Voronoï summation for GLn: collusion between level and modulus

Abstract:We investigate the Vorono\"{\i} summation problem for ${\rm GL}_n$ in the level aspect for $n\geq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified, a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give classical refinements of our formula and study the $p$-adic generalisations of the Bessel transform.