Algebraic points on curves

Let C be an algebraic curve over Q, in other words, a complex Riemann surface that is described by a system of polynomial equations of rational coefficients.

of genus at least 2, i.e., a 1-dimensional negatively curved complex manifold defined by polynomial equations with rational coefficients.
A celebrated result of Faltings implies that, despite the hyperbolicity of C, all algebraic points on C are organized into families of nonnegative curvature. We explore how these families provide structure to access the arithmetic of C and give some applications to the study of elliptic curves. This talk is based in part on joint work with A. Bourdon, Ö. Ejder, Y. Liu, and F. Odumodu, with I. Vogt, and with I. Balçik, S. Chan, and Y. Liu.