The Gross-Zagier formula on Shimura curves [[electronic resource] /] / Xinyi Yuan, Shou-wu Zhang, and Wei Zhang

• Record Nr.: UNINA9910817785603321
• Autore: Yuan Xinyi <1981->
• Pubbl/distr/stampa: Princeton, : Princeton University Press, 2012, c2013
• ISBN: 9786613883919; 1-4008-4564-5; 1-283-57146-3
• Edizione: [Course Book]
• Descrizione fisica: 1 online resource (267 p.)
• Collana: Annals of mathematics studies ; ; no. 184
• Altri autori (Persone): ZhangShouwu; ZhangWei <1981->
• Disciplina: 516.3/52
• Soggetti: Shimura varieties; Arithmetical algebraic geometry; Automorphic forms; Quaternions
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Frontmatter -- Contents -- Preface -- Chapter One. Introduction and Statement of Main Results -- Chapter Two. Weil Representation and Waldspurger Formula -- Chapter Three. Mordell-Weil Groups and Generating Series -- Chapter Four. Trace of the Generating Series -- Chapter Five. Assumptions on the Schwartz Function -- Chapter Six. Derivative of the Analytic Kernel -- Chapter Seven. Decomposition of the Geometric Kernel -- Chapter Eight. Local Heights of CM Points -- Bibliography -- Index
• Sommario/riassunto: 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.