基本信息
- 原书名：Six Short Chapters on Automorphic Forms and L-functions
内容简介
书籍
数学书籍
Six Short Chapters on Automorphic Forms and L-functions treats the period conjectures of Shimura and the moment conjecture. These conjectures are of central importance in contemporary number theory, but have hitherto remained little discussed in expository form. The book is divided into six short and relatively independent chapters, each with its own theme, and presents a motivated and lively account of the main topics, providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant literature.
数学书籍
Six Short Chapters on Automorphic Forms and L-functions treats the period conjectures of Shimura and the moment conjecture. These conjectures are of central importance in contemporary number theory, but have hitherto remained little discussed in expository form. The book is divided into six short and relatively independent chapters, each with its own theme, and presents a motivated and lively account of the main topics, providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant literature.
作译者
Ze-Li Dou and Qiao Zhang are both associate professors of Mathematics at Texas Christian University, USA.
目录
《自守形式与L-函数简明六章(英文版)》
Chapter 1 Modular forms and the Shimura-Taniyama Conjecture
1.1 Elliptic functions
1.2 Modular forms
1.3 Examples
1.4 Hecke operators and eigenforms
1.5 L-functions
1.6 Modular forms of higher level
1.7 Elliptic curves
1.8 Conjectures, and the theorem of Wiles, et al.
References
Chapter 2 Periods of automorphic forms
2.1 Automorphic forms
2.2 Periods arising from Dirichlet series
2.3 Periods arising from elliptic curves
2.4 Periods arising from cohomology theory
2.5 Recapitulation
2.6 Hilbert modular forms
2.7 Automorphic forms with respect to a quaternion algebra
2.8 Adelic automorphic forms
Chapter 1 Modular forms and the Shimura-Taniyama Conjecture
1.1 Elliptic functions
1.2 Modular forms
1.3 Examples
1.4 Hecke operators and eigenforms
1.5 L-functions
1.6 Modular forms of higher level
1.7 Elliptic curves
1.8 Conjectures, and the theorem of Wiles, et al.
References
Chapter 2 Periods of automorphic forms
2.1 Automorphic forms
2.2 Periods arising from Dirichlet series
2.3 Periods arising from elliptic curves
2.4 Periods arising from cohomology theory
2.5 Recapitulation
2.6 Hilbert modular forms
2.7 Automorphic forms with respect to a quaternion algebra
2.8 Adelic automorphic forms
前言
This modest tract offers a little light reading on a heavy subject. In six short and relatively independent chapters, we provide motivated discussion on a few topics in number theory that axe tremendously active and rapidly progressing today.
Both the content and style of this book had their origin in invited talks we gave in the last few years, in China, Turkey, and the United States.Our audiences consisted in mathematicians and graduate students whose backgrounds were rather diverse. Although all were interested in the topics in general, many were not experts in the field. It was natural, therefore,that we endeavored to give motivation on each main topic before delving into deeper material. Also, we attempted to keep the lectures as independent from one another as possible, and to make at least part of each lecture comprehensible to the entire audience. Finally, in order not to have the main ideas eclipsed by the subordinate (but often heavily technical) material, as a rule we omitted detailed proofs. Of course, we should point out that the lectures do form an integral whole as well: they evolve around the moment conjecture for L-functions and the period conjectures of Shimura.
When the invitation to provide a written account of these lectures came,we at first felt the temptation of generating a fuller and more coherent work.Upon further consideration, however, we have decided against this idea. To undertake such a project would require a vast amount of time and energy,and we are not certain that we can afford to do so at the present time. On the other hand, we also believe that a motivated--and as nontechnical as possiblc introduction of the main topics can be useful, since it can help bridge the gap between the basics and certain specialized research areas, and perhaps also to serve as a road map for relevant literature. Consequently,what the reader now has in hand is an extended version of the original talks: revised, somewhat fleshed out, but otherwise retaining most of the characteristics of the original lectures themselves. Although this work is by no means complete or comprehensive, it is our hope that, through this, the reader can gain an overall appreciation for the main topics and conjectures considered herein. In the meantime, to the reader who intends to study these topics in depth, we hope that we have offered a motivated guide to the literature.
A rough idea for the content and organization of this book can easily be gained by glancing at the table of content, and we shall therefore not elab-orate too much here. During the preparation for our lectures as well as for this manuscript, we have benefited from many existing resources in the liter-ature. For the more introductory parts (which form the early parts of most of the chapters), we wish to mention especially Goro Shimura's Introduction to the Arithmetic Theory of Automorphic Functions, Danie L Bump's Au- tomorphic Forms and Representations, and Henryk Iwaniec and Emmanuel Kowalski's Analytic Number Theory. In several places, the arrangement of concepts and the choice of certain examples have been influenced by these excellent books. It is impossible to acknowledge in detail the numerous works we have consulted for the rest of the book, and the reader must be referred to the Bibliography. Some work by the authors themselves on the conjectures have been briefly (and incompletely) summarized here as well,mainly in the two final chapters.
