A converse theorem for epsilon factors
The authors' goal is to find a certain inverse theorem to results of \textit{H. Saito} [Compos. Math. 85, 99-108 (1993; Zbl 0795.22009)]. Saito gives a property of epsilon factors of base change lifts of irreducible representations of \(Gl_2(F)\), where the lift is to \(Gl_2(K)\) and \(K/F\) is a quadratic extension. The authors prove the following result (and we cite from the abstract):NEWLINENEWLINENEWLINE``Let \(F\) be a non-Archimedean local field of characteristic zero and \(K\) a quadratic extension of \(F\). Let \(S\) be the set of characters of \(K^\times\) trivial on \(F^\times\). Let \(\chi_1\) and \(\chi_2\) be two characters of \(K^\times\) trivial on \(F^\times\). Let \(\chi_1\) and \(\chi_2\) be two characters of \(K^\times\) such that \(\chi_{1|F^\times}= \chi_{2|F^\times}\neq 1\). Let \(\psi\) be a nontrivial additive character of \(F\) and \(\psi_K= \psi\circ \text{tr}_{K/F}\). If \(\varepsilon(\chi_1\lambda, \psi_K)= \varepsilon(\chi_2\lambda, \psi_K)\) for all \(\lambda\in S\) then \(\chi_1\) and \(\chi_2\) agree on all units in the ring of integers in \(K\) and on all elements of trace zero. If, in addition, the conductor of \(\chi_{1|F^\times}\) is not zero then \(\chi_1= \chi_2\).''NEWLINENEWLINENEWLINEThis result is close to a modification of a conjecture by D. Prasad. This conjecture would enable the authors to apply a characterisation of distinguished representations by \textit{J. Hakim} [Duke Math. J. 62, 1-22 (1991; Zbl 0724.22016)] and would imply that a principal series representation \(\Pi= \Pi(\chi, \chi^{-1})\) with \(\omega_{K/F}(-1)=1\) and \(\chi_{|{\mathcal U}_F}\neq 1\) is a base change lift if \(\varepsilon(\chi\lambda, \psi_K)= \varepsilon(\chi^\sigma\lambda, \psi_K)\) for all \(\lambda\in S\), where \({\mathcal U}_F\) is the group of units of the ring of integers of \(F\), \(\sigma\) is the nontrivial element of the Galois group of \(K/F\) and \(\omega_{K/F}\) is the character of \(F^\times\) corresponding to the extension \(K/F\) by class field theory.
scientific article; zbMATH DE number 1641595
scientific article