Legendre’s Singular Modulus

Abstract

In 1811, Legendre published an identity between integrals involving the constant that inspired Abel to create his brilliant theory of complex multiplication. Then k reappeared as the eccentricity of an ellipse whose arclength Ramanujan computed explicitly in terms of gamma functions with rational arguments. Finally, the constant appeared as a consequence of the three-body choreography along Bernoulli’s lemniscate. We develop these results in detail as well as mentioning random walks on a cubic lattice and the renormalization of the period of the simple pendulum. All this shows the wonderful unity underlying seemingly different branches of mathematics and physics.