Period Functions for Maass Wave Forms and Cohomology

Period Functions for Maass Wave Forms and Cohomology

Author: Roelof W. Bruggeman

Publisher:

Published: 2015

Total Pages: 132

ISBN-13: 9781470425036

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We construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups Γ ⊂ PSL2(R). In the case that Γ is the modular group PSL2(Z) this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J.[[,]]Lewis and D.[[,]]Zagier in Ann. Math. 153 (2001), 191–258, where a bijection was given between cuspidal Maass forms and period functions. We introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables us to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all Γ-invariant eigenfunctions of the Laplace operator. For spaces of Maass cusp forms we also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. We use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.


Book Synopsis Period Functions for Maass Wave Forms and Cohomology by : Roelof W. Bruggeman

Download or read book Period Functions for Maass Wave Forms and Cohomology written by Roelof W. Bruggeman and published by . This book was released on 2015 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: We construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups Γ ⊂ PSL2(R). In the case that Γ is the modular group PSL2(Z) this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J.[[,]]Lewis and D.[[,]]Zagier in Ann. Math. 153 (2001), 191–258, where a bijection was given between cuspidal Maass forms and period functions. We introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables us to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all Γ-invariant eigenfunctions of the Laplace operator. For spaces of Maass cusp forms we also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. We use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.

Period Functions for Maass Wave Forms and Cohomology

Period Functions for Maass Wave Forms and Cohomology

Author: R. Bruggeman

Publisher: American Mathematical Soc.

Published: 2015-08-21

Total Pages: 150

ISBN-13: 1470414074

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The authors construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups Γ⊂PSL2(R). In the case that Γ is the modular group PSL2(Z) this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191-258, where a bijection was given between cuspidal Maass forms and period functions. The authors introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables them to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all Γ-invariant eigenfunctions of the Laplace operator. For spaces of Maass cusp forms the authors also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. They use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.


Book Synopsis Period Functions for Maass Wave Forms and Cohomology by : R. Bruggeman

Download or read book Period Functions for Maass Wave Forms and Cohomology written by R. Bruggeman and published by American Mathematical Soc.. This book was released on 2015-08-21 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups Γ⊂PSL2(R). In the case that Γ is the modular group PSL2(Z) this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191-258, where a bijection was given between cuspidal Maass forms and period functions. The authors introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables them to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all Γ-invariant eigenfunctions of the Laplace operator. For spaces of Maass cusp forms the authors also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. They use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.

On the Theory of Maass Wave Forms

On the Theory of Maass Wave Forms

Author: Tobias Mühlenbruch

Publisher: Springer Nature

Published: 2020-05-06

Total Pages: 527

ISBN-13: 3030404757

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This textbook provides a rigorous analytical treatment of the theory of Maass wave forms. Readers will find this unified presentation invaluable, as it treats Maass wave forms as the central area of interest. Subjects at the cutting edge of research are explored in depth, such as Maass wave forms of real weight and the cohomology attached to Maass wave forms and transfer operators. Because Maass wave forms are given a deep exploration, this book offers an indispensable resource for those entering the field. Early chapters present a brief introduction to the theory of classical modular forms, with an emphasis on objects and results necessary to fully understand later material. Chapters 4 and 5 contain the book’s main focus: L-functions and period functions associated with families of Maass wave forms. Other topics include Maass wave forms of real weight, Maass cusp forms, and weak harmonic Maass wave forms. Engaging exercises appear throughout the book, with solutions available online. On the Theory of Maass Wave Forms is ideal for graduate students and researchers entering the area. Readers in mathematical physics and other related disciplines will find this a useful reference as well. Knowledge of complex analysis, real analysis, and abstract algebra is required.


Book Synopsis On the Theory of Maass Wave Forms by : Tobias Mühlenbruch

Download or read book On the Theory of Maass Wave Forms written by Tobias Mühlenbruch and published by Springer Nature. This book was released on 2020-05-06 with total page 527 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a rigorous analytical treatment of the theory of Maass wave forms. Readers will find this unified presentation invaluable, as it treats Maass wave forms as the central area of interest. Subjects at the cutting edge of research are explored in depth, such as Maass wave forms of real weight and the cohomology attached to Maass wave forms and transfer operators. Because Maass wave forms are given a deep exploration, this book offers an indispensable resource for those entering the field. Early chapters present a brief introduction to the theory of classical modular forms, with an emphasis on objects and results necessary to fully understand later material. Chapters 4 and 5 contain the book’s main focus: L-functions and period functions associated with families of Maass wave forms. Other topics include Maass wave forms of real weight, Maass cusp forms, and weak harmonic Maass wave forms. Engaging exercises appear throughout the book, with solutions available online. On the Theory of Maass Wave Forms is ideal for graduate students and researchers entering the area. Readers in mathematical physics and other related disciplines will find this a useful reference as well. Knowledge of complex analysis, real analysis, and abstract algebra is required.

