Faltings heights and l-functions: Minicourse given

Shou Wu Zhang, Ziyang Gao, Rafael Von Kanel, Lucia Mocz

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish (US)
Title of host publicationAnnals of Mathematics Studies
PublisherPrinceton University Press
Pages102-172
Number of pages71
Edition202
StatePublished - Jan 1 2019

Publication series

NameAnnals of Mathematics Studies
Number202
Volume2019-January
ISSN (Print)0066-2313

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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  • Cite this

    Zhang, S. W., Gao, Z., Kanel, R. V., & Mocz, L. (2019). Faltings heights and l-functions: Minicourse given. In Annals of Mathematics Studies (202 ed., pp. 102-172). (Annals of Mathematics Studies; Vol. 2019-January, No. 202). Princeton University Press.