On sup-norms of cusp forms of powerful level

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue \lambda. It is shown that |f|_\infty <<_{\lambda, \epsilon} N^{-1/12 + \epsilon} for any \epsilon>0. The exponent is further improved in the case when N is not divisible by "small squares". Our work extends and generalizes previously known results in the special case of N squarefree.Comment: Final version, to appear in JEMS. Please also note that the results of this paper have been significantly improved in my recent paper arXiv:1509.07489 which uses a fairly different methodolog

Absolute convergence of the twisted Arthur-Selberg trace formula

We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven when the group is split. The result extends the work of Finis-Lapid (and M\"uller, spectral side) to the twisted setting. We use the absolute convergence to give a geometric interpretation of sums of residues of certain Rankin-Selberg L-functions.Comment: Accepted to be published in Mathematische Zeitschrift. Removed proof of RCL for base change; Section 8 now requires Assumption 8.1. Also, minor correction