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Arithmetic and GeometryTen Years in Alpbach (AMS-202)$
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Gisbert Wüstholz and Clemens Fuchs

Print publication date: 2019

Print ISBN-13: 9780691193779

Published to Princeton Scholarship Online: May 2020

DOI: 10.23943/princeton/9780691193779.001.0001

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

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

(p.102) Chapter Four Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang
Arithmetic and Geometry

Ziyang Gao

Rafael von Känel

Lucia Mocz

Princeton University Press

This chapter explores Shou-Wu Zhang's minicourse on Faltings heights and L-functions. It essentially consists of three parts. The first part discusses conjectures and results in the literature which give bounds, or formulae in terms of L-functions, for “Faltings heights.” The authors also mention various applications of such conjectures and results. The second part is devoted to the work of Yuan–Zhang in which they proved the averaged Colmez conjecture. Here, the authors detail the main ideas and concepts used in their proof. The third part focuses on the work of Yuan–Zhang in the function field world. Therein they compute special values of higher derivatives of certain automorphic L-functions in terms of self-intersection numbers of Drinfeld–Heegner cycles on the moduli stack of shtukas. The result of Yuan–Zhang might be viewed as a higher Gross–Zagier/Chowla–Selberg formula in the function field setting. The authors then motivate and explain the philosophy that Chowla–Selberg type formulae are special cases of Gross–Zagier type formulae coming from identities between geometric and analytic kernels.

Keywords:   Shou-Wu Zhang, Faltings heights, L-functions, Colmez conjecture, Drinfeld–Heegner cycles, Gross–Zagier formula, Chowla–Selberg formula, function field

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