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An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces
 
Wayne Aitken Harvard University, Cambridge, MA
An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces
eBook ISBN:  978-1-4704-0158-0
Product Code:  MEMO/120/573.E
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces
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An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces
Wayne Aitken Harvard University, Cambridge, MA
eBook ISBN:  978-1-4704-0158-0
Product Code:  MEMO/120/573.E
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1201996; 174 pp
    MSC: Primary 14; 11

    The first half of this work gives a treatment of Deligne's functorial intersection theory tailored to the needs of this paper. This treatment is intended to satisfy three requirements: 1) that it be general enough to handle families of singular curves, 2) that it be reasonably self-contained, and 3) that the constructions given be readily adaptable to the process of adding norms and metrics such as is done in the second half of the paper.

    The second half of the work is devoted to developing a class of intersection functions for singular curves that behaves analogously to the canonical Green's functions introduced by Arakelov for smooth curves. These functions are called intersection functions since they give a measure of intersection over the infinite places of a number field. The intersection over finite places can be defined in terms of the standard apparatus of algebraic geometry.

    Finally, the author defines an intersection theory for arithmetic surfaces that includes a large class of singular arithmetic surfaces. This culminates in a proof of the arithmetic Riemann-Roch theorem.

    Readership

    Graduate students and research mathematicians interested in algebraic geometry and number theory.

  • Table of Contents
     
     
    • Chapters
    • 1. The intersection pairing for one-dimensional schemes
    • 2. The intersection pairing for families of one-dimensional schemes
    • 3. The Riemann-Roch isomorphism
    • 4. Intersection functions on complex curves
    • 5. The arithmetic Riemann-Roch isomorphism
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1201996; 174 pp
MSC: Primary 14; 11

The first half of this work gives a treatment of Deligne's functorial intersection theory tailored to the needs of this paper. This treatment is intended to satisfy three requirements: 1) that it be general enough to handle families of singular curves, 2) that it be reasonably self-contained, and 3) that the constructions given be readily adaptable to the process of adding norms and metrics such as is done in the second half of the paper.

The second half of the work is devoted to developing a class of intersection functions for singular curves that behaves analogously to the canonical Green's functions introduced by Arakelov for smooth curves. These functions are called intersection functions since they give a measure of intersection over the infinite places of a number field. The intersection over finite places can be defined in terms of the standard apparatus of algebraic geometry.

Finally, the author defines an intersection theory for arithmetic surfaces that includes a large class of singular arithmetic surfaces. This culminates in a proof of the arithmetic Riemann-Roch theorem.

Readership

Graduate students and research mathematicians interested in algebraic geometry and number theory.

  • Chapters
  • 1. The intersection pairing for one-dimensional schemes
  • 2. The intersection pairing for families of one-dimensional schemes
  • 3. The Riemann-Roch isomorphism
  • 4. Intersection functions on complex curves
  • 5. The arithmetic Riemann-Roch isomorphism
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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