Miller, Stephen D.Schmid, Wilfried2023-02-132012Miller, Stephen D., and Wilfried Schmid. 2012. On the Rapid Decay of Cuspidal Automorphic Forms. Advances in Mathematics 231, no. 2: 940–964.1088-41650001-8708https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37374346Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets. We also state and prove the decay for smooth but not necessarily K-finite cusp forms. We shall use this to establish new results about the analytic continuation of L-functions. Our decay statements also close a gap in the literature on L-functions.en-USautomorphic formsrapid decaycusp formsL-functionsRankin-Selbergintegral representationsuniformly moderate growthJournal Article2023-02-1310.1016/j.aim.2012.06.001