Positivity of quadratic base change \(L\)-functions. (Q2773539)

In the paper under review, the authors prove that the value of certain quadratic base change \(L\)-functions at the center of symmetry is nonnegative. One case of their result is the following. Let \(F\) be a number field and let \(E\) be a quadratic extension of \(F\). Let \(\pi\) be an automorphic cuspidal representation of \(GL_2(\mathbb{A}_F)\) where \(\mathbb{A}_F\) is the adele ring of \(F\). Assume that the central character of \(\pi\) is trivial. Assume also that the base change \(\Pi\) of \(\pi\) from \(F\) to \(E\) is cuspidal. (\(\Pi\) is an automorphic cuspidal representation of \(GL_2 (\mathbb{A}_E)\).) Then NEWLINE\[NEWLINEL(1/2,\Pi)\geq 0.NEWLINE\]NEWLINE This result was first proved by \textit{J. Guo} [Duke Math J. 83, 157--189 (1996; Zbl 0861.11032)]. The authors extend Guo's result by considering automorphic representations \(\pi\) with nontrivial central character and by twisting the representation \(\Pi\) with a character. There are some natural restrictions on the central character and the twisted character.NEWLINENEWLINE Such positivity results have been proved by many authors in many important cases. Many of them are mentioned in the bibliography of the paper under review. Recently, \textit{E. Lapid} and \textit{S. Rallis} [Ann. Math. (2) 157, No. 3, 891--917 (2003; Zbl 1067.11026)] proved a rather general result of this type which covers most or all of the known examples.NEWLINENEWLINE In the paper under review, the idea of the proof is to factor a global distribution of positive type into a finite product of local distributions of positive type times the above mentioned \(L\)-value. (Some other explicit positive constants involving epsilon factors and other \(L\)-values also appear). Now the positivity of the distributions will imply the positivity of the \(L\)-value. The factorization is proved by the use of a relative trace formulae which was developed by Jacquet for this base change.NEWLINENEWLINE It is important to note that the authors do much more than ``use'' Jacquet's trace formula. They make a local comparison of the distributions that appear in the trace formula. This comparison allows the global factorization mentioned above. The factorization expresses in a precise way the heuristic statement that the value of the \(L\)-function is the square of a period integral. This type of factorization is likely to be quite general.NEWLINENEWLINE The authors also revise the analysis of the continuous spectrum which in their words was ``somewhat deficient'' in previous papers. This involves a new form of truncation. Quoting from the introduction: `Thus we take care at once of a gap in several papers''.