Faltings heights and l-functions: Minicourse given

I his chapter essentially consists of three parts. In the first part (94.1), we discuss conjectures and results in the literature which give bounds, or formulae in terms of ^-functions, for "Fallings 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 used in their proof, and we explain in detail various constructions in [60]. In the third part (§4.4), we go into the function field world and we discuss the work of Yun-Zhang [61]. Therein they compute special values of higher derivatives of certain automor-phic L-functions in terms of self-intersection numbers of Drinfeld-Heegner cycles on the moduli stack of shtukas. The result of Yun-Zhang might be viewed as a higher Gross-Zagier/Chowla-Selberg formula in the function field setting. In fact, throughout §4.3 and §4.4 we try to motivate and explain the philosophy that Chowla-Selberg type formulae (such as the averaged Colmez conjecture) are special cases of Gross-Zagier type formulae coming from identities between geometric and analytic kernels. We tried to make this chapter accessible for non-specialists. In particular, we conducted quite some effort to add throughout explanations, discussions, background material, and references which were not part of the minicourse. We would like to thank Shou-Wu Zhang for answering questions, for giving useful explanations, and for sharing his insights.