Modular Forms and Special Cycles on Shimura Curves
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A01=Michael Rapoport
A01=Stephen S. Kudla
A01=Tonghai Yang
Addition
Adjunction formula
Algebraic number theory
Arakelov theory
Author_Michael Rapoport
Author_Stephen S. Kudla
Author_Tonghai Yang
Automorphism
Bijection
Borel subgroup
Calculation
Category=PBH
Category=PBKF
Coefficient
Cohomology
Combinatorics
Complex multiplication
Cup product
Deformation theory
Dimension
Divisor
Eigenfunction
Eigenvalues and eigenvectors
Eisenstein series
Elliptic curve
Endomorphism
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equation
Explicit formulae (L-function)
Fourier series
Fundamental matrix (linear differential equation)
Galois group
Generating function
Green's function
Induced representation
Intersection (set theory)
Intersection number
Irreducible component
Isomorphism class
L-function
Laurent series
Line bundle
Mathematics
Metaplectic group
Modular curve
Modular form
Modularity (networks)
Moduli space
Multiple integral
Number theory
Numerical integration
P-adic number
Pairing
Prime factor
Prime number
Pullback (category theory)
Pullback (differential geometry)
Quadratic form
Quadratic residue
Quaternion algebra
Riemann zeta function
Shimura variety
Siegel Eisenstein series
Siegel modular form
SL2(R)
Special case
Standard L-function
Summation
Tensor product
Theorem
Topology
Trace (linear algebra)
Triangular matrix
Uniformization
Valuative criterion
Whittaker function
Product details
- ISBN 9780691125510
- Weight: 539g
- Dimensions: 152 x 235mm
- Publication Date: 24 Apr 2006
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soule arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations.
The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.
Stephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universitat, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.