It remains our most pleasant duty to acknowledge the many people from whom we have received much help and support, mathematically and other-wise. First and foremost, we thank Professor Goro Shimura and Professor Dorian Goldfeld, our thesis advisors. We are grateful to Professor Shou-Wu Zhang of Columbia University, Professor Hongwen Lu of Tongji University,Professor K. Ilhan Ikeda of Istanbul Bilgi University, Professor Juping Wang of Fudan University, and Professor Tianze Wang of Henan University for their kind invitations and their steadfast support. We are much indebted to Professor Paula Cohen Tretkoff of Texas A&M University and Profes-sor Yingchun Cai of Tongji University for their encouragement and advice.With gratitude we acknowledge the kindly and efficient assistance from the editors, Liping Wang and Lisa Libin Fan. Finally, we give our heartfelt thanks to all members of our families, and especially to our wives, Susan Staples and Jing Tian, for their longsuffering and their love.
Ze-Li Dou and Qiao Zhang
Fort Worth, TX, USA
October 11, 2011
Both the content and style of this book had their origin in invited talks we gave in the last few years, in China, Turkey, and the United States.Our audiences consisted in mathematicians and graduate students whose backgrounds were rather diverse. Although all were interested in the topics in general, many were not experts in the field. It was natural, therefore,that we endeavored to give motivation on each main topic before delving into deeper material. Also, we attempted to keep the lectures as independent from one another as possible, and to make at least part of each lecture comprehensible to the entire audience. Finally, in order not to have the main ideas eclipsed by the subordinate (but often heavily technical) material, as a rule we omitted detailed proofs. Of course, we should point out that the lectures do form an integral whole as well: they evolve around the moment conjecture for L-functions and the period conjectures of Shimura.
When the invitation to provide a written account of these lectures came,we at first felt the temptation of generating a fuller and more coherent work.Upon further consideration, however, we have decided against this idea. To undertake such a project would require a vast amount of time and energy,and we are not certain that we can afford to do so at the present time. On the other hand, we also believe that a motivated--and as nontechnical as possiblc introduction of the main topics can be useful, since it can help bridge the gap between the basics and certain specialized research areas, and perhaps also to serve as a road map for relevant literature. Consequently,what the reader now has in hand is an extended version of the original talks: revised, somewhat fleshed out, but otherwise retaining most of the characteristics of the original lectures themselves. Although this work is by no means complete or comprehensive, it is our hope that, through this, the reader can gain an overall appreciation for the main topics and conjectures considered herein. In the meantime, to the reader who intends to study these topics in depth, we hope that we have offered a motivated guide to the literature.
A rough idea for the content and organization of this book can easily be gained by glancing at the table of content, and we shall therefore not elab-orate too much here. During the preparation for our lectures as well as for this manuscript, we have benefited from many existing resources in the liter-ature. For the more introductory parts (which form the early parts of most of the chapters), we wish to mention especially Goro Shimura's Introduction to the Arithmetic Theory of Automorphic Functions, Danie L Bump's Au- tomorphic Forms and Representations, and Henryk Iwaniec and Emmanuel Kowalski's Analytic Number Theory. In several places, the arrangement of concepts and the choice of certain examples have been influenced by these excellent books. It is impossible to acknowledge in detail the numerous works we have consulted for the rest of the book, and the reader must be referred to the Bibliography. Some work by the authors themselves on the conjectures have been briefly (and incompletely) summarized here as well,mainly in the two final chapters.
It remains our most pleasant duty to acknowledge the many people from whom we have received much help and support, mathematically and other-wise. First and foremost, we thank Professor Goro Shimura and Professor Dorian Goldfeld, our thesis advisors. We are grateful to Professor Shou-Wu Zhang of Columbia University, Professor Hongwen Lu of Tongji University,Professor K. Ilhan Ikeda of Istanbul Bilgi University, Professor Juping Wang of Fudan University, and Professor Tianze Wang of Henan University for their kind invitations and their steadfast support. We are much indebted to Professor Paula Cohen Tretkoff of Texas A&M University and Profes-sor Yingchun Cai of Tongji University for their encouragement and advice.With gratitude we acknowledge the kindly and efficient assistance from the editors, Liping Wang and Lisa Libin Fan. Finally, we give our heartfelt thanks to all members of our families, and especially to our wives, Susan Staples and Jing Tian, for their longsuffering and their love.
Ze-Li Dou and Qiao Zhang
Fort Worth, TX, USA
October 11, 2011
