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The Gross-Zagier Formula on Shimura Curves$
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Xinyi Yuan, Shou-wu Zhang, and Wei Zhang

Print publication date: 2012

Print ISBN-13: 9780691155913

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691155913.001.0001

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Decomposition of the Geometric Kernel

Decomposition of the Geometric Kernel

(p.206) Chapter Seven Decomposition of the Geometric Kernel
The Gross-Zagier Formula on Shimura Curves

Xinyi Yuan

Shou-Wu Zhang

Wei Zhang

Princeton University Press

This chapter describes the decomposition of the geometric kernel. It considers the assumptions on the Schwartz function and decomposes the height series into local heights using arithmetic models. The intersections with the Hodge bundles are zero, and a decomposition to a sum of local heights by standard results in Arakelov theory is achieved. The chapter proceeds by reviewing the definition of the Néeron–Tate height and shows how to compute it by the arithmetic Hodge index theorem. When there is no horizontal self-intersection, the height pairing automatically decomposes to a summation of local pairings. The chapter proves that the contribution of the Hodge bundles in the height series is zero. It also compares two kernel functions and states the computational result. It concludes by deducing the kernel identity.

Keywords:   geometric kernel, Schwartz function, height series, Hodge bundle, local height, Arakelov theory, Néeron–Tate height, Hodge index theorem, kernel function, kernel identity

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