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This monograph treats one case of a series of conjectures by S. Kudla, whose goal is to show that Fourier of Eisenstein series encode information about the Arakelov intersection theory of special cycles on Shimura varieties of orthogonal and unitary type. Here, the Eisenstein series is a Hilbert modular form of weight one over a real quadratic field, the Shimura variety is a classical Hilbert modular surface, and the special cycles are complex multiplication points and the Hirzebruch–Zagier divisors. By developing new techniques in deformation theory, the authors successfully compute the Arakelov intersection multiplicities of these divisors, and show that they agree with the Fourier coefficients of derivatives of Eisenstein series.

Mathematics --- Physical Sciences & Mathematics --- Algebra --- Shimura varieties. --- Eisenstein series. --- Algebraic cycles. --- Arakelov theory. --- Arakelov geometry --- Cycles, Algebraic --- Series, Eisenstein --- Varieties, Shimura --- Mathematics. --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Math --- Science --- Automorphic functions --- Arithmetical algebraic geometry --- Geometry, Algebraic

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Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. Bringing together contributions from areas that are not usually interacting with each other, this volume introduces diverse users of Eisenstein series to a variety of important applications. With this juxtaposition of perspectives, the reader obtains deeper insights into the arithmetic of Eisenstein series. The exposition focuses on the common structural properties of Eisenstein series occurring in many related applications that have arisen in several recent developments in arithmetic: Arakelov intersection theory on Shimura varieties, special values of L-functions and Iwasawa theory, and equidistribution of rational/integer points on homogeneous varieties. Key questions that are considered include: Is it possible to identify a class of Eisenstein series whose Fourier coefficients (resp. special values) encode significant arithmetic information? Do such series fit into p-adic families? Are the Eisenstein series that arise in counting problems of this type? Contributors include: B. Brubaker, D. Bump, J. Franke, S. Friedberg, W.T. Gan, P. Garrett, M. Harris, D. Jiang, S.S. Kudla, E. Lapid, K. Prasanna, A. Raghuram, F. Shahidi, R. Takloo-Bighash.

Eisenstein series. --- Mathematics. --- Math --- Science --- Series, Eisenstein --- Automorphic functions --- Number theory. --- Geometry. --- Geometry, algebraic. --- Topological Groups. --- Number Theory. --- Applications of Mathematics. --- Algebraic Geometry. --- Topological Groups, Lie Groups. --- Groups, Topological --- Continuous groups --- Algebraic geometry --- Geometry --- Mathematics --- Euclid's Elements --- Number study --- Numbers, Theory of --- Algebra --- Applied mathematics. --- Engineering mathematics. --- Algebraic geometry. --- Topological groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Engineering --- Engineering analysis --- Mathematical analysis