Modular Forms & Special Cycles on Shimura Curves

by Kudla, Stephen S.; Rapoport, Michael; Yang, Tonghai

• ISBN13: 9780691125503
• ISBN10: 0691125503
• Format: Hardcover
• Copyright: 2006-05-01
• Publisher: Princeton Univ Pr

Summary

Modular Forms and Special Cycles on Shimura Curvesis 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-Souleacute; 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.

Table of Contents

Acknowledgments	ix
Introduction	1	(26)
Bibliography	21	(6)
Arithmetic intersection theory on stacks	27	(18)
The one-dimensional case	27	(3)
Pic (M), CH1/Z(M), and CH2/Z(M)	30	(1)
Green functions	31	(3)
Pic (M), CH1/Z(M), and CH2/Z(M)	34	(2)
The pairing CH1 (M) X CH (M) → CH2 (M)	36	(2)
Arakelov heights	38	(1)
The arithmetic adjunction formula	39	(6)
Bibliography	43	(2)
Cycles on Shimura curves	45	(26)
Shimura curves	46	(1)
Uniformization	47	(2)
The Hodge bundle	49	(2)
Special endomorphisms	51	(5)
Green functions	56	(1)
Special 0-cycles	57	(14)
Bibliography	68	(3)
An arithmetic theta function	71	(34)
The structure of arithmetic Chow groups	71	(6)
The arithmetic theta function	77	(2)
The vertical component: definite theta functions	79	(8)
The analytic component: Maass forms	87	(7)
The Mordell-Weil component	94	(2)
Borcherds' generating function	96	(4)
An intertwining property	100	(5)
Bibliography	102	(3)
The central derivative of a genus two Eisenstein series	105	(62)
Genus two Eisenstein series	105	(6)
Nonsingular Fourier coefficients	111	(10)
The Siegel-Weil formula	121	(16)
Singular coefficients	137	(2)
Eisenstein series of genus one	139	(1)
BT	140	(4)
WT	144	(10)
The central derivative---the rank one case	154	(7)
The constant term	161	(6)
Bibliography	165	(2)
The generating function for 0-cycles	167	(14)
The case T > 0 with Diff(T, B) = {p} for p | D(B)	169	(3)
The case T > 0 with Diff(T, B) = {p} for p | D(B)	172	(3)
The case of nonsingular T with sig(T) = (1.1) or (0, 2)	175	(2)
Singular terms, T of rank 1	177	(2)
The constant term, T = 0	179	(2)
Bibliography	180	(1)
Appendix. The case p = 2,p | D(B)	181	(24)
Statement of the result	181	(5)
Review of the special cycles Z(j), for q(j) ε, \ {0}	186	(2)
Configurations	188	(3)
Calculations	191	(10)
The first nondiagonal case	201	(4)
Bibliography	204	(1)
An inner product formula	205	(60)
Statement of the main result	206	(2)
The case t1t2 is not a square	208	(4)
A weakly admissible Green function	212	(9)
A finer decomposition of special cycles	221	(4)
Application of adjunction	225	(6)
Contributions for p | D(B)	231	(7)
Contributions for p | D(B)	238	(7)
Computation of the discriminant terms	245	(7)
Comparison for the case t1, t2 > 0, and t1t2 = m2	252	(7)
The case t1, t2 < 0 with t1t2 = m2	259	(3)
The constant terms	262	(3)
Bibliography	264	(1)
On the doubling integral	265	(86)
The global doubling integral	266	(3)
Review of Waldspurger's theory	269	(10)
An explicit doubling formula	279	(6)
Local doubling integrals	285	(35)
Appendix: Coordinates on metaplectic groups	320	(31)
Bibliography	346	(5)
Central derivatives of L-functions	351	(20)
The arithmetic theta lift	351	(5)
The arithmetic inner product formula	356	(9)
The relation with classical newforms	365	(6)
Bibliography	369	(2)
Index	371