On the Voronoĭ formula for GL(n)

We prove a general Vorono\u{\i} formula for cuspidal automorphic representations of ${\rm GL}(n)$ over number fields. This generalizes recent work by Miller-Schmid and Goldfeld-Li on Maass forms. Our method follows closely the adelic framework of integral representations of $L$-functions. The proof is flexible enough to allow ramification and we propose possible variants. For example the assumption that the additive twist is trivial at places where the representation is ramified is sufficient to obtain an explicit final statement with hyper-Kloosterman sums.