You are previewing The Riemann Hypothesis for Function Fields.
O'Reilly logo
The Riemann Hypothesis for Function Fields

Book Description

This book provides a lucid exposition of the connections between non-commutative geometry and the famous Riemann Hypothesis, focusing on the theory of one-dimensional varieties over a finite field. The reader will encounter many important aspects of the theory, such as Bombieri's proof of the Riemann Hypothesis for function fields, along with an explanation of the connections with Nevanlinna theory and non-commutative geometry. The connection with non-commutative geometry is given special attention, with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space, and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes. The exposition is accessible at the graduate level and above, and provides a wealth of motivation for further research in this area.

Table of Contents

  1. Cover
  2. Series
  3. Title
  4. Copyright
  5. Dedication
  6. Contents
  7. List of illustrations
  8. Preface
  9. Introduction
  10. 1 Valuations
    1. 1.1 Trace and norm
      1. 1.1.1 The canonical pairing
    2. 1.2 Valued fields
      1. 1.2.1 Norms on a vector space
      2. 1.2.2 Discrete valuations
      3. 1.2.3 Different and ramification
      4. 1.2.4 Inseparable extensions
    3. 1.3 Valuations of F[sub(q)](T)
    4. 1.4 Global fields
      1. 1.4.1 Constants and nonconstant functions
      2. 1.4.2 Ramification
  11. 2 The local theory
    1. 2.1 Additive character and measure
      1. 2.1.1 Characters of F[sub(q)](T)
      2. 2.1.2 Characters of K
      3. 2.1.3 Fourier transform
    2. 2.2 Multiplicative character and measure
    3. 2.3 Local zeta function
    4. 2.4 Functional equation
  12. 3 The zeta function
    1. 3.1 Additive theory
      1. 3.1.1 Divisors
      2. 3.1.2 Riemann–Roch
    2. 3.2 Multiplicative theory
      1. 3.2.1 Divisor classes
      2. 3.2.2 Coarse idele classes
    3. 3.3 The zeta function
      1. 3.3.1 Constant field extensions
      2. 3.3.2 Shifted zeta function
    4. 3.4 Computation
    5. 3.5 Semi-local theory
    6. 3.6 Two-variable zeta function
      1. 3.6.1 The polynomial L[sub(C)](X, Y)
  13. 4 Weil positivity
    1. 4.1 Functions on the coarse idele classes
    2. 4.2 Zeros of Λ[sub(C)]
    3. 4.3 Explicit formula
  14. 5 The Frobenius flow
    1. 5.1 Heuristics for the Riemann hypothesis
    2. 5.2 Orbits of Frobenius
      1. 5.2.1 The projective line
      2. 5.2.2 Example: elliptic curves
      3. 5.2.3 The curve C
    3. 5.3 Galois covers
    4. 5.4 The Riemann hypothesis for C
      1. 5.4.1 The Frobenius flow
      2. 5.4.2 Frobenius as symmetries
    5. 5.5 Comparison with the Riemann hypothesis
  15. 6 Shift operators
    1. 6.1 The Hilbert spaces Z and H
      1. 6.1.1 The truncated shift on Z
      2. 6.1.2 The trace using a kernel
    2. 6.2 Shift operators
      1. 6.2.1 Averaging spaces
    3. 6.3 Local trace
      1. 6.3.1 Order of the cutoffs
      2. 6.3.2 Direct computation
      3. 6.3.3 Trace using the kernel
    4. 6.4 How to prove the Riemann hypothesis for C?
    5. 6.5 The operators M, A, C, F*, and E
      1. 6.5.1 Restricting the support
      2. 6.5.2 Making the measure additive
      3. 6.5.3 Smoothing the oscillations
      4. 6.5.4 Restricting to the coarse idele classes
    6. 6.6 Semi-local and global trace
    7. 6.7 The kernel on the analytic space
  16. 7 Epilogue
    1. 7.1 Archimedean translation
    2. 7.2 Global translation
    3. 7.3 The space of adele classes
  17. References
  18. Index of notation
  19. Index