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On a theta correspondence with respect to a quadratic extension. (English) Zbl 0943.11025

Let \(E\) be a totally real quadratic extension of a totally real number field \(F\), and let \(B_E\) be a quaternion algebra over \(E\) equipped with an \(F\)-linear automorphism \(\tau\) with \(\tau^2 = 1_{B_E}\) and \(\tau |_E \neq 1_E\). Given a suitably defined automorphic form \(h\) with respect to the quaternion algebra \(B_E\), the author constructs a Hilbert modular form \(I(z,h)\) defined with respect to the field \(F\) by using a convolution of \(h\) with a theta function. He then obtains an explicit formula for the Fourier coefficients of \(I(z,h)\).

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F55 Other groups and their modular and automorphic forms (several variables)
11F30 Fourier coefficients of automorphic forms
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References:

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