A note on Fourier coefficients of Poincaré series. (English) Zbl 1220.11063
The authors give a direct and more transparent proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series. As a consequence, they deduce a strong approximation property for \(\text{GL}(2)\)-cusp forms which was obtained before by Serre in the case of holomorphic forms and by Sarnak in the case of Maass forms. Similar results are derived as well for the space of Siegel modular forms.
Reviewer: Emre Alkan (Istanbul)
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
References:
| [1] | Iwaniec, Analytic Number Theory (2004) · doi:10.1090/coll/053 |
| [2] | DOI: 10.1007/BF01342938 · Zbl 0088.28903 · doi:10.1007/BF01342938 |
| [3] | DOI: 10.2307/2371774 · doi:10.2307/2371774 |
| [4] | DOI: 10.1017/CBO9780511619878 · doi:10.1017/CBO9780511619878 |
| [5] | Sarnak, Analytic Number Theory and Diophantine Problems pp 75– (1987) |
| [6] | DOI: 10.1007/BF02054945 · Zbl 0042.32001 · doi:10.1007/BF02054945 |
| [7] | DOI: 10.1090/S0894-0347-97-00220-8 · Zbl 0871.11032 · doi:10.1090/S0894-0347-97-00220-8 |
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