The Conference on L-Functions : Fukuoka, Japan, 18-23 February 2006

This invaluable volume collects papers written by many of the world's top experts on L-functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole. The contributions reflect the latest, most advanced and most important aspects of L-functions. In particular, it contains Hida's lecture notes at the conference and at the Eigen variety semester in Harvard University and Weng's detailed account of his works on high rank zeta functions and non-abelian L-functions. Contents......Page 10 Preface......Page 6 List of Participants......Page 8 Quantum Maass Forms......Page 12 1 Quantum Maass forms associated to Maass cusp forms and Eisenstein series......Page 13 2 Quantum Maass forms associated to invariant eigenfunctions......Page 20 References......Page 26 Introduction......Page 28 1 Lecture 1: Galois deformation and L-invariant......Page 35 2 Lecture 2: Elliptic curves with multiplicative reduction......Page 42 3 Lecture 3: L-invariants of CM fields......Page 48 4 Appendix: Differential and adjoint square Selmer group......Page 55 References......Page 61 Siegel Modular Forms of Weight Three and Conjectural Correspondence of Shimura Type and Langlands Type......Page 66 1 Definition of Siegel modular forms......Page 67 2 Conjectures on dimensions of weight 3......Page 68 3 Conjecture on Eichler type correspondence......Page 71 4 Geometric interpretation......Page 75 5 Conjecture on Shimura type correspondence......Page 76 References......Page 78 0 Introduction......Page 82 1 An arithmetic formula for Fourier coefficients......Page 84 2 Applications to convolutions......Page 89 References......Page 96 On an Extension of the Derivation Relation for Multiple Zeta Values......Page 100 References......Page 105 1 Symmetric fourth......Page 106 2 Symmetric mth powers......Page 109 3 First occurences of poles of symmetric power L-functions......Page 114 4 Descent to cuspidal representations on classical groups......Page 116 5 Remark on the images of functorial lift......Page 119 References......Page 122 0 Introduction......Page 126 1 Zeta functions of root systems......Page 129 2 Structural background of functional relations......Page 132 3 Functional relations for S3(s; A3)......Page 136 References......Page 149 1 Automorphic forms......Page 152 2 Sum formulas......Page 155 3 The inversion problem......Page 158 4 Proof (1)......Page 160 5 Proof (2)......Page 165 6 Concluding remarks......Page 170 References......Page 172 1 The Selberg class......Page 176 2 The Lindelof class......Page 178 References......Page 184 0 Introduction......Page 186 1 The idea of the proof......Page 188 2 The frame of the proof......Page 190 3 Proof of Theorem 1......Page 193 4 Proof of Lemma 1......Page 203 5 Proof of Lemma 5......Page 206 References......Page 209 0 Introduction......Page 212 1 Setting the stage......Page 216 2 Elliptic curves associated with J2(n)......Page 218 3 Geometric interpretation of the differential equation for W2(T)......Page 220 4 Modular properties......Page 222 5 Closing remarks......Page 225 References......Page 228 A Geometric Approach to L-Functions......Page 230 1 High Rank Zetas for Number Fields......Page 234 2 Non-Abelian L-Functions......Page 265 3 Geometric and Analytic Truncations......Page 275 4 Rankin-Selberg & Zagier Method......Page 307 5 High Rank Zetas and Eisenstein Series......Page 326 6 Stability and Distance to Cusps......Page 332 7 Explicit Formulas for Rank Two Zetas......Page 342 8 Zeros of Rank Two Zetas......Page 347 9 A Rank Three Zeta and Its Zeros......Page 353 REFERENCES......Page 376 Collects papers written by many of the world's top experts on L-functions. This work not only covers a range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole. It also includes contributions that reflect the most important aspects of L-functions.