UID:
almafu_9958091962802883
Format:
1 online resource (384 p.)
Edition:
Course Book
ISBN:
1-299-40102-3
,
1-4008-3716-2
Series Statement:
Annals of mathematics studies ; no. 161
Content:
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.
Note:
Description based upon print version of record.
,
Frontmatter --
,
Contents --
,
Acknowledgments --
,
Chapter 1. Introduction --
,
Chapter 2. Arithmetic intersection theory on stacks --
,
Chapter 3. Cycles on Shimura curves --
,
Chapter 4. An arithmetic theta function --
,
Chapter 5. The central derivative of a genus two Eisenstein series --
,
Chapter 6. The generating function for 0-cycles --
,
Chapter 6 Appendix --
,
Chapter 7. An inner product formula --
,
Chapter 8. On the doubling integral --
,
Chapter 9. Central derivatives of L-functions --
,
Index
,
Issued also in print.
,
English
Additional Edition:
ISBN 0-691-12550-3
Additional Edition:
ISBN 0-691-12551-1
Language:
English
Subjects:
Mathematics
DOI:
10.1515/9781400837168
Bookmarklink