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© 2012

Intersections of Hirzebruch–Zagier Divisors and CM Cycles

Benefits

  • Develops new methods in explicit arithmetic intersection theory

  • Develops new techniques for the study of Shimura varieties and automorphic forms, central objects in modern number theory

  • Proves new cases of conjectures of S. Kudla

Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2041)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Benjamin Howard, Tonghai Yang
    Pages 1-9
  3. Benjamin Howard, Tonghai Yang
    Pages 11-24
  4. Benjamin Howard, Tonghai Yang
    Pages 25-41
  5. Benjamin Howard, Tonghai Yang
    Pages 43-63
  6. Benjamin Howard, Tonghai Yang
    Pages 65-84
  7. Benjamin Howard, Tonghai Yang
    Pages 85-133
  8. Back Matter
    Pages 135-140

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA
  2. 2.Department of MathematicsUniversity of Wisconsin, MadisonMadisonUSA

Bibliographic information

  • Book Title Intersections of Hirzebruch–Zagier Divisors and CM Cycles
  • Authors Benjamin Howard
    Tonghai Yang
  • Series Title Lecture Notes in Mathematics
  • DOI https://doi.org/10.1007/978-3-642-23979-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2012
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-642-23978-6
  • eBook ISBN 978-3-642-23979-3
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages VIII, 140
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
  • Buy this book on publisher's site
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Reviews

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)