Linear relations among Poincaré series (Q1927671)

In this paper the authors study linear relations between Poincaré series of different indices with arbitrary but fixed level and weight. They show that, under some mild assumptions, two Poincaré series having distinct indices are not identically equal unless they are both identically zero. From this result they deduce that, if the space of cusp forms of weight \(k\) and level \(q\) is not spanned by forms with complex multiplication, where the weight is at least 4, then a positive proportion of Poincaré series of weight \(k\) and level \(q\) are pairwise distinct when they are ordered in terms of their indices. The proofs require a nice adaptation of an approach of Rankin together with bounds on Kloosterman sums due to Weil, bounds on Fourier coefficients of cusp forms due to Jenkins and Rouse and nonvanishing results for Fourier coefficients due to Balog and Ono.