Klein Quartic Research Articles

Deformations of klein's curve and representations of its group of 168 automorphisms

The quotient of the upper half plane H by the group of integral linear transformations congruent to the identity modulo 7, gives as quotient a curve admitting a group of 168 automorphisms. We study here the Hodge decomposition and its connection with Torelli's theorem since it affords insight into the relations between geodesics in H and the equation of the curve in projective space. We obtain explicit formulae for the differentials of the first and second kind, the linear representations of the group, and the conics that give back the equation.

Fixed Points of Automorphisms of Compact Riemann Surfaces

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

XI—The Identification of Klein's Quartic

SummaryThere is a mode of specialising a quartic polynomial which causes a binary quartic to become equianharmonic and a ternary quartic to become a Klein quartic, admitting a group of 168 linear self-transformations. The six relations which must be satisfied by the coefficients of the ternary quartic were given by Coble forty years ago, but their true significance was never suspected and they have remained until now an isolated curiosity. In § 2 we give, in terms of a quadric and a Veronese surface, the geometrical interpretation of the six relations; we also give, in terms of the adjugate of a certain matrix, their algebraical interpretation. Both these interpretations make it abundantly clear that this set of relations specialising a ternary quartic has analogues for quartic polynomials in any number of variables, and point unmistakably to what these analogues are.That a ternary quartic is, when so specialised, a Klein quartic is proved in §§ 4–6. The proof bifurcates after (5.3); one branch leads instantly to the standard form of the Klein quartic while the other leads to another form which, on applying a known test, is found also to represent a Klein quartic. One or two properties of the curve follow from this new form of its equation. In §§ 8–10 some properties of a Veronese surface are established which are related to known properties of plane quartic curves; and these considerations lead to a discussion, in § 11, of certain hexads of points associated with a Klein curve.