### Abstract

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.

Original language | English (US) |
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Title of host publication | Annals of Mathematics Studies |

Publisher | Princeton University Press |

Pages | 102-172 |

Number of pages | 71 |

Edition | 202 |

State | Published - Jan 1 2019 |

### Publication series

Name | Annals of Mathematics Studies |
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Number | 202 |

Volume | 2019-January |

ISSN (Print) | 0066-2313 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Annals of Mathematics Studies*(202 ed., pp. 102-172). (Annals of Mathematics Studies; Vol. 2019-January, No. 202). Princeton University Press.