TY - CHAP

T1 - Faltings heights and l-functions

T2 - Minicourse given

AU - Zhang, Shou Wu

AU - Gao, Ziyang

AU - Kanel, Rafael Von

AU - Mocz, Lucia

N1 - Publisher Copyright:
© 2019 Princeton University Press. All rights reserved.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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M3 - Chapter

AN - SCOPUS:85075951730

T3 - Annals of Mathematics Studies

SP - 102

EP - 172

BT - Annals of Mathematics Studies

PB - Princeton University Press

ER -