Period relations for automorphic induction and applications, I

Lin, Jie

Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 95-100.

Soit K un corps quadratique imaginaire. Soit Π (resp. ${\Pi }^{\prime }$) une représentation cuspidale régulière algébrique de ${\mathit{GL}}_n({\mathbb{A}}_K)$ (resp. ${\mathit{GL}}_{n-1}({\mathbb{A}}_K)$), qui est, de plus, cohomologique et auto-duale. Si Π est une induction automorphe cyclique d'un caractère de Hecke χ sur un corps CM, on montre les relations entre les périodes automorphes de Π définies par Harris et celles de χ. Par conséquent, on affine une formule de Grobner et Harris pour les valeurs critiques de $L(s,\Pi \times {\Pi }^{\prime })$, L étant la fonction de Rankin–Selberg. Cela complète la démonstration d'une version automorphe de la conjecture de Deligne dans certains cas.

Let K be a quadratic imaginary field. Let Π (resp. ${\Pi }^{\prime }$) be a regular algebraic cuspidal representation of ${\mathit{GL}}_n({\mathbb{A}}_K)$ (resp. ${\mathit{GL}}_{n-1}({\mathbb{A}}_K)$), which is moreover cohomological and conjugate self-dual. When Π is a cyclic automorphic induction of a Hecke character χ over a CM field, we show relations between automorphic periods of Π defined by Harris and those of χ. Consequently, we refine a formula given by Grobner and Harris for critical values of the Rankin–Selberg L-function $L(s,\Pi \times {\Pi }^{\prime })$. This completes the proof of an automorphic version of Deligne's conjecture in certain cases.

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