## Details

Original language | English |
---|---|

Title of host publication | Arithmetic and Geometry |

Subtitle of host publication | Ten Years in Alpbach |

Editors | Gisbert Wüstholz, Clemens Fuchs |

Publisher | Princeton University Press |

Chapter | 4 |

Pages | 102-174 |

Number of pages | 71 |

ISBN (electronic) | 9780691197548 |

Publication status | Published - 8 Oct 2019 |

## Publication series

Name | Annals of Mathematics Studies |
---|---|

Publisher | Princeton University Press |

Volume | 202 |

ISSN (Print) | 0066-2313 |

## Abstract

## ASJC Scopus subject areas

## Cite this

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**Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang.**/ Gao, Ziyang; Kanel, Rafael Von; Mocz, Lucia.

Arithmetic and Geometry: Ten Years in Alpbach. ed. / Gisbert Wüstholz; Clemens Fuchs. Princeton University Press, 2019. p. 102-174 (Annals of Mathematics Studies; Vol. 202).

Research output: Chapter in book/report/conference proceeding › Contribution to book/anthology › Research › peer review

*Arithmetic and Geometry: Ten Years in Alpbach.*Annals of Mathematics Studies, vol. 202, Princeton University Press, pp. 102-174. https://doi.org/10.1515/9780691197548-005

*Arithmetic and Geometry: Ten Years in Alpbach*(pp. 102-174). (Annals of Mathematics Studies; Vol. 202). Princeton University Press. https://doi.org/10.1515/9780691197548-005

}

TY - CHAP

T1 - Faltings Heights and L-functions

T2 - Minicourse Given by Shou-Wu Zhang

AU - Gao, Ziyang

AU - Kanel, Rafael Von

AU - Mocz, Lucia

PY - 2019/10/8

Y1 - 2019/10/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85075951730&partnerID=8YFLogxK

U2 - 10.1515/9780691197548-005

DO - 10.1515/9780691197548-005

M3 - Contribution to book/anthology

AN - SCOPUS:85075951730

T3 - Annals of Mathematics Studies

SP - 102

EP - 174

BT - Arithmetic and Geometry

A2 - Wüstholz, Gisbert

A2 - Fuchs, Clemens

PB - Princeton University Press

ER -