Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

The dimensions of spaces of holomorphic second-order automorphic forms and their cohomology

Author(s): Nikolaos Diamantis; Cormac O'Sullivan
Journal: Trans. Amer. Math. Soc. 360 (2008), 5629-5666.
MSC (2000): Primary 11F12; Secondary 11F72, 11F75
Posted: June 19, 2008
MathSciNet review: 2425686
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: In this paper we answer a question of Zagier and find the dimensions of spaces of holomorphic second-order forms of even weight. We also establish a cohomological interpretation and prove an Eichler-Shimura-type isomorphism.

References:

[B]
K. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag (1982). MR 672956 (83k:20002)

[CDO]
G. Chinta, N. Diamantis, C. O'Sullivan, Second order modular forms, Acta Arith. 103 (2002), 209-223. MR 1905087 (2003b:11037)

[CO]
G. Chinta, C. O'Sullivan, Poincaré series constructed from period polynomials (to appear).

[DI]
F. Diamond, J. Im, Modular forms and modular curves., In Seminar on Fermat's Last Theorem, Providence, RI, 1995, pp. 39-133.. MR 1357209 (97g:11044)

[DKMO]
N. Diamantis, M. Knopp, G. Mason, C. O'Sullivan, L-functions of second-order cusp forms, Ramanujan J. 12 (2006), 327-347. MR 2293794 (2008c:11073)

[DO]
N. Diamantis, C. O'Sullivan, Hecke theory of series formed with modular symbols and relations among convolution $ L$-functions, Mathematische Annalen 318 (1) (2000), 85-105. MR 1785577 (2002e:11063)

[F]
D. Farmer, Converse theorems and second order modular forms, AMS sectional meeting talk, Salt Lake City, 2002.

[FW]
D. Farmer, K. Wilson, Converse theorems assuming a partial Euler product, arXiv:math. NT/0408221v1 (2004).

[G]
D. Goldfeld, Zeta functions formed with modular symbols, Proc. of the Symposia in Pure Math. 66 (1999), 111-122. MR 1703748 (2000g:11039)

[GO]
D. Goldfeld, C. O'Sullivan, Estimating additive character sums for Fuchsian groups, Ramanujan J. 7 (2003), 241-267. MR 2035805 (2005c:11060)

[Gu]
R. C. Gunning, The Eichler cohomology groups and automorphic forms, Trans. Amer. Math. Soc. 100 (1961), 44-62. MR 0140126 (25:3549)

[I1]
H. Iwaniec, Spectral methods of automorphic forms, 2nd ed., vol. 53, Graduate Studies in Mathematics, Amer. Math. Soc., 2002. MR 1942691 (2003k:11085)

[I2]
H. Iwaniec, Topics in Classical Automorphic Forms, vol. 17, Graduate Studies in Mathematics, Amer. Math. Soc., 1997. MR 1474964 (98e:11051)

[I3]
H. Iwaniec, Fourier coefficients of modular forms and Kloosterman sums, Unpublished lecture notes, Rutgers University (1987).

[IM]
O. Imamoglu, Y. Martin, A converse theorem for second-order modular forms of level $ N$, Acta Arith. 123 4 (2006), 361-376. MR 2262250 (2008a:11061)

[JO]
J. Jorgenson, C. O'Sullivan., Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series, Nagoya Math. J. 179 (2005), 47-102. MR 2164401 (2006k:11080)

[K]
M. Knopp, Some new results on the Eichler cohomology of automorphic forms, Bull. Amer. Math. Soc. 80 (1974), 607-632. MR 0344454 (49:9193)

[KZ]
P. Kleban, D. Zagier, Crossing probabilities and modular forms, J. Stat. Phys. 113 (2003), 431-454. MR 2013692 (2004j:82018)

[O1]
C. O'Sullivan, Properties of Eisenstein series formed with modular symbols, J. Reine Angew. Math. 518 (2000), 163-186. MR 1739405 (2000j:11073)

[O2]
C. O'Sullivan, Identities from the holomorphic projection of modular forms, Number Theory for the Millennium III (2002), 87-106. MR 1956270 (2003m:11064)

[PR]
Y. N. Petridis, M. S. Risager, Modular symbols have a normal distribution, Geom. Funct. Anal. 14 (2004), 1013-1043. MR 2105951 (2005h:11101)

[Ra]
J. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, 1994. MR 1299730 (95j:57011)

[Ru]
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1964. MR 0166310 (29:3587)

[Sa]
P. Sarnak, Some Applications of Modular Forms, Cambridge Tracts in Math. 99, Cambridge Univ. Press, 1990. MR 1102679 (92k:11045)

[Se]
A. Selberg, On Discontinuous Groups in Higher-dimensional Spaces, Tata Institute, Bombay (1960). MR 0130324 (24:A188)

[Sh]
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, Princeton, NJ, 1971. MR 0314766 (47:3318)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11F12, 11F72, 11F75

Retrieve articles in all Journals with MSC (2000): 11F12, 11F72, 11F75

Additional Information:

Nikolaos Diamantis
Affiliation: Department of Mathematics, University of Nottingham, Nottingham, England

Cormac O'Sullivan
Affiliation: Department of Mathematics and Computer Science, Bronx Community College, Bronx, New York 10453

DOI: 10.1090/S0002-9947-08-04755-7
PII: S 0002-9947(08)04755-7
Received by editor(s): February 24, 2005
Posted: June 19, 2008
Additional Notes: The first author was partially supported by EPSRC grant EP/D032350/1
The second author was partially supported by a grant from the City University of New York PSC-CUNY Research Award Program
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia