Introduction

Overview. In this expository paper wedescribe some of the remarkable properties of theKlein quartic thatare of particular interest in number theory. TheKlein quartic X is theunique curve of genus 3 over ℂ with an automorphismgroup 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 itsrepresentation on the three-dimensional space V inwhose projectivization ℙ (V) = ℙ2 the Klein quarticlives. The first section is devoted to thisrepresentation and its invariants, starting over ℂand then considering arithmetical questions offields of definition and integral structures. Therewe also encounter a G-Iattice that later occurs as both theperiod lattice and a Mordell-Weillattice for X. In the second section weintroduce X and investigate it as a Riemann surfacewith automorphisms by G.

In the third section we consider the arithmetic ofX: rational points,relations with the Fermat curve and Fermat's “LastTheorem” for exponent 7, and some extremalproperties of the reduction of X modulo the primes2,3,7 dividing #G.