On the positivity of the central critical values of automorphic \(L\)-functions for \(GL(2)\)
Let \(F\) be a number field, \(A\) be the adeles of \(F\), and let \(\pi\) be a cuspidal automorphic representation of \(GL(2,A)\) with trivial central character. Let \(\eta= \prod \eta_v\) be a quadratic idele class character of \(F\) such that \(\eta_v=1\) at all archimedean places \(v\). Then the author shows that the \(L\)-value \(L(1/2,\pi) L(1/2, \pi \otimes \eta)\) is nonnegative. Here \(1/2\) is the center of the critical strip. As the author notes, combining this theorem with results on the positivity of an average of \(L(1/2,\pi \otimes \eta)\) as \(\eta\) varies, which should be obtained by combining standard analytic methods with the results of the reviewer and \textit{J. Hoffstein} [Ann. Math., II. Ser. 142, 385-423 (1995; Zbl 0847.11026)], this theorem implies that the central value \(L(1/2,\pi)\) itself is nonnegative. Nonnegativity is consistent with the Riemann Hypothesis for the \(L\)-function \(L(s,\pi)\) -- indeed, if \(L(1/2, \pi)\) were negative, then there would be a real zero of \(L(s, \pi)\) strictly between \(1/2\) and 1. The main ingredient in the proof of this theorem is an identity which expresses the value \(L(1/2, \pi) L(1/2, \pi \otimes \eta)\), up to a product of local factors at bad places, as the square of the norm of certain period integrals. These integrals are periods, over non-split tori coming from the quadratic extension \(E\) of \(F\) determined by \(\eta\), of the cusp forms in certain Jacquet-Langlands correspondents of \(\pi\). This kind of formula was first obtained by \textit{J.-L. Waldspurger} [Compos. Math. 54, 173-242 (1985; Zbl 0567.10021)], using the theory of the Weil representation. However, the author notes that Waldspurger's result is not sufficient for this application. Instead, the author establishes his result by further developing the relative trace formula introduced by \textit{H. Jacquet} [Ann. Sci. Éc. Norm. Supér, IV. Ser. 19, 185-229 (1986; Zbl 0605.10015)] to study Waldspurger's result. On one side of this formula one has roughly a suitable kernel function in two variables integrated in each variable over the maximal torus of \(G = GL_2\), with one of the integrals including a twist by \(\eta\). This gives the \(L\)-value studied times a product of values of local distributions arising from the integrals at the bad places. On the other side one has a sum of period integrals of matching kernel functions defined on the inner forms of \(G\); each period is taken over the maximal torus corresponding to \(E^*\). In Jacquet [loc. cit.] a fundamental lemma was proved matching these two when the test functions built into the kernels were constructed from the corresponding elements of the local Hecke algebras. In this paper the matching is extended to pairs of smooth functions, and the bad prime local distributions are analyzed. More precisely, given a smooth function of compact support \(f\) on \(G(F_v)\) for some finite place \(v\), the author first shows that there is a set of functions on the inner forms of \(G\) of the form \(f_1* f_1^*\) with matching orbital integrals. The distributions arising on this side of the relative trace formula are of positive type, hence positive on such functions. The author then computes the remaining factors at the bad places on the \(G\)-side of the identity, linking them to \(\varepsilon\)-factors, and shows that their product is positive. His result follows.
positivity of the central critical values
automorphic \(L\)-functions for \(GL(2)\)
scientific article; zbMATH DE number 912124
scientific article