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Table of contents Preface vii 1 Irreducibility and Cuspidality Dinakar Ramakrishnan 1 1 Preliminaries 5 2 The first step in the proof 15 3 The second step in the proof 16 4 Galois representations attached to regular, selfdual cusp forms on GL(4) 18 5 Two useful lemmas on cusp forms on GL(4) 20 6 Finale 21 References 25 2 On Liftings of Holomorphic Modular Forms Tamotsu Ikeda 29 1 Basic facts 29 2 Fourier coefficients of the Eisenstein series 30 3 Kohnen plus space 32 4 Lifting of cusp forms 33 5 Outline of the proof 34 6 Relation to the Saito-Kurokawa lifts 35 7 Hermitian modular forms and hermitian Eisensetein series 37 8 The case m = 2n + 1 39 9 The case m = 2n 40 10 L-functions 40 11 The case m = 2 41 References 42 3 Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs Toshiyuki Kobayashi 45 1 Introduction and statement of main results 45 2 Mainmachinery fromcomplex geometry 56 3 Proof of Theorem A 61 4 Proof of Theorem C 68 5 Uniformly bounded multiplicities — Proof of Theorems B and D 70 6 Counterexamples 77 7 Finite-dimensional cases — Proof of Theorems E and F 83 8 Generalization of the Hua-Kostant-Schmid formula 88 9 Appendix: Associated bundles on Hermitian symmetric spaces 103 References 105 4 The Rankin-Selberg Method for Automorphic Distributions Stephen D. Miller and Wilfried Schmid 111 1 Introduction 111 2 Standard L-functions for SL(2) 115 3 Pairings of automorphic distributions 121 4 The Rankin-Selberg L-function for GL(2) 128 5 Exterior square on GL(4) 137 References 148 5 Langlands Functoriality Conjecture and Number Theory Freydoon Shahidi 151 1 Introduction 151 2 Modular forms, Galois representations and Artin L-functions 152 3 Lattice point problems and the Selberg conjecture 156 4 Ramanujan conjecture for Maass forms 158 5 Sato-Tate conjecture 159 6 Functoriality for symmetric powers 161 7 Functoriality for classical groups 163 8 Ramanujan conjecture for classical groups 164 9 The method 166 References 169 6 Discriminant of Certain K3 Surfaces Ken-Ichi Yoshikawa 175 1 Introduction — Discriminant of elliptic curves 175 2 K3 surfaces with involution and their moduli spaces 178 3 Automorphic forms on the moduli space 180 4 Equivariant analytic torsion and 2-elementary K3 surfaces 182 5 The Borcherds products 184 6 Borcherds products for odd unimodular lattices 186 7 K3 surfaces of Matsumoto-Sasaki-Yoshida 188 8 Discriminant of quartic surfaces 200 References 209
© Toshiyuki Kobayashi