The dimensions of spaces of holomorphic secondorder automorphic forms and their cohomology
Authors:
Nikolaos Diamantis and Cormac O'Sullivan
Journal:
Trans. Amer. Math. Soc. 360 (2008), 56295666
MSC (2000):
Primary 11F12; Secondary 11F72, 11F75
Published electronically:
June 19, 2008
MathSciNet review:
2425686
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: In this paper we answer a question of Zagier and find the dimensions of spaces of holomorphic secondorder forms of even weight. We also establish a cohomological interpretation and prove an EichlerShimuratype isomorphism.
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 [CDO]
 G. Chinta, N. Diamantis, C. O'Sullivan, Second order modular forms, Acta Arith. 103 (2002), 209223. 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. 39133.. MR 1357209 (97g:11044)
 [DKMO]
 N. Diamantis, M. Knopp, G. Mason, C. O'Sullivan, Lfunctions of secondorder cusp forms, Ramanujan J. 12 (2006), 327347. MR 2293794 (2008c:11073)
 [DO]
 N. Diamantis, C. O'Sullivan, Hecke theory of series formed with modular symbols and relations among convolution functions, Mathematische Annalen 318 (1) (2000), 85105. 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), 111122. MR 1703748 (2000g:11039)
 [GO]
 D. Goldfeld, C. O'Sullivan, Estimating additive character sums for Fuchsian groups, Ramanujan J. 7 (2003), 241267. MR 2035805 (2005c:11060)
 [Gu]
 R. C. Gunning, The Eichler cohomology groups and automorphic forms, Trans. Amer. Math. Soc. 100 (1961), 4462. 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 H. Iwaniec, Fourier coefficients of modular forms and Kloosterman sums, Unpublished lecture notes, Rutgers University (1987).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 secondorder modular forms of level , Acta Arith. 123 4 (2006), 361376. MR 2262250 (2008a:11061)
 [JO]
 J. Jorgenson, C. O'Sullivan., Convolution Dirichlet series and a Kronecker limit formula for secondorder Eisenstein series, Nagoya Math. J. 179 (2005), 47102. MR 2164401 (2006k:11080)
 [K]
 M. Knopp, Some new results on the Eichler cohomology of automorphic forms, Bull. Amer. Math. Soc. 80 (1974), 607632. MR 0344454 (49:9193)
 [KZ]
 P. Kleban, D. Zagier, Crossing probabilities and modular forms, J. Stat. Phys. 113 (2003), 431454. MR 2013692 (2004j:82018)
 [O1]
 C. O'Sullivan, Properties of Eisenstein series formed with modular symbols, J. Reine Angew. Math. 518 (2000), 163186. MR 1739405 (2000j:11073)
 [O2]
 C. O'Sullivan, Identities from the holomorphic projection of modular forms, Number Theory for the Millennium III (2002), 87106. MR 1956270 (2003m:11064)
 [PR]
 Y. N. Petridis, M. S. Risager, Modular symbols have a normal distribution, Geom. Funct. Anal. 14 (2004), 10131043. MR 2105951 (2005h:11101)
 [Ra]
 J. Ratcliffe, Foundations of Hyperbolic Manifolds, SpringerVerlag, New York, 1994. MR 1299730 (95j:57011)
 [Ru]
 W. Rudin, Principles of Mathematical Analysis, McGrawHill, 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)
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 A. Selberg, On Discontinuous Groups in Higherdimensional Spaces, Tata Institute, Bombay (1960). MR 0130324 (24:A188)
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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:
http://dx.doi.org/10.1090/S0002994708047557
PII:
S 00029947(08)047557
Received by editor(s):
February 24, 2005
Published electronically:
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 PSCCUNY Research Award Program
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
