On the central value of Rankin $L$-functions for self-dual algebraic representations of linear groups over totally real fields

Abstract

Deligne has formulated extremely influential conjectures about certain special values of the $L$-functions of (Grothendieck) motives over a number field $F$. Given the conjectural dictionary between motives and 'algebraic' automorphic representations of $\textrm{GL}(N, {\mathbb A}_F)$, where ${\mathbb A}_F$ denotes the ad\`eles of $F$, they translate into conjectures concerning the $L$-functions of these automorphic representations. These complex representations, when they are 'regular', can be conjugated by the automorphisms of the complex field ${\mathbb C}$. It then follows, as a weak consequence of Deligne's conjectures, that the vanishing at critical points (integers of half-integers) of the automorphic $L$-functions should be invariant by automorphisms of ${\mathbb C}$. If $F$ is totally imaginary, this has been proven by Moeglin, for standard or Rankin $L$-functions. Here we extend the result to Rankin $L$-fuctions for totally real fields $F$, under a parity and a regularity assumption. The proof relies on Eisenstein cohomology and the Zucker conjecture (a theorem of Looijenga and Saper-Stern.)