Abstract: The talk is intended to be accessible to a wide audience. I will begin by recalling an ancient problem, known as the "cattle problem" and attributed to Archimedes. Its resolution, that I will explain in details, is essentially equivalent to the one of Pell's equation. It naturally leads to the construction of certain manifolds associated to quadratic forms which I will describe in specific cases. In dimension 5 and higher these manifolds are essentially the only known examples of manifolds that can carry a metric of negative sectional curvature. The "arithmetic" nature of their construction makes the study of the topology of these manifolds difficult. However, they have the particularity to contain numerous totally geodesic submanifolds: the special cycles. If time permits I will finally explain how these cycles can be used to shed some light on the topology of these arithmetic manifolds, like the hyperplane sections or algebraic cycles illuminate the topology of complex projective varieties. These recent works are drawn from joint work with L. Clozel, on the one hand, and with J. Millson and C. Moeglin on the other.