| Klein's quartic curve, X , is the genus 3 curve canonically modelled in P2 by the equation XY3+YZ3+ZX3=0. X has 168 automorphisms, the maximum for a curve of genus 3, making X the smallest genus Hurwitz curve. It also has a split Jacobian, and in fact Jac( X ) is not just isogenous, but isomorphic over Qe2pi/7 to the product of three isomorphic elliptic curves.;We examine three distinct aspects of this curve. The first is the most geometric, where we will look at subvarieties of moduli spaces. A Weierstrass point is a point on an algebraic curve where the canonical divisor contains a divisor of unusually high order. The moduli space of curves with automorphisms can be stratified by these points. Higher-order Weierstrass points are the natural generalization of this concept to the pluricanonical series. Using connections between the Weierstrass points and the fixed points of non-trivial automorphisms, we find that exactly two of the three types of fixed points of X are higher-order Weierstrass points.;The second is more transcendental, using the special symmetries of the period lattice for this curve in considering a genus 3 arithmetic-geometric mean for X . The classical arithmetic-geometric mean (AGM) for elliptic curves has recently been generalized to genus 3, the last genus for which such an algorithm is possible. We compare the curve resulting from this algorithm with a construction for the AGM image of X using the curve's split Jacobian and the elliptic curve AGM.;Finally, the unusual properties of the elliptic curves in Jac ( X ) leads to a need for a greater understanding of the genus 1 arithmetic-geometric mean. The relation between elliptic curves and the AGM was discovered by Lagrange and Gauss as a method of calculating elliptic integrals. This classical approach is limited to cases where the elliptic curve is given by a cubic equation with three real roots. We define an AGM for all elliptic curves over C for all of the normalizations of their algebraic equations, and the period 2 points to which these AGM's correspond. |
