The Gross-Zagier Formula on Shimura Curves Hardback
Part of the Annals of Mathematics Studies series
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.
- Format: Hardback
- Pages: 272 pages
- Publisher: Princeton University Press
- Publication Date: 02/12/2012
- Category: Algebraic geometry
- ISBN: 9780691155913
- Paperback from £52.75
- EPUB from £70.16