Construction of automorphic forms and integrals

Niebur, Douglas

doi:10.1090/S0002-9947-1974-0344196-8

Construction of automorphic forms and integrals
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by Douglas Niebur PDF

Trans. Amer. Math. Soc. 191 (1974), 373-385 Request permission

Abstract:

It is well known that modular forms of positive dimension have Fourier coefficients given by certain infinite series involving Kloostermann sums and the modified Bessel function of the first kind. In this paper a functional equation which characterizes all such Fourier series is found. It is also shown that these Fourier series have a construction similar to that of Poincaré series of negative dimension.

References

Marvin Isadore Knopp, Construction of automorphic forms on $H$-groups and supplementary Fourier series, Trans. Amer. Math. Soc. 103 (1962), 168–188. MR 136735, DOI 10.1090/S0002-9947-1962-0136735-9
Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033, DOI 10.1090/surv/008
Hans Petersson, Zur analytischen Theorie der Grenzkreisgruppen, Math. Ann. 115 (1938), no. 1, 175–204 (German). MR 1513184, DOI 10.1007/BF01448938

On the partition function

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Hans Rademacher, The Fourier Series and the Functional Equation of the Absolute Modular Invariant J($\tau$), Amer. J. Math. 61 (1939), no. 1, 237–248. MR 1507375, DOI 10.2307/2371403
Hans Rademacher and Herbert S. Zuckerman, On the Fourier coefficients of certain modular forms of positive dimension, Ann. of Math. (2) 39 (1938), no. 2, 433–462. MR 1503417, DOI 10.2307/1968796