# The Gross-Zagier Formula on Shimura Curves

**Xinyi Yuan, Shou-wu Zhang & Wei Zhang**

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.

**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.