Critical values of l functions for gl3 215 gl1 over a number field

Abstract

We prove an algebraicity result for all the critical values of L functions forGL3 215 GL1 over a totally real field and a CM field separately These L functions are attached to a cohomological cuspidal automorphic representationof GL3 having cohomology with respect to a general coefficient systemand an algebraic Hecke character of GL1 This is derived from the theory ofRankin Selberg L functions attached to pairs of automorphic representationson GL3 215 GL2 Our results are a generalization and refinement of the resultsof Mahnkopf 26 and Geroldinger 14 The resulting expressions for criticalvalues of the Rankin Selberg L functions are compatible with Deligne sconjecture As an application we obtain algebraicity results for symmetricsquare L functions newline newline