most curves with automorphisms do not even come close to this. For instance, in genus 3, the Hurwitz bound is 168, but only five curves have automorphism groups satisfying #G > 24. We will call an automorphism group large if #G > 4(g − 1). By Riemann-Hurwitz, this implies that the quotient curve is P1, and the quotient morphism branches over 3 or 4 points. Using Fuchsian group theory, it is possible to set up computer searches for large automorphism groups. Algebraists have carried this out [2,12] and published lists of pairs (G, g) such that there is a curve of genus g with large automorphism group G, but we don’t know equations of all these curves. In other words, there is no published algorithm yet which given the group produces equations of the curve. In this note I will outline a non-algorithmic strategy for producing equations of Riemann surfaces with automorphisms under their canonical embeddings (or for hyperelliptic curves, an equation of the form y2 = f(x) instead). Surprisingly, although my strategy is non-algorithmic, the algorithmic steps generally get us close enough to the answer that we can finish them by hand. Section 2 presents an example which is very typical in this sense. In particular, I am able to find all of the genus 4 and 5 Riemann surfaces with large automorphism groups, and all but two of the genus 6 Riemann surfaces with large automorphism groups. Many of these example are classically known, but others may be new, and to my knowledge, this is the first time they are all collected in one place. In Section 1 I outline the strategy in more detail. In Section 2 I apply the strategy to one example; this section serves both as an illustration of the strategy and a mini-tutorial on how to use my Magma code. Section 3 is intended to be a reference work. In the final section, Section 4, I will discuss an idea for how the final step could be made algorithmic, too; in theory, this could lead to an algorithm for finding equations of any Riemann surface with automorphisms. (The missing ingredient is knowing how to compute a so-called flattening stratification.) 0.1. Acknowledgements. I would like to thank the undergraduates from my Introductory VIGRE Research Group held in Fall 2010 at the University of Georgia: Eddie Beck, Zachary Freeland, Tyler Johnson, Malik Obeidin, Jacob Rooney, and Lev Tovstopyat-Nelip, where I began the main programming of this work was begun. I am also very grateful to the computational algebra group at Date: August 8, 2011.