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The first half of this work gives a treatment of Delignes 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 singularMoreThe first half of this work gives a treatment of Delignes 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 Greens 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. An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces by Wayne Aitken