## Algebra & Number Theory

### Higher-order Maass forms

#### Abstract

The spaces of Maass forms of even weight and of arbitrary order are studied. It is shown that, if we allow exponential growth at the cusps, these spaces are as large as algebraic restrictions allow. These results also apply to higher-order holomorphic forms of even weight.

#### Article information

Source
Algebra Number Theory, Volume 6, Number 7 (2012), 1409-1458.

Dates
Received: 29 April 2011
Revised: 20 September 2011
Accepted: 27 October 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729890

Digital Object Identifier
doi:10.2140/ant.2012.6.1409

Mathematical Reviews number (MathSciNet)
MR3007154

Zentralblatt MATH identifier
1286.11055

#### Citation

Bruggeman, Roelof; Diamantis, Nikolaos. Higher-order Maass forms. Algebra Number Theory 6 (2012), no. 7, 1409--1458. doi:10.2140/ant.2012.6.1409. https://projecteuclid.org/euclid.ant/1513729890

#### References

• R. W. Bruggeman, “Modular forms of varying weight, III”, J. Reine Angew. Math. 371 (1986), 144–190.
• R. W. Bruggeman, Families of automorphic forms, Monographs in Mathematics 88, Birkhäuser, Boston, MA, 1994.
• J. H. Bruinier, K. Ono, and R. C. Rhoades, “Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues”, Math. Ann. 342:3 (2008), 673–693.
• K.-t. Chen, “Algebras of iterated path integrals and fundamental groups”, Trans. Amer. Math. Soc. 156 (1971), 359–379.
• G. Chinta, N. Diamantis, and C. O'Sullivan, “Second order modular forms”, Acta Arith. 103:3 (2002), 209–223.
• A. Deitmar, “Higher order invariants, cohomology, and automorphic forms”, preprint, 2008.
• A. Deitmar, “Higher order group cohomology and the Eichler–Shimura map”, J. Reine Angew. Math. 629 (2009), 221–235.
• N. Diamantis, “Special values of higher derivatives of $L$-functions”, Forum Math. 11:2 (1999), 229–252.
• N. Diamantis and C. O'Sullivan, “The dimensions of spaces of holomorphic second-order automorphic forms and their cohomology”, Trans. Amer. Math. Soc. 360:11 (2008), 5629–5666.
• N. Diamantis and D. Sim, “The classification of higher-order cusp forms”, J. Reine Angew. Math. 622 (2008), 121–153.
• N. Diamantis and R. Sreekantan, “Iterated integrals and higher order automorphic forms”, Comment. Math. Helv. 81:2 (2006), 481–494.
• D. Goldfeld, “Special values of derivatives of $L$-functions”, pp. 159–173 in Number theory (Halifax, 1994), edited by K. Dilcher, CMS Conf. Proc. 15, American Mathematical Society, Providence, RI, 1995.
• H. Iwaniec, Introduction to the spectral theory of automorphic forms, Revista Matemática Iberoamericana, Madrid, 1995.
• M. Koecher and A. Krieg, Elliptische Funktionen und Modulformen, Springer, Berlin, 1998.
• I. Kra, “On cohomology of kleinian groups”, Ann. of Math. $(2)$ 89 (1969), 533–556.
• S. Lang, ${\rm SL}_{2}({\bf R})$, Addison-Wesley, Reading, MA, 1975.
• J. Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys 8, American Mathematical Society, Providence, R.I., 1964.
• H. Maass, Lectures on modular functions of one complex variable, 2nd ed., Tata Inst. Fund. Res. Lectures on Math. and Phys. 29, Tata Institute of Fundamental Research, Bombay, 1983.
• Y. I. Manin, “Parabolic points and zeta functions of modular curves”, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66. In Russian; translated in Math. USSR-Izv. 6 (1972), 19–64.
• H. Petersson, “Über den Bereich absoluter Konvergenz der Poincaréschen Reihen”, Acta Math. 80 (1948), 23–63.
• Y. N. Petridis and M. S. Risager, “Modular symbols have a normal distribution”, Geom. Funct. Anal. 14:5 (2004), 1013–1043.
• W. Roelcke, “Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I”, Math. Ann. 167:4 (1966), 292–337.
• W. Roelcke, “Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, II”, Math. Ann. 168 (1967), 261–324.
• L. J. Slater, Confluent hypergeometric functions, Cambridge University Press, New York, 1960.
• R. Sreekantan, “Higher order modular forms and mixed Hodge theory”, Acta Arith. 139:4 (2009), 321–340.