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 ...
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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.
Günter Harder and A. Raghuram
- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.003.0007
- Subject:
- Mathematics, Number Theory
This chapter turns to L-functions. It first covers motivic and cohomological L-functions. There is a well-known conjectural dictionary between cohomological cuspidal automorphic representations of ...
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This chapter turns to L-functions. It first covers motivic and cohomological L-functions. There is a well-known conjectural dictionary between cohomological cuspidal automorphic representations of GLn and pure rank n motives. The chapter briefly reviews this dictionary while recasting it in the context of strongly inner Hecke summands on the one hand and pure effective motives on the other. Afterward, the critical points for L-functions and the combinatorial lemma are explored. In particular, the chapter reviews the Rankin–Selberg L-functions. A proof of combinatorial lemma is also given. The chapter then provides the main result on special values of L-functions. It concludes with some remarks.Less
This chapter turns to L-functions. It first covers motivic and cohomological L-functions. There is a well-known conjectural dictionary between cohomological cuspidal automorphic representations of GLn and pure rank n motives. The chapter briefly reviews this dictionary while recasting it in the context of strongly inner Hecke summands on the one hand and pure effective motives on the other. Afterward, the critical points for L-functions and the combinatorial lemma are explored. In particular, the chapter reviews the Rankin–Selberg L-functions. A proof of combinatorial lemma is also given. The chapter then provides the main result on special values of L-functions. It concludes with some remarks.
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.
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.
Anantharam Raghuram and Günter Harder
- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.001.0001
- Subject:
- Mathematics, Number Theory
This book studies the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of ...
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This book studies the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel–Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin–Selberg L-functions for GL(n) × GL(m), where n + m = N. The book carries through the entire program with an eye toward generalizations. The book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.Less
This book studies the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel–Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin–Selberg L-functions for GL(n) × GL(m), where n + m = N. The book carries through the entire program with an eye toward generalizations. The book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.
Jon P. Keating
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198797319
- eISBN:
- 9780191838774
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797319.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and ...
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The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.Less
The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.
Günter Harder and A. Raghuram
- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.003.0001
- Subject:
- Mathematics, Number Theory
This introductory chapter presents the general principle that the cohomology of arithmetic groups and the L-functions L(s, π, r) attached to irreducible “pieces” π have a strong symbiotic ...
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This introductory chapter presents the general principle that the cohomology of arithmetic groups and the L-functions L(s, π, r) attached to irreducible “pieces” π have a strong symbiotic relationship with each other. The symbiosis goes in both directions. The first is that expressions in the special values L(k, π, r) enter in the transcendental description of the cohomology. Since the cohomology is defined over ℚ one can deduce rationality (algebraicity) results for these expressions in special values. Next, these special values in turn influence the structure of the cohomology as a Hecke module; prime numbers dividing these values occur in the denominators of Eisenstein classes.Less
This introductory chapter presents the general principle that the cohomology of arithmetic groups and the L-functions L(s, π, r) attached to irreducible “pieces” π have a strong symbiotic relationship with each other. The symbiosis goes in both directions. The first is that expressions in the special values L(k, π, r) enter in the transcendental description of the cohomology. Since the cohomology is defined over ℚ one can deduce rationality (algebraicity) results for these expressions in special values. Next, these special values in turn influence the structure of the cohomology as a Hecke module; prime numbers dividing these values occur in the denominators of Eisenstein classes.
Uwe Weselmann
- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.003.0009
- Subject:
- Mathematics, Number Theory
This chapter generalizes an identity conjectured by G. Harder and proved by D. Zagier from the case of GL3(ℝ)-representations to the case of general GLN(ℝ)-representations. These are useful in ...
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This chapter generalizes an identity conjectured by G. Harder and proved by D. Zagier from the case of GL3(ℝ)-representations to the case of general GLN(ℝ)-representations. These are useful in applying the results of Harder and Anantharam Raghuram on quotients of special values of L-functions. To begin, the chapter provides the general setting for this analysis—the group theoretic data and the induced representations. It then discusses the intertwining operators as well as the J-admissible permutations. The chapter goes on to discuss representations and L-functions before turning to the main theorem on Archimedean intertwining operator. Finally, the chapter discusses some applications to cohomology.Less
This chapter generalizes an identity conjectured by G. Harder and proved by D. Zagier from the case of GL3(ℝ)-representations to the case of general GLN(ℝ)-representations. These are useful in applying the results of Harder and Anantharam Raghuram on quotients of special values of L-functions. To begin, the chapter provides the general setting for this analysis—the group theoretic data and the induced representations. It then discusses the intertwining operators as well as the J-admissible permutations. The chapter goes on to discuss representations and L-functions before turning to the main theorem on Archimedean intertwining operator. Finally, the chapter discusses some applications to cohomology.