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The Gross-Zagier Formula on Shimura Curves
Xinyi Yuan, Shou-wu Zhang, Wei Zhang, et al.
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Local field, Homomorphism, Coefficient, Divisor, Uniformization, Connected component (graph theory), Fourier series, Scientific notation, Big O notation, Base change, Shou-Wu Zhang, Functional equation, Cusp form, Pairing, Quaternion algebra, Automorphism, Shimura variety, Canonical map, Tensor product, Theta function, Bijection, Characteristic function (probability theory), Surjective function, Irreducible representation, Degeneracy (mathematics), Double coset, Theorem, Dimension (vector space), Connected space, Endomorphism, Change of variables, Summation, Constant term, Automorphic form, L-function, Linearity, Determinant, Whittaker function, Modularity (networks), Reciprocity law, Maximal compact subgroup, Quadratic form, Cardinality, Existential quantification, Eigenvalues and eigenvectors, Canonical bundle, Embedding, Linear combination, Smoothness, Haar measure, Function space, Irreducible component, Intersection theory, Isomorphism class, Polynomial, Fourier transform, Morphism, Abelian variety, Coset, Subset, Support (mathematics), Iwasawa decomposition, One-dimensional space, Poisson summation formula, Subgroup, Computation, Analytic continuation, Adele ring, Equivalence class, Eisenstein series