自守形式與L-函數(shù)簡明六章

內容概要

Six Short Chapters on Automorphic Forms and L-functions treats the period conjectures ofShimura and the moment conjecture.These conjectures are of central importance in contemporary number theory,but have hitherto remained little disussed 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 riew of the conjectures and providing researchers intending to specialize in the area a guide to the relevant liter
ature.

作者簡介

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

書籍目錄

Chapter 1 Modular forms and the Shimura-Taniyama Conjecture1.1 Elliptic functions1.2 Modular forms1.3 Examples1.4 Hecke operators and eigenforms1.5 L-functions1.6 Modular forms of higher level1.7 Elliptic curves1.8 Conjectures,and the theorem of Wiles,et alReferencesChapter 2 Periods of automorphic forms2.1 Automorphic forms2.2 Periods arising from Dirichlet series2.3 Periods arising from elliptic curves2.4 Periods arising from cohomology theory2.5 Recapitulation2.6 Hilbert modular forms2.7 Automorphic forms with respect to a quaternion algebra2.8 Adelic automorphic forms2.9 Hecke operators2.10 The standard L-function for Hilbert modular forms and the V periods2.11 The inner product and the Q periods2.12 The Eichler-Shimura Isomorphism and the P periodsReferencesChapter 3 Lifting of automorphic forms3.1 Dirichlet series3.2 Hecke characters with respect to a real quadratic number field3.3 L-series and the base change lift for characters3.4 Hilbert modular forms over a real quadratic field and the Doi-Naganuma lift3.5 Applications of base change3.6 Shimura's period conjectures3.7 Yoshida's workReferencesChapter 4 Zeros of L-functions4.1 Trivial zeros4.2 The Grand Riemann Hypothesis4.3 Zero-free regions4.4 Siegel zeros4.5 Correlations of zeros4.6 Random matrix theoryReferencesChapter 5 Special values of L-functions5.1 The Birch-Swinnerton-Dyer conjecture5.2 The Gross-Zagier formula5.3 The class number problem5.4 The Lindel?f conjecture and subconvexity bounds5.5 Discrete moment conjecture of L-functions5.6 Integral moment conjectures of L-functionsReferencesChapter 6 Theta lifts and periods with respect to a quadratic extension6.1 Setting6.2 Automorphic forms with respect to BE6.3 Theta function6.4 Transformation formulas6.5 Theta correspondence6.6 Reduction of the integral6.7 Two technical lemmata6.8 Main theorems on theta correspondence6.9 Adelic forms on *′A6.10 Embeddings and inner product6.11 Theta correspondence of adelic forms6.12 Hecke Operators on GL2(F)6.13 The action of Hecke operators on I(z;C;r;h)6.14 Hecke operators on adelic forms6.15 Relation between actions of Hecke operators6.16 Further ReadingReferences

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