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 $\text{GL}(N,{\mathbb{A}}_F)$, where ${\mathbb{A}}_F$ denotes the adèles 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.)

References in corpus (1)

• Local-global compatibility and the action of monodromy on nearby cycles