TY - BOOK ID - 77890651 TI - The Gross-Zagier Formula on Shimura Curves AU - Yuan, Xinyi, AU - Zhang, Shou-wu AU - Zhang, Wei PY - 2012 SN - 9786613883919 1400845645 1283571463 9781400845644 0691155925 9780691155920 0691155917 9780691155913 9780691155913 9780691155920 9781283571463 PB - Princeton, NJ DB - UniCat KW - Shimura varieties. KW - Arithmetical algebraic geometry. KW - Automorphic forms. KW - Quaternions. KW - Algebra, Universal KW - Algebraic fields KW - Curves KW - Surfaces KW - Numbers, Complex KW - Vector analysis KW - Automorphic functions KW - Forms (Mathematics) KW - Algebraic geometry, Arithmetical KW - Arithmetic algebraic geometry KW - Diophantine geometry KW - Geometry, Arithmetical algebraic KW - Geometry, Diophantine KW - Number theory KW - Varieties, Shimura KW - Arithmetical algebraic geometry KW - Arakelov theory. KW - Benedict Gross. KW - Don Zagier. KW - EichlerГhimura theory. KW - Eisenstein series. KW - GrossКagier formula. KW - Heegner point. KW - Hodge bundle. KW - Hodge index theorem. KW - L-series. KW - MordellЗeil group. KW - NeronДate height. KW - RankinГelberg L-function. KW - Schwartz function. KW - Shimizu lifting. KW - Shimura curve. KW - Shimura curves. KW - SiegelЗeil formula. KW - Waldspurger formula. KW - Weil representation. KW - abelian varieties. KW - analytic kernel function. KW - analytic kernel. KW - degenerate Schwartz function. KW - discrete series. KW - generating series. KW - geometric kernel. KW - height series. KW - holomorphic kernel function. KW - holomorphic projection. KW - incoherent Eisenstein series. KW - incoherent automorphic representation. KW - incoherent quaternion algebra. KW - kernel function. KW - kernel identity. KW - local height. KW - modular curve. KW - modularity. KW - multiplicity function. KW - non-archimedean local field. KW - non-degenerate quadratic space. KW - ordinary component. KW - orthogonal space. KW - projector. KW - pull-back formula. KW - ramified quadratic extension. KW - supersingular component. KW - superspecial component. KW - theta function. KW - theta liftings. KW - theta series. KW - trace identity. KW - un-normalized kernel function. KW - unramified quadratic extension. UR - https://www.unicat.be/uniCat?func=search&query=sysid:77890651 AB - 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. ER -