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Under endoscopic assumptions about $L$-packets of unitary groups, we provethe local Gan-Gross-Prasad conjecture for tempered representations of unitarygroups over $p$-adic fields. Roughly, this conjecture says that branching lawsfor $U(n-1)\subset U(n)$ can be computed using epsilon factors.

Endoscopie et conjecture raffin\'ee de Gan-Gross-Prasad pour les groupes unitaires