Intersections of Hirzebruch–Zagier Divisors and CM Cycles

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   • Benjamin Howard, Tonghai Yang

   Pages 1-9

3. Linear Algebra

   • Benjamin Howard, Tonghai Yang

   Pages 11-24

4. Moduli Spaces of Abelian Surfaces

   • Benjamin Howard, Tonghai Yang

   Pages 25-41

5. Eisenstein Series

   • Benjamin Howard, Tonghai Yang

   Pages 43-63

6. The Main Results

   • Benjamin Howard, Tonghai Yang

   Pages 65-84

7. Local Calculations

   • Benjamin Howard, Tonghai Yang

   Pages 85-133

8. Back Matter

   Pages 135-140

   PDF

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.

• 11-XX
• Arakelov geometry
• Hilbert modular surfaces
• arithmetic intersection theory
• automorphic forms

From the reviews:

“The reviewer recommends this beautiful monograph to anyone interested in the circle of conjecture proposed by Kudla et al., particularly from the point of view of arithmetic geometry. The work contains many useful references and intricate proofs that do not appear elsewhere, and is likely to be extremely useful to future progress in the area.” (Jeanine Van Order, Zentralblatt MATH, Vol. 1238, 2012)

• Department of Mathematics, Boston College, Chestnut Hill, USA

  Benjamin Howard

• Department of Mathematics, University of Wisconsin, Madison, Madison, USA

  Tonghai Yang

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