Modular Forms & Special Cycles on Shimura Curves (Annals of Mathematics Studies, Band 161)

Kudla Stephen, S., Tonghai Yang und Michael Rapoport:

ISBN 10: 0691125503 / ISBN 13: 9780691125503

Published by Princeton University Press, 2006

About this Item

373 Seiten; 9780691125503.3 Gewicht in Gramm: 1.

Seller Inventory # 1065110

Bibliographic Details

• Title: Modular Forms & Special Cycles on Shimura Curves (Annals of Mathematics Studies, Band 161)
• Publisher: Princeton University Press
• Publication Date: 2006
• Language: English
• ISBN 10: 0691125503
• ISBN 13: 9780691125503
• Binding: hardcover
• Condition: Gut
• Weight: 1 gram
• Seller catalogs:
  • Varia

About this title

Synopsis

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-Soulé 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.

About the Author

Stephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universität, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.