On a Curve of Genus 7

Abstract

The theory of Riemann surfaces with biholomorphic transformations into themselves, or, what is essentially the same thing, curves with birational self-transformations, was begun very long ago. Klein's quartic curve (6) of genus 3 with its' group of 168 automorphisms (as we shall call birational self-transformations) was the object of much study, and Klein's discovery is commemorated by the name 'Klein's group' for the simple group of order 168, though the group, as LF(2,7), must certainly have been known to Galois.When g = 3, 84(gr -1) = 168, and Hurwitz, in his fundamental paper (5), showed that a Riemann surface of genus g ^ 2 cannot have more than 84(gr -1) automorphisms, and he gave a simple group-theoretic criterion for a finite group to occur as a group of 84(gr-1) automorphisms of a curve of genus g. However, this simple and direct lead was not followed Up until 1961, when I inferred the existence of infinitely many curves for which the Hurwitz bound is attained (10). Using the methods of classical algebraic geometry, Wiman (12) had in the meantime obtained fairly exhaustive results about genus g = 2, 3, 4, 5, and 6, establishing in particular that only for g = 3 is the maximum Si(g-1) attained. Had he gone one step further, to genus 7, he, would have found a curve whose group of 504 automorphisms is isomorphic to LF(2,2 3 ). This curve is the subject of the present paper.Indeed the existence of such a curve is immediately deducible from Hurwitz's criterion and the presentation of the group LF(2,2 3 ) given in Coxeter and Moser's book ((3) 97). However, Hurwitz's method only tells us how to obtain the Riemann surface of the curve as an abstract Riemann surface, and does not give any algorithm for obtaining equations for the curve. In general, no algorithm is known for deriving equations of curves with known groups of automorphisms, but if the automorphism group is soluble, and the field of invariant functions is the field of a rational curve, then, by the theory of Galois, the field of the curve is an extension by radicals, and if enough is known about the branch-points this will enable us to obtain equations defining the curve. Now the group LF(2,2 3 ) which we are considering has a soluble subgroup of Proe. London Math. Soc. (3) IS (1965) 527-42 528 A. M. MACBEATH

On Riemann surfaces with maximal automorphism groups

“…For q = 1 this surface is found in Macbeath [6], and, if we make use of the methods of Macbeath [5], we obtain the surfaces for*/ > 1.…”

Mirrors of reflections of regular maps

“…Let S be the Riemann surface of genus 7 admitting 504 conformal automorphisms. This surface is known as the Fricke-Macbeath surface; see [8,14]. It is known that S underlies a regular map M of type {3, 7}, which is called the Fricke-Macbeath map.…”

Characteristic Classes on Symmetric Products of Curves

“…Let G 2 = h and let G 1 be the normalizer of G 2 in ‫ސ‬SL 2 (F 7 ). One can imitate the steps followed by Macbeath in [5] to compute the equation of X and then one notices that X can be constructed by adding a seventh root of a polynomial available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089504001946 q(z) = (z − a) 4 (ωz − a) 2 (ω 2 z − a) (where ω = e 2πi/3 ) to ‫(ރ‬z) (one can also use the formula (2.2) from [3]).…”