On the central value of Rankin $L$-functions for self-dual algebraic representations of linear groups over totally real fields
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| Format: | Preprint |
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2023
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| author | Clozel, Laurent Kret, Arno |
| author_facet | Clozel, Laurent Kret, Arno |
| contents | 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 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.) |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_05049 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the central value of Rankin $L$-functions for self-dual algebraic representations of linear groups over totally real fields Clozel, Laurent Kret, Arno Number Theory 11F67 (Primary), 11F75 (Secondary) 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 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.) |
| title | On the central value of Rankin $L$-functions for self-dual algebraic representations of linear groups over totally real fields |
| topic | Number Theory 11F67 (Primary), 11F75 (Secondary) |
| url | https://arxiv.org/abs/2306.05049 |