Critical values of Rankin–Selberg L-functions for GL_n × GL_n-1 and the symmetric cube L-functions for GL₂

Abstract

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL_n × GL_n-1 over ℚ assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for L-functions for GL_n × GL_n-1 over any number field F while using certain period relations proved by Freydoon Shahidi and the author, and some additional inputs as will be explained below. Thanks to some recent work of Binyong Sun, the nonvanishing hypothesis has now been proved. The results of this article are unconditional. Applying this to GL₃ × GL₂, new unconditional algebraicity results for the special values of symmetric cube L-functions for GL₂ over F have been proved. Previously, algebraicity results for the critical values of symmetric cube L-functions for GL₂ have been known only in special cases by the works of Garrett–Harris, Kim–Shahidi, Grobner–Raghuram, and Januszewski.