The Gross-Zagier formula on Shimura curves

Xinyi Yuan, Shou-Wu Zhang, Wei Zhang

Research output: Book/ReportBook

3 Scopus citations

Abstract

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.

Original languageEnglish (US)
PublisherPrinceton University Press
Number of pages256
ISBN (Electronic)9781400845644
ISBN (Print)9780691155920
StatePublished - Nov 11 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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    Yuan, X., Zhang, S-W., & Zhang, W. (2012). The Gross-Zagier formula on Shimura curves. Princeton University Press.