CRITICAL VALUES OF RANKIN–SELBERG L-FUNCTIONS FOR GLn ×GLn−1 AND THE SYMMETRIC CUBE L-FUNCTIONS FOR GL2

In a previous article [35] an algebraicity result for the central critical value for L-functions for GLn × GLn−1 over Q was proved assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize [35, Thm. 1.1] for all critical values for L-functions for GLn×GLn−1 over any number field F while using the period relations of [37] and some additional inputs as will be explained below. Thanks to a recent preprint of Binyong Sun [44], the nonvanishing hypothesis has now been proved, and so one may claim that the results of this article are unconditional. Using such results for GL3 ×GL2, new unconditional algebraicity result 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: see Garrett–Harris [12], Kim–Shahidi [25], Grobner–Raghuram [17], and Januszewski [23].