Introduction

Overview. In this expository paper we describe some of the remarkable properties of the Klein quartic that are of particular interest in number theory. The Klein quartic X is the unique curve of genus 3 over ℂ with an automorphism group G of size 168, the maximum for its genus. Since G is central to the story, we begin with a detailed description of G and its representation on the three-dimensional space V in whose projectivization ℙ (V) = ℙ2 the Klein quartic lives. The first section is devoted to this representation and its invariants, starting over ℂ and then considering arithmetical questions of fields of definition and integral structures. There we also encounter a G-Iattice that later occurs as both the period lattice and a Mordell-Weillattice for X. In the second section we introduce X and investigate it as a Riemann surface with automorphisms by G.

In the third section we consider the arithmetic of X: rational points, relations with the Fermat curve and Fermat's “Last Theorem” for exponent 7, and some extremal properties of the reduction of X modulo the primes 2,3,7 dividing #G.