# An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces

Share this page
*Wayne Aitken*

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.

#### Table of Contents

# Table of Contents

## An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces

- Contents v6 free
- Chapter 1. The Intersection Pairing for One-Dimensional Schemes 110 free
- 1. Preliminaries 211
- 2. The Determinant of Cohomology Functor 413
- 3. The Norm Functor for Zero-Dimensional Schemes 1625
- 4. The Definition of the Intersection Pairing 2029
- 5. The Intersection Pairing and Norms 2837
- 6. The Norm Functor for Divisors 3241
- 7. Other Properties of the Intersection Pairing 3847
- 8. Extensions of the Base Field 4453

- Chapter 2. The Intersection Pairing for Families of One-dimensional Schemes 4756
- Chapter 3. The Riemann-Roch Isomorphism 8089
- Chapter 4. Intersection Functions on Complex Curves 101110
- 1. Motivation: The Non-Archimedean Situation 101110
- 2. The Archimedean Case: Basic Definitions 109118
- 3. Intersection Functions on Nonsingular Curves 115124
- 4. Intersection Functions on Singular Curves: The Existence Theorem 123132
- 5. Classification of Intersection Functions: Preliminaries 128137
- 6. Classification of Intersection Functions 137146
- 7. Chern Forms of Intersection Functions 144153
- 8. Chern Forms of Intersection Functions: Proofs 147156
- 9. Chern Forms of Intersection Functions: Another Interpretation 153162

- Chapter 5. The Arithmetic Riemann-Roch Isomorphism 161170
- Bibliography 174183