1 (p1): Chapter 1 Modular forms and the Shimura-Taniyama Conjecture 1 (p2): 1.1 Elliptic functions 3 (p3): 1.2 Modular forms 5 (p4): 1.3 Examples 8 (p5): 1.4 Hecke operators and eigenforms 10 (p6): 1.5 L-functions 12 (p7): 1.6 Modular forms of higher level 13 (p8): 1.7 Elliptic curves 15 (p9): 1.8 Conjectures,and the theorem of Wiles,et al 16 (p10): References 17 (p11): Chapter 2 Periods of automorphic forms 17 (p12): 2.1 Automorphic forms 19 (p13): 2.2 Periods arising from Dirichlet series 20 (p14): 2.3 Periods arising from elliptic curves 22 (p15): 2.4 Periods arising from cohomology theory 23 (p16): 2.5 Recapitulation 23 (p17): 2.6 Hilbert modular forms 25 (p18): 2.7 Automorphic forms with respect to a quaternion algebra 27 (p19): 2.8 Adelic automorphic forms 29 (p20): 2.9 Hecke operators 32 (p21): 2.10 The standard L-function for Hilbert modular forms and the V periods 35 (p22): 2.11 The inner product and the Q periods 36 (p23): 2.12 The Eichler-Shimura Isomorphism and the P periods 42 (p24): References 43 (p25): Chapter 3 Lifting of automorphic forms 43 (p26): 3.1 Dirichlet series 44 (p27): 3.2 Hecke characters with respect to a real quadratic number field 46 (p28): 3.3 L-series and the base change lift for characters 48 (p29): 3.4 Hilbert modular forms over a real quadratic field and the Doi-Naganuma lift 51 (p30): 3.5 Applications of base change 52 (p31): 3.6 Shimura's period conjectures 54 (p32): 3.7 Yoshida's work 55 (p33): References 57 (p34): Chapter 4 Zeros of L-functions 57 (p35): 4.1 Trivial zeros 60 (p36): 4.2 The Grand Riemann Hypothesis 62 (p37): 4.3 Zero-free regions 67 (p38): 4.4 Siegel zeros 71 (p39): 4.5 Correlations of zeros 73 (p40): 4.6 Random matrix theory 75 (p41): References 77 (p42): Chapter 5 Special values of L-functions 77 (p43): 5.1 The Birch-Swinnerton-Dyer conjecture...

zlibzh/no-category/Ze-Li Dou， Qiao Zhang, Ze Li Dou/自守形式与L-函数简明六章 英文_42572388.pdf

SIX SHORT CHAPTERS ON AUTOMORPHIC FPRMS AND L-FUNCTIONS=自守形式与L-函数简明六章

Six Short Chapters on Automorphic Forms and L-functions

Dou, Ze-Li, Zhang, Qiao

Ze Li Dou; Qiao Zhang

Spektrum Akademischer Verlag. in Springer-Verlag GmbH

Steinkopff. in Springer-Verlag GmbH

Springer ; Science Press Beijing

Springer London, Limited

Science Press ; Springer

SCIENCE PREESS BEIJING

China, People's Republic, China

Springer Nature, Berlin, 2012

Heidelberg, New York, 2012

Beijing, Berlin, ©2012

Germany, Germany

Dec 11, 2012

Bookmarks: p1 (p1): Chapter 1 Modular forms and the Shimura-Taniyama Conjecture
p2 (p1): 1.1 Elliptic functions
p3 (p3): 1.2 Modular forms
p4 (p5): 1.3 Examples
p5 (p8): 1.4 Hecke operators and eigenforms
p6 (p10): 1.5 L-functions
p7 (p12): 1.6 Modular forms of higher level
p8 (p13): 1.7 Elliptic curves
p9 (p15): 1.8 Conjectures,and the theorem of Wiles,et al
p10 (p16): References
p11 (p17): Chapter 2 Periods of automorphic forms
p12 (p17): 2.1 Automorphic forms
p13 (p19): 2.2 Periods arising from Dirichlet series
p14 (p20): 2.3 Periods arising from elliptic curves
p15 (p22): 2.4 Periods arising from cohomology theory
p16 (p23): 2.5 Recapitulation
p17 (p23): 2.6 Hilbert modular forms
p18 (p25): 2.7 Automorphic forms with respect to a quaternion algebra
p19 (p27): 2.8 Adelic automorphic forms
p20 (p29): 2.9 Hecke operators
p21 (p32): 2.10 The standard L-function for Hilbert modular forms and the V periods
p22 (p35): 2.11 The inner product and the Q periods
p23 (p36): 2.12 The Eichler-Shimura Isomorphism and the P periods
p24 (p42): References
p25 (p43): Chapter 3 Lifting of automorphic forms
p26 (p43): 3.1 Dirichlet series
p27 (p44): 3.2 Hecke characters with respect to a real quadratic number field
p28 (p46): 3.3 L-series and the base change lift for characters
p29 (p48): 3.4 Hilbert modular forms over a real quadratic field and the Doi-Naganuma lift
p30 (p51): 3.5 Applications of base change
p31 (p52): 3.6 Shimura's period conjectures
p32 (p54): 3.7 Yoshida's work
p33 (p55): References
p34 (p57): Chapter 4 Zeros of L-functions
p35 (p57): 4.1 Trivial zeros
p36 (p60): 4.2 The Grand Riemann Hypothesis
p37 (p62): 4.3 Zero-free regions
p38 (p67): 4.4 Siegel zeros
p39 (p71): 4.5 Correlations of zeros
p40 (p73): 4.6 Random matrix theory
p41 (p75): References
p42 (p77): Chapter 5 Special values of L-functions
p43 (p77): 5.1 The Birch-Swinnerton-Dyer conjecture
p44 (p80): 5.2 The Gross-Zagier formula
p45 (p82): 5.3 The class number problem
p46 (p85): 5.4 The Lindel？f conjecture and subconvexity bounds
p47 (p89): 5.5 Discrete moment conjecture of L-functions
p48 (p94): 5.6 Integral moment conjectures of L-functions
p49 (p97): References
p50 (p99): Chapter 6 Theta lifts and periods with respect to a quadratic extension
p51 (p99): 6.1 Setting
p52 (p100): 6.2 Automorphic forms with respect to BE
p53 (p103): 6.3 Theta function
p54 (p106): 6.4 Transformation formulas
p55 (p107): 6.5 Theta correspondence
p56 (p107): 6.6 Reduction of the integral
p57 (p109): 6.7 Two technical lemmata
p58 (p110): 6.8 Main theorems on theta correspondence
p59 (p110): 6.9 Adelic forms on ？
p60 (p111): 6.10 Embeddings and inner product
p61 (p113): 6.11 Theta correspondence of adelic forms
p62 (p115): 6.12 Hecke Operators on GL2(F)
p63 (p117): 6.13 The action of Hecke operators on I(z;C;r;h)
p64 (p118): 6.14 Hecke operators on adelic forms
p65 (p120): 6.15 Relation between actions of Hecke operators
p66 (p121): 6.16 Further Reading
p67 (p122): References

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"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. Ze-Li Dou and Qiao Zhang are both associate professors of Mathematics at Texas Christian University, USA.

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. Book jacket

This treatment of the period conjectures of Shimura and the moment conjecture covers two little-discussed, yet vital topics in number theory. It also provides a guide to the frontiers of research on these themes, as well as an overview of relevant literature.

2024-06-13