Holomorphic Automorphic Forms and Cohomology

Holomorphic Automorphic Forms and Cohomology

Author: Roelof Bruggeman

Publisher: American Mathematical Soc.

Published: 2018-05-29

Total Pages: 167

ISBN-13: 1470428555

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Book Synopsis Holomorphic Automorphic Forms and Cohomology by : Roelof Bruggeman

Download or read book Holomorphic Automorphic Forms and Cohomology written by Roelof Bruggeman and published by American Mathematical Soc.. This book was released on 2018-05-29 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume

Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume

Author: Roelof Bruggeman

Publisher: American Mathematical Society

Published: 2023-07-31

Total Pages: 186

ISBN-13: 1470465450

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View the abstract.


Book Synopsis Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume by : Roelof Bruggeman

Download or read book Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume written by Roelof Bruggeman and published by American Mathematical Society. This book was released on 2023-07-31 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Modular Forms

Modular Forms

Author: Henri Cohen

Publisher: American Mathematical Soc.

Published: 2017-08-02

Total Pages: 714

ISBN-13: 0821849476

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The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms. Some “gems” of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms. This book is essentially self-contained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.


Book Synopsis Modular Forms by : Henri Cohen

Download or read book Modular Forms written by Henri Cohen and published by American Mathematical Soc.. This book was released on 2017-08-02 with total page 714 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms. Some “gems” of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms. This book is essentially self-contained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.

Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities

Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities

Author: Bart Bories

Publisher: American Mathematical Soc.

Published: 2016-06-21

Total Pages: 146

ISBN-13: 147041841X

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In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f∈Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerate with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f−1(0)⊂C3 close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerate surface singularity.


Book Synopsis Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities by : Bart Bories

Download or read book Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities written by Bart Bories and published by American Mathematical Soc.. This book was released on 2016-06-21 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f∈Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerate with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f−1(0)⊂C3 close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerate surface singularity.

The Conference on L-Functions

The Conference on L-Functions

Author: Lin Weng

Publisher: World Scientific

Published: 2007

Total Pages: 383

ISBN-13: 981270504X

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This invaluable volume collects papers written by many of the world's top experts on L-functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole. The contributions reflect the latest, most advanced and most important aspects of L-functions. In particular, it contains Hida's lecture notes at the conference and at the Eigenvariety semester in Harvard University and Weng's detailed account of his works on high rank zeta functions and non-abelian L-functions.


Book Synopsis The Conference on L-Functions by : Lin Weng

Download or read book The Conference on L-Functions written by Lin Weng and published by World Scientific. This book was released on 2007 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: This invaluable volume collects papers written by many of the world's top experts on L-functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole. The contributions reflect the latest, most advanced and most important aspects of L-functions. In particular, it contains Hida's lecture notes at the conference and at the Eigenvariety semester in Harvard University and Weng's detailed account of his works on high rank zeta functions and non-abelian L-functions.

Higher Moments of Banach Space Valued Random Variables

Higher Moments of Banach Space Valued Random Variables

Author: Svante Janson

Publisher: American Mathematical Soc.

Published: 2015-10-27

Total Pages: 124

ISBN-13: 1470414651

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The authors define the :th moment of a Banach space valued random variable as the expectation of its :th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. The authors study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals.


Book Synopsis Higher Moments of Banach Space Valued Random Variables by : Svante Janson

Download or read book Higher Moments of Banach Space Valued Random Variables written by Svante Janson and published by American Mathematical Soc.. This book was released on 2015-10-27 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors define the :th moment of a Banach space valued random variable as the expectation of its :th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. The authors study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals.

On Non-Topological Solutions of the $A_{2}$ and $B_{2}$ Chern-Simons System

On Non-Topological Solutions of the $A_{2}$ and $B_{2}$ Chern-Simons System

Author: Weiwei Ao

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 100

ISBN-13: 1470415437

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Click here to view the abstract. IntroductionProof of Theorem 1.1 in the caseProof of Theorem 1.1 in the caseAppendixBibliography


Book Synopsis On Non-Topological Solutions of the $A_{2}$ and $B_{2}$ Chern-Simons System by : Weiwei Ao

Download or read book On Non-Topological Solutions of the $A_{2}$ and $B_{2}$ Chern-Simons System written by Weiwei Ao and published by American Mathematical Soc.. This book was released on 2016-01-25 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: Click here to view the abstract. IntroductionProof of Theorem 1.1 in the caseProof of Theorem 1.1 in the caseAppendixBibliography