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 on overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant liter ature.
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目录
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 the Doi-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 Lindel?f 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
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