The Klein quartic in number theory (Q2702549)

Klein's quartic, i.e. the plane complex projective curve defined by the equation \(x^3y+ y^3z+z^3x=0\), is the unique Riemann surface of genus three with a holomorphic automorphism group of size 168, the maximum for this genus. In 1878 Felix Klein gave a construction of this surface as a quotient of the upper complex half-plane \({\mathfrak H}\) by the simple group \(G= \text{PSL}_2 (\mathbb{F}_7)\), via uniformization theory, and thereby he exhibited some of its remarkable geometric properties. Since Klein's discovery, more than a century ago, the structure of this quartic surface has loomed up in various mathematical contexts, among them being the theory of modular curves in algebraic number theory. NEWLINENEWLINENEWLINEIn the article under review, the author re-describes Klein's quartic \(X\), from a modern point of view, and highlights some of its remarkable properties that are of particular interest in number theory. NEWLINENEWLINENEWLINEThe first section is entitled ``The group \(G\) and its representation \((V,\rho)\)'' and provides a detailed description of the modular group \(G\) (admitting 168 elements) and its representation on the 3-dimensional vector space \(V\) in whose projectivization \(\mathbb{P}(V)= \mathbb{P}^2\) the Klein quartic appears. Furthermore, the author exhibits a \(G\)-lattice \(L\) in \(V\), which later occurs as both the period lattice and a Mordell-Weil lattice for Klein's quartic \(X\). NEWLINENEWLINENEWLINESome extremal properties in characteristics 2, 3, and 7, the primes dividing the order of \(G\), are also explained by means of the isomorphisms \(G\cong \text{SL}_3(\mathbb{F}_2)\) and \(\Aut(G)= \text{SO}_3 (\mathbb{F}_7)\). This topic is completed later in Section 3. NEWLINENEWLINENEWLINESection 2 discusses \(X\) from the viewpoint of the theory of Riemann surfaces. Klein's quartic is shown to be the simplest Hurwitz curve in genus three, and its Jacobian is investigated as a cube of an elliptic curve with complex multiplication by \(O_{\mathbb{Q}(\sqrt{-7})} \subset \mathbb{Q}(\sqrt{-7})\). This interpretation, which is due to T. Ekedahl and J.-P. Serre (1993), allows the period lattice of Klein's quartic to be described as a certain Mordell-Weil lattice. NEWLINENEWLINENEWLINESection 3 deals with the arithmetic geometry of Klein's quartic. The author examines its rational points, its relations with the corresponding Fermat curve and ``Fermat's Last Theorem'' for exponent 7, and its reductions modulo the prime numbers 2, 3, 7 dividing the order of \(G\). NEWLINENEWLINENEWLINEIn the fourth and last section, \(X\) is explicitly identified with the modular curve \(X_0(7)\). Along these lines, some quotients of Klein's quartic are described as classical modular curves, followed by a brief report on two more recent results concerning M. A. Kenku's proof of the solution of the so-called ``class number one problem'' (Theorem of Stark-Heegner), on the one hand, and the recognition of Klein's quartic as a Shimura modular curve, on the other hand. NEWLINENEWLINENEWLINEThis paper provides an excellent overview of old and new arithmetic aspects of the celebrated Klein quartic curve. NEWLINENEWLINENEWLINEFor the entire collection see Zbl 0941.00006 and Zbl 0991.00005 for the review of the paperback edition.