Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang

These are notes from the minicourse given by Shou-Wu Zhang (Princeton University). The notes were worked out and extended by Ziyang Gao, Rafael von K¨anel,¹ and Lucia Mocz. This chapter essentially consists of three parts. In the first part (§4.1), we discuss conjectures and results in the literature which give bounds, or formulae in terms of L-functions, for “Faltings heights.” We also mention various applications of such conjectures and results. The second part (§4.2 and §4.3) is devoted to the work of Yuan-Zhang [60] in which they proved the averaged Colmez conjecture. Here we discuss the main ideas and concepts