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Gross-Zagier Formula on Shimura Curves

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English

By (author): Shou-wu Zhang Wei Zhang Xinyi Yuan

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. See more
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A01=Shou-wu ZhangA01=Wei ZhangA01=Xinyi YuanAbelian varietyAdele ringAge Group_UncategorizedAnalytic continuationAuthor_Shou-wu ZhangAuthor_Wei ZhangAuthor_Xinyi Yuanautomatic-updateAutomorphic formAutomorphismBase changeBig O notationBijectionCanonical bundleCanonical mapCardinalityCategory1=Non-FictionCategory=PBMWChange of variablesCharacteristic function (probability theory)CoefficientComputationConnected component (graph theory)Connected spaceConstant termCOP=United StatesCosetCusp formDegeneracy (mathematics)Delivery_Delivery within 10-20 working daysDeterminantDimension (vector space)DivisorDouble cosetEigenvalues and eigenvectorsEisenstein seriesEmbeddingEndomorphismeq_isMigrated=2Equivalence classExistential quantificationFourier seriesFourier transformFunction spaceFunctional equationHaar measureHomomorphismIntersection theoryIrreducible componentIrreducible representationIsomorphism classIwasawa decompositionL-functionLanguage_EnglishLinear combinationLinearityLocal fieldMaximal compact subgroupModularity (networks)MorphismOne-dimensional spacePA=AvailablePairingPoisson summation formulaPolynomialPrice_€100 and abovePS=ActiveQuadratic formQuaternion algebraReciprocity lawScientific notationShimura varietyShou-Wu ZhangSmoothnesssoftlaunchSubgroupSubsetSummationSupport (mathematics)Surjective functionTensor productTheoremTheta functionUniformizationWhittaker function
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Product Details
  • Weight: 397g
  • Dimensions: 152 x 235mm
  • Publication Date: 02 Dec 2012
  • Publisher: Princeton University Press
  • Publication City/Country: US
  • Language: English
  • ISBN13: 9780691155920

About Shou-wu ZhangWei ZhangXinyi Yuan

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.

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