On Deligne's Conjecture for Certain Automorphic $L$-Functions for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ and $\mathrm{GL}(4)$

Shih-Yu Chen; Yao Cheng

doi:10.4171/dm/725

On Deligne's Conjecture for Certain Automorphic $L$ -Functions for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ and $\mathrm{GL}(4)$

Shih-Yu Chen
Institute of Mathematics, Academia Sinica, No. 1, Sec. 4, Roosevelt R., Taipei 10617, Taiwan, ROC
Yao Cheng
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge R., Singapore 119076

$On Deligne's Conjecture for Certain Automorphic $L$-Functions for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ and $\mathrm{GL}(4)$ cover$

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Abstract

We prove Deligne's conjecture for certain automorphic $L$ -functions for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ and $\mathrm{GL}(4)$ . The proof is based on rationality results for central critical values of triple product $L$ -functions, which follow from establishing explicit Ichino's formulae for trilinear period integrals for Hilbert cusp forms on totally real étale cubic algebras over $\mathbb{Q}$ .

Cite this article

Shih-Yu Chen, Yao Cheng, On Deligne's Conjecture for Certain Automorphic $L$ -Functions for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ and $\mathrm{GL}(4)$ . Doc. Math. 24 (2019), pp. 2241–2297

DOI 10.4171/DM/725