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Eichler cohomology theorem for generalized modular forms. (English) Zbl 1232.11064
In the paper under review W. Raji proves an Eichler cohomology theorem (ECT) for weakly parabolic generalized modular forms (GMFs). GMFs extend the notion of a modular form on a subgroup of finite index in $$\text{SL}(2,\mathbb Z)$$ by relaxing the restriction on the associated multiplier system (character). In the extension the character is no longer required to be unitary (i.e., its range is not necessarily contained in the unit circle). This generalization, appearing innocuous at first glance, in fact has led to some interesting developments [see, e.g., H. Petersson, Math. Ann. 115, 175–204 (1938; Zbl 0018.06202, JFM 64.0318.03); M. Knopp and G. Mason, J. Number Theory 99, No. 1, 1–28 (2003; Zbl 1074.11025)].
A proof of the ECT for GMFs has been published before, by the author, the reviewer and J. Lehner [Int. J. Number Theory 5, No. 6, 1049–1059 (2009; Zbl 1236.11052)]. Essential to the earlier proof is Stokes’s theorem and a result on the characters of entire weight 0 GMFs [M. Knopp and G. Mason, Int. J. Number Theory 5, No. 5, 845–857 (2009; Zbl 1236.11054), Theorem 4.2]. By contrast, the proof that Raji gives here depends upon his generalization of a method developed much earlier, which is based upon the venerable circle method (as developed by Rademacher and his school) and, as well, upon a method applied by S. Husseini and the reviewer to prove the standard ECT (i.e., treating automorphic forms with unitary character) [Ill. J. Math. 15, 565–577 (1971; Zbl 0224.10024)]. In turn, the latter article derives its approach to Eichler cohomology from the circle method and a 1939 paper of H. Rademacher [Am. J. Math. 61, 237–248 (1939; Zbl 0020.22003)].
A basic ingredient the author requires here is his extension of the circle method to obtain explicit expressions for the Fourier coefficients of GMFs of negative weight $$-k$$, for $$k$$ sufficiently large [Int. J. Number Theory 5, No. 1, 153–160 (2009; Zbl 1214.11053)]. The portion of the proof based upon the Illinois Journal article (cited above) is to be found in sections 2 and 3 of the present paper.

##### MSC:
 11F75 Cohomology of arithmetic groups 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F11 Holomorphic modular forms of integral weight 11F30 Fourier coefficients of automorphic forms
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##### References:
 [1] Eichler M., Acta Arith. 11 pp 169– [2] Husseini S., Illinois J. Math 15 pp 565– [3] Knopp M., Trans. Amer. Math. Soc. 103 pp 168– [4] Knopp M., Illinois J. Math. 48 pp 1345– [5] DOI: 10.1142/S1793042109002547 · Zbl 1236.11052 [6] DOI: 10.1142/S179304211000340X · Zbl 1201.11054 [7] Lehner J., Michigan Math. J. 4 pp 265– [8] DOI: 10.1007/BF01448938 · Zbl 0018.06202 [9] DOI: 10.1142/S1793042109002006 · Zbl 1214.11053
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