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| Six Short Chapters on Automorphic Forms and / functions treats the period conjectures of Shimura and the moment conjecture. These conjectures are of centralimportance in contemporary number theory， but have hitherto remained littlediscussed in expository form. The book is divided into six short and relativelyindependent chapters， each with its own theme， and presents a motivated and livelyaccount ofthe main topics， providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevantliterature. |
| 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 Autornorphic 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 numbereld 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 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 G'A 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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