Computing the cohomology of Shimura curves in quasi-linear time

Computing spaces of modular and automorphic forms is an important problem in algorithmic number theory. The case of classical modular forms was studied by Manin and Birch independently using Hecke modules of modular symbols, namely the $\mathbb Z$-module spanned by homology classes of geodesic paths between cusps of a modular curve. For Hilbert modular forms, using the Jacquet-Langlands correspondence, Greenberg and Voight gave an algorithm to compute Hecke modules of Hilbert modular forms appearing in the orbifold cohomology of Shimura curves. However, while practical their method relies on general linear algebra to find a basis of such cohomology, yielding a complexity that of linear algebra. In this talk, relying on work of Imbert, we explain how to use the geometry of the underlying Shimura curve to build such a basis in quasi-linear time in the area of the curve.