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Elementary Theory of L-functions and Eisenstein Series / Haruzo Hida.

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An elementary but detailed insight into the theory of L-functions. The presentation is self contained and concise.

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Bibliographic Details
Online Access: Full Text (via Cambridge)
Main Author: Hida, Haruzo
Format: Electronic eBook
Language:English
Published: Cambridge : Cambridge University Press, 1993.
Series:London Mathematical Society student texts ; no. 26.
Subjects:
L-functions.
Eisenstein series.
Table of Contents:
  • Cover
  • Title
  • Copyright
  • Contents
  • Preface
  • Suggestions to the reader
  • Chapter 1. Algebraic Number Theory
  • 1.1. Linear algebra over rings
  • 1.2. Algebraic number fields
  • 1.3. p-adic numbers
  • Chapter 2. Classical L-functions and Eisenstein series
  • 2.1. Euler's method
  • 2.2. Analytic continuation and the functional equation
  • 2.3. Hurwitz and Dirichlet L-functions
  • 2.4. Shintani L-functions
  • 2.5. L-functions of real quadratic field
  • 2.6. L-functions of imaginary quadratic fields
  • 2.7. Hecke L-functions of number fields.
  • Chapter 3. p-adic Hecke L-functions
  • 3.1. Interpolation series
  • 3.2. Interpolation series in p-adic fields
  • 3.3. p-adic measures on Zp
  • 3.4. The p-adic measure of the Riemann zeta function
  • 3.5. p-adic Dirichlet L-functions
  • 3.6. Group schemes and formal group schemes
  • 3.7. Toroidal formal groups and p-adic measures
  • 3.8. p-adic Shintani L-functions of totally real fields
  • 3.9. p-adic Hecke L-functions of totally real fields
  • Chapter 4. Homological Interpretation
  • 4.1. Cohomology groups on Gm(C)
  • 4.2. Cohomological interpretation of Dirichlet L-values.
  • 4.3. p-adic measures and locally constant functions
  • 4.4. Another construction of p-adic Dirichlet L-functions
  • Chapter 5. Elliptic modular forms and their L-functions
  • 5.1. Classical Eisenstein series of GL(2)/Q
  • 5.2. Rationality of modular forms
  • 5.3. Hecke operators
  • 5.4. The Petersson inner product and the Rankin product
  • 5.5. Standard L-functions of holomorphic modular forms
  • Chapter 6. Modular forms and cohomology groups
  • 6.1. Cohomology of modular groups
  • 6.2. Eichler-Shimura isomorphisms
  • 6.3. Hecke operators on cohomology groups.
  • 6.4. Algebraicity theorem for standard L-functions of GL(2)
  • 6.5. Mazur's p-adic Mellin transforms
  • Chapter 7. Ordinary A-adic forms, two variable p-adic Rankin products and Galois representations
  • 7.1. p-Adic families of Eisenstein series
  • 7.2. The projection to the ordinary part
  • 7.3. Ordinary A-adic forms
  • 7.4. Two variable p-adic Rankin product
  • 7.5. Ordinary Galois representations into GL2(ZP[[X]])
  • 7.6. Examples of A-adic forms
  • Chapter 8. Functional equations of Hecke L-functions
  • 8.1. Adelic interpretation of algebraic number theory.
  • 8.2. Hecke characters as continuous idele characters
  • 8.3. Self-duality of local fields
  • 8.4. Haar measures and the Poisson summation formula
  • 8.5. Adelic Haar measures
  • 8.6. Functional equations of Hecke L-functions
  • Chapter 9. Adelic Eisenstein series and Rankin products
  • 9.1. Modular forms on GL2(FA)
  • 9.2. Fourier expansion of Eisenstein series
  • 9.3. Functional equation for Eisenstein series
  • 9.4. Analytic continuation of Rankin products
  • 9.5. Functional equations for Rankin products
  • Chapter 10. Three variable p-adic Rankin products.

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