*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the ...
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This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the Appell hypergeometric function. Appell introduced four series F1, F2, F3, F4 in two complex variables, each of which generalizes the classical Gauss hypergeometric series and satisfies its own system of two linear second order partial differential equations. The solution spaces of the systems corresponding to the series F2, F3, F4 all have dimension 4, whereas that of the system corresponding to the series F1 has dimension 3. This chapter focuses on the F1-system whose monodromy group, under certain conditions, acts on the complex 2-ball. It first considers the action of S5 on the blown-up projective plane before turning to Appell hypergeometric functions, arithmetic monodromy groups, and an invariant known as the signature.Less

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the Appell hypergeometric function. Appell introduced four series *F*1, *F*2, *F*3, *F*4 in two complex variables, each of which generalizes the classical Gauss hypergeometric series and satisfies its own system of two linear second order partial differential equations. The solution spaces of the systems corresponding to the series *F*2, *F*3, *F*4 all have dimension 4, whereas that of the system corresponding to the series *F*1 has dimension 3. This chapter focuses on the *F*1-system whose monodromy group, under certain conditions, acts on the complex 2-ball. It first considers the action of *S*5 on the blown-up projective plane before turning to Appell hypergeometric functions, arithmetic monodromy groups, and an invariant known as the signature.

*Mark Green, Phillip A. Griffiths, and Matt Kerr*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.001.0001
- Subject:
- Mathematics, Analysis

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive ...
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Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it is an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The book gives the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. It also indicates that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.Less

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it is an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The book gives the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. It also indicates that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

*Graham Ellis*

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0006
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical ...
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This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical groups, graphs of groups, special linear groups, triangle groups, generalized triangle groups, Coxeter groups, Artin groups, and arithmetic groups.Less

This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical groups, graphs of groups, special linear groups, triangle groups, generalized triangle groups, Coxeter groups, Artin groups, and arithmetic groups.

*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.