Introduction 1 1. Notation and conventions 8 2. Shimura varieties for unitary groups 13 3. Modular forms on unitary groups 23 4. Hida theory for unitary groups 36 5. Ordinary p-adic Eisenstein series on U(2, 1) 52 6. Constant terms of the p-adic Eisenstein series 68 7. Eisenstein ideal and p-adic L-functions 76 8. Application to the main conjecture for CM fields 93 Acknowledgments 107 Reference...

 Environmental enrichment is one of the methods for improving fish growth performance. In the present trial, the effects of both light color and music on growth performance and survival of goldfish (Carassius auratus) (initial mean weight: 4.15±0.08 g) were investigated for 2 months. Two light color treatments (red light and white light) and three music treatments (M0: without music, M30:30 min...

Contents 0 Introduction 2 1 Notations and preliminaries 5 2 Invariant diierential operators 9 3 Deenition of the Eisenstein integral 13 4 Relation with the principal series 14 5 Finite dimensional class (1,1) representations 20 6 Functions of S-polynomial growth 23 7 S-genericity 25 8 Projection along innnitesimal characters 25 9 Estimates for j 30 10 Initial estimates for Eisenstein integrals ...

Eisenstein series play a critical role in number theory. For two hundred years they have been an essential tool in the analysis of automorphic L-functions and in studying properties of quadratic forms in one and several variables. The construction is clear and straightforward, while their properties are sometimes very surprising. The arithmetic of their Fourier coefficients, and their analytic ...

Part 3. Geometric Construction of Loop Eisenstein Series 31 11. Bloch’s Map 31 12. Adeleic Loop groups and G-bundles on a Punctured Surface 33 13. Affine flag varieties and extensions of G-bundles 38 14. Relative Chern classes and Central extensions 41 15. Loop Eisenstein Series and Geometric Generating Functions 47 16. Ribbons and a Formal Analogue 48 17. Example: Loop Eisenstein Series on P1. 53

Found in the collected works of Eisenstein are twenty continued fraction expansions. The expansions have since emerged in the literature in various forms, although a complete historical account and self-contained treatment has not been given. We provide one here, motivated by the fact that these expansions give continued fraction expansions for modular forms. Eisenstein himself did not record p...