Special Values of L-functions for GL(n) Over a CM Field

Abstract

Abstract We prove a Galois-equivariant algebraicity result for the ratios of successive critical values of $L$-functions for ${\textrm GL}(n)/F,$ where $F$ is a totally imaginary quadratic extension of a totally real number field $F^+$. The proof uses (1) results of Arthur and Clozel on automorphic induction from ${\textrm GL}(n)/F$ to ${\textrm GL}(2n)/F^+$, (2) results of my work with Harder on ratios of critical values for $L$-functions of ${\textrm GL}(2n)/F^+$, and (3) period relations amongst various automorphic and cohomological periods for ${\textrm GL}(2n)/F^+$ using my work with Shahidi. The reciprocity law inherent in the algebraicity result is exactly as predicted by Deligne’s conjecture on the special values of motivic $L$-functions.