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.

Original languageEnglish (US)
PublisherPrinceton University Press
ISBN (Print)9780691155913
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.