Appears in collection : 7ème Séminaire Itzykson : « Résurgence et quantification »

Why do modular forms tend to have integer coefficients in their q-expansion? Are they counting something? And, can we "categorify" a modular form by representing it as a graded Euler characteristic of some homology theory? There are several ways to answer these questions, e.g. the one based on classical theory of modular forms relates them to counting points on elliptic curves and motives of higher-dimensional varieties. In this talk, we will ask these questions for mock modular forms, introduced by Ramanujan. They also exhibit q-expansion with integer coefficients and, surprisingly, the answer to these question is based on resurgent analysis of a different power series, which at first shows no signs of integrality!