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

Authors:
Nikolaos Diamantis and Cormac O'Sullivan

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5629-5666

MSC (2000):
Primary 11F12; Secondary 11F72, 11F75

Published electronically:
June 19, 2008

MathSciNet review:
2425686

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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.

**[B]**Kenneth S. Brown,*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR**672956****[CDO]**G. Chinta, N. Diamantis, and C. O’Sullivan,*Second order modular forms*, Acta Arith.**103**(2002), no. 3, 209–223. MR**1905087**, 10.4064/aa103-3-2**[CO]**G. Chinta, C. O'Sullivan,*Poincaré series constructed from period polynomials*(to appear).**[DI]**Fred Diamond and John Im,*Modular forms and modular curves*, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133. MR**1357209****[DKMO]**N. Diamantis, M. Knopp, G. Mason, and C. O’Sullivan,*𝐿-functions of second-order cusp forms*, Ramanujan J.**12**(2006), no. 3, 327–347. MR**2293794**, 10.1007/s11139-006-0147-2**[DO]**Nikolaos Diamantis and Cormac O’Sullivan,*Hecke theory of series formed with modular symbols and relations among convolution 𝐿-functions*, Math. Ann.**318**(2000), no. 1, 85–105. MR**1785577**, 10.1007/s002080000112**[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]**Dorian Goldfeld,*Zeta functions formed with modular symbols*, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 111–121. MR**1703748****[GO]**Dorian Goldfeld and Cormac O’Sullivan,*Estimating additive character sums for Fuchsian groups*, Ramanujan J.**7**(2003), no. 1-3, 241–267. Rankin memorial issues. MR**2035805**, 10.1023/A:1026255414488**[Gu]**R. C. Gunning,*The Eichler cohomology groups and automorphic forms*, Trans. Amer. Math. Soc.**100**(1961), 44–62. MR**0140126**, 10.1090/S0002-9947-1961-0140126-3**[I1]**Henryk Iwaniec,*Spectral methods of automorphic forms*, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR**1942691****[I2]**Henryk Iwaniec,*Topics in classical automorphic forms*, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR**1474964****[I3]**H. Iwaniec,*Fourier coefficients of modular forms and Kloosterman sums*, Unpublished lecture notes, Rutgers University (1987).**[IM]**Özlem Imamoḡlu and Yves Martin,*A converse theorem for second-order modular forms of level 𝑁*, Acta Arith.**123**(2006), no. 4, 361–376. MR**2262250**, 10.4064/aa123-4-5**[JO]**Jay Jorgenson and Cormac O’Sullivan,*Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series*, Nagoya Math. J.**179**(2005), 47–102. MR**2164401****[K]**Marvin I. Knopp,*Some new results on the Eichler cohomology of automorphic forms*, Bull. Amer. Math. Soc.**80**(1974), 607–632. MR**0344454**, 10.1090/S0002-9904-1974-13520-2**[KZ]**Peter Kleban and Don Zagier,*Crossing probabilities and modular forms*, J. Statist. Phys.**113**(2003), no. 3-4, 431–454. MR**2013692**, 10.1023/A:1026012600583**[O1]**Cormac O’Sullivan,*Properties of Eisenstein series formed with modular symbols*, J. Reine Angew. Math.**518**(2000), 163–186. MR**1739405**, 10.1515/crll.2000.003**[O2]**Cormac O’Sullivan,*Identities from the holomorphic projection of modular forms*, Number theory for the millennium, III (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 87–106. MR**1956270****[PR]**Y. N. Petridis and M. S. Risager,*Modular symbols have a normal distribution*, Geom. Funct. Anal.**14**(2004), no. 5, 1013–1043. MR**2105951**, 10.1007/s00039-004-0481-8**[Ra]**John G. Ratcliffe,*Foundations of hyperbolic manifolds*, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR**1299730****[Ru]**Walter Rudin,*Principles of mathematical analysis*, Second edition, McGraw-Hill Book Co., New York, 1964. MR**0166310****[Sa]**Peter Sarnak,*Some applications of modular forms*, Cambridge Tracts in Mathematics, vol. 99, Cambridge University Press, Cambridge, 1990. MR**1102679****[Se]**Atle Selberg,*On discontinuous groups in higher-dimensional symmetric spaces*, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR**0130324****[Sh]**Goro Shimura,*Introduction to the arithmetic theory of automorphic functions*, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR**0314766**

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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/S0002-9947-08-04755-7

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 PSC-CUNY Research Award Program

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.