Fourier expansion of Arakawa lifting. I: An explicit formula and examples of non-vanishing lifts. (English) Zbl 1306.11035
Summary: Given an elliptic cusp form \(f\) and an automorphic form \(f'\) on a definite quaternion algebra over \(\mathbb Q\), there is a theta lifting from \((f, f')\) to an automorphic form \(\mathcal L(f, f')\) on the quaternion unitary group \(\mathrm{GSp}(1, 1)\) generating quaternionic discrete series at the Archimedean place. The aim of this paper is to provide an explicit formula for Fourier coefficients of \(\mathcal L(f, f')\) in terms of periods of \(f\) and \(f'\) with respect to a unitary character \(\chi\) of an imaginary quadratic field. As an application, we show the existence of \((f, f')\) with \(\mathcal L(f, f') \not\equiv 0\).
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11F32 | Modular correspondences, etc. |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
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