Determination of cusp forms on \(\mathrm{GL}(2)\) by coefficients restricted to quadratic subfields (with an appendix by Dipendra Prasad and Dinakar Ramakrishnan)
This paper contains a description of the fibers of the Asai transfer in the dihedral case. The result complements the author's previous findings [Int. Math. Res. Not. 2003, No. 41, 2221--2254 (2003; Zbl 1048.11042)].NEWLINENEWLINEFix a quadratic number field extension \(E\mathop{/}F\) with Galois involution \(\theta\). Given two cuspidal automorphic representations \(\pi\) and \(\pi'\) of \(\mathrm{GL}_2(\mathbb{A}_E)\) with isomorphic Asai transfers, the author proves existence of an idele class character \(\chi\) of \(E\), trivial on the ideles of \(F\), such that \(\pi \otimes \chi \cong \pi'\) or \(\pi^\theta \otimes \chi \cong \pi'\).NEWLINENEWLINEAs main chief application, the author presents a result on Fourier coefficients of newforms over \(E\). The statement of the relevant Theorem 4.0.3 is self contained:NEWLINENEWLINE`` Let \(E/F\) be a quadratic extension of totally real fields. Let \(\chi\) and \(\chi'\) be finite order Hecke characters of \(E\) such taht \(\chi|_{C_F} = \chi'_{C_F}\). Suppose that \(f \in \mathcal{S}_k(\chi)\), \(f' \in \mathcal{S}_k(\chi')\), are normalized newforms, defined with respect to the field~\(E\), whose Fourier coefficients satisfy the relation NEWLINE\[NEWLINE\begin{gathered} c\big( \mathfrak{n} \mathfrak{o}_E,\, f \big) = c\big( \mathfrak{n} \mathfrak{o}_E,\, f' \big) \end{gathered}NEWLINE\]NEWLINE for all non-zero integral ideals \(\mathfrak{n} \subset \mathfrak{o}_F\). Then there is a finite order Hecke character \(\nu\) of \(C_E\) such that \(\nu|_{C_F} = 1\) and NEWLINE\[NEWLINE\begin{gathered} c\big( \gamma(\mathfrak{m}),\, f \big) = \nu(\mathfrak{m})\, c\big( \mathfrak{m},\, f \big) \text{,}\quad \forall \mathfrak{m} \subset \mathfrak{o}_E, \text{ for some \(\gamma \in \mathrm{Gal}(E \mathop{/} F)\).''} \end{gathered}NEWLINE\]NEWLINE The reader should pay attention to some interesting historical remark on page~1375 and the appendix by D.~Prasad and D. Ramakrishnan. The cuspidality criterion for the Asai transfer was established in the appendix to [\textit{H. Jacquet} and \textit{J. A. Shalika}, Am. J. Math. 103, 777--815 (1981; Zbl 0491.10020)]. However, it was incorrect in the dihedral case. The authors of the appendix give the correct one, accompanied by two different proofs. We recite the relevant part of Theorem~B, while the reader is referred to the appendix for details on notation:NEWLINENEWLINE`` Let \(F\) be a number field or a local field, \(K \mathop{/} F\) a quadratic extension with non-trivial automorphism \(\theta\), and \(\pi\) a cuspidal representation in \(\mathcal{A}_2(K)\). Denote by \(\Pi\) the Asai transfer \(\mathrm{As}_{K \mathop{/} F}(\pi)\), which is in \(\mathcal{A}_4(F)\). Then we have the following:NEWLINENEWLINE[...] If \(\pi\) dihedral, then \(\Pi\) is non-cuspidal iff \(\pi\) is induced from a quadratic extension \(M\) of \(K\) which is biquadratic over~\(F\). [...] ''
scientific article; zbMATH DE number 6031103
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