Six Short Chapters on Automorphic Forms and L_fubctions

自守形式与L—函数简明六章
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Author: Ze_LiDou and Qiao Zhang
Language: Chinese
ISBN/ISSN: 9787030331687
Published on: 2012-01
Hardcover

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.



《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

.2.9 hecke operators
2.10 the standard l-function for hilbert modular forms and the v periods
2.11 the inner product and the q periods
2.12 the eichler-shimura isomorphism and the p periods
references
chapter 3 lifting of automorphic forms
3.1 dirichlet series
3.2 hecke characters with respect to a real quadratic number field
3.3 l-series and the base change lift for characters
3.4 hilbert modular forms over a real quadratic field and thedoi-naganuma lift
3.5 applications of base change
3.6 shimura's period conjectures
3.7 yoshida's work
references
chapter 4 zeros of l-functions
4.1 trivial zeros
4.2 the grand riemann hypothesis
4.3 zero-free regions
4.4 siegel zeros
4.5 correlations of zeros
4.6 random matrix theory
references
chapter 5 special values of l-functions
5.1 the birch-swinnerton-dyer conjecture
5.2 the gross-zagier formula
5.3 the class number problem
5.4 the lindelsf conjecture and subconvexity bounds
5.5 discrete moment conjecture of l-functions
5.6 integral moment conjectures of l-functions
references
chapter 6 theta lifts and periods with respect to a quadratic extension
6.1 setting
6.2 automorphic forms with respect to be
6.3 theta function
6.4 transformation formulas
6.5 theta correspondence
6.6 reduction of the integral
6.7 two technical lemmata
6.8 main theorems on theta correspondence
6.9 adelic forms on ga
6.10 embeddings and inner product
6.11 theta correspondence of adelic forms
6.12 hecke operators on gl2(f)
6.13 the action of hecke operators on i(z; c; r; h)
6.14 hecke operators on adelic forms
6,15 relation between actions of hecke operators
6.16 further reading
references


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