Determination of cusp forms on \(\mathrm{GL}(2)\) by coefficients restricted to quadratic subfields (with an appendix by Dipendra Prasad and Dinakar Ramakrishnan)
type: Item
instance of: scholarly article
title: Determination of cusp forms on \(\mathrm{GL}(2)\) by coefficients restricted to quadratic subfields (with an appendix by Dipendra Prasad and Dinakar Ramakrishnan)
zbMATH Open document ID: 1294.11057
published in: Journal of Number Theory
publication date: 2012-05-07T00:00:00Z
reviewed by: Martin Raum
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 6031103
zbMATH Keywords: cusp form
zbMATH Keywords: \(L\)-function
zbMATH Keywords: functoriality
zbMATH Keywords: quadratic field
zbMATH Keywords: Asai transfer
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.09.014
OpenAlex ID: W2091070892
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DOI: 10.1016/J.JNT.2011.09.014
author: M. Krishnamurthy
review text: 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…
description: scientific article; zbMATH DE number 6031103