Intersections of Hirzebruch–Zagier Divisors and CM Cycles (Lecture Notes in Mathematics Book 2041)

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. Annotation 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 Front Matter....Pages i-viii Introduction....Pages 1-9 Linear Algebra....Pages 11-24 Moduli Spaces of Abelian Surfaces....Pages 25-41 Eisenstein Series....Pages 43-63 The Main Results....Pages 65-84 Local Calculations....Pages 85-133 Back Matter....Pages 135-140