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Mathematics - Number Theory and Discrete Mathematics | Intersections of Hirzebruch–Zagier Divisors and CM Cycles

Intersections of Hirzebruch–Zagier Divisors and CM Cycles

Series: Lecture Notes in Mathematics, Vol. 2041

Howard, Benjamin, Yang, Tonghai

2012, VIII, 140p.

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  • 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
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.

Content Level » Research

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

Related subjects » Number Theory and Discrete Mathematics

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