The Gross-Zagier Formula on Shimura Curves

Princeton University Press
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This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations.

The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas.

The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it.

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About the author

Xinyi Yuan is assistant professor of mathematics at Princeton University. Shou-wu Zhang is professor of mathematics at Princeton University and Columbia University. Wei Zhang is assistant professor of mathematics at Columbia University.
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Additional Information

Publisher
Princeton University Press
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Published on
Nov 11, 2012
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Pages
272
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ISBN
9781400845644
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Language
English
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Genres
Mathematics / Number Theory
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Content Protection
This content is DRM protected.
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Available on Android devices
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Eligible for Family Library

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