Black holes and modular forms in string theory

The study of black holes in string theory has led to the discovery of deep and surprising connections between black holes and modular forms -- which are two classical, a priori unrelated, subjects. This article explains the main physical and mathematical ideas behind these connections. It is known from the pioneering work of J.Bekenstein and S.Hawking in the 1970s that black holes have thermodynamic entropy, and should therefore be made up of a collection of microscopic quantum states. Superstring theory provides a framework wherein we can associate a number of microscopic states that make up the quantum-statistical system underlying a black hole, thus explaining their thermodynamic behavior from a more fundamental point of view. %The above-mentioned connections arise from the observation that, i The basic connection to modular forms arises from the observation that, in the simplest superstring-theoretic construction, the generating function of the number of microscopic states is a modular form. In one direction, modular symmetry acts as a powerful guide to the calculation of quantum-gravitational effects on the black hole entropy. In the other direction, the connection has led to the discovery of surprising relations between Ramanujan's mock modular forms and a class of string-theoretic black holes, thus providing an infinite number of new examples of mock modular forms.