*Ziyang Gao, Rafael von Känel, and Lucia Mocz*

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0004
- Subject:
- Mathematics, Geometry / Topology

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 ...
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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.Less

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.

*Gisbert Wüstholz and Clemens Fuchs (eds)*

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0001
- Subject:
- Mathematics, Geometry / Topology

This introductory chapter provides an overview of the three topics discussed in this book: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and ...
More

This introductory chapter provides an overview of the three topics discussed in this book: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. These topics were covered during the Alpbach Summerschool 2016, the celebration of the tenth session with outstanding speakers covering very different research areas in arithmetic and Diophantine geometry. The first course was given by Peter Scholze on local Shimura varieties and features recent results concerning the local Langlands conjecture. It considers the unpublished theorem which states that for each local Shimura datum, there exists a so-called local Shimura variety, which is a (pro-)rigid analytic space. The second course was given by Umberto Zannier and deals with a rather classical theme but from a modern point of view. His course is on hyperelliptic continued fractions and generalized Jacobians, using the classical Pell equation as the starting point. The third course was given by Shou-Wu Zhang and originates in the famous Chowla–Selberg formula, which was taken up by Gross and Zagier in 1984 to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Building on this work, X. Yuan, Shou-Wu Zhang, and Wei Zhang succeeded in proving the Gross–Zagier formula on Shimura curves and shortly later they verified the Colmez conjecture on average. In the course, Zhang presents new interesting aspects of the formula.Less

This introductory chapter provides an overview of the three topics discussed in this book: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and *L*-functions. These topics were covered during the Alpbach Summerschool 2016, the celebration of the tenth session with outstanding speakers covering very different research areas in arithmetic and Diophantine geometry. The first course was given by Peter Scholze on local Shimura varieties and features recent results concerning the local Langlands conjecture. It considers the unpublished theorem which states that for each local Shimura datum, there exists a so-called local Shimura variety, which is a (pro-)rigid analytic space. The second course was given by Umberto Zannier and deals with a rather classical theme but from a modern point of view. His course is on hyperelliptic continued fractions and generalized Jacobians, using the classical Pell equation as the starting point. The third course was given by Shou-Wu Zhang and originates in the famous Chowla–Selberg formula, which was taken up by Gross and Zagier in 1984 to relate values of the *L*-function for elliptic curves with the height of Heegner points on the curves. Building on this work, X. Yuan, Shou-Wu Zhang, and Wei Zhang succeeded in proving the Gross–Zagier formula on Shimura curves and shortly later they verified the Colmez conjecture on average. In the course, Zhang presents new interesting aspects of the formula.

*Gisbert Wüstholz and Clemens Fuchs (eds)*

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the ...
More

This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. The first course contains recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. The second course addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. The third course originates in the Chowla–Selberg formula and relates values of the L-function for elliptic curves with the height of Heegner points on the curves. It proves the Gross–Zagier formula on Shimura curves and verifies the Colmez conjecture on average.Less

This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and *L*-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. The first course contains recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. The second course addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. The third course originates in the Chowla–Selberg formula and relates values of the *L*-function for elliptic curves with the height of Heegner points on the curves. It proves the Gross–Zagier formula on Shimura curves and verifies the Colmez conjecture on average.