Linear independence of Poincaré series of exponential type via non-analytic methods

Abstract

Given a finite set { T } \{T\} of symmetric, positive definite, half-integral n n by n n matrices over Z \mathbf {Z} which are inequivalent under the action of G L ( n , Z ) \mathrm {GL}(n, \mathbf {Z}) , we show that the corresponding set of Poincaré series { P k n ( T ) } \{ P_k^n(T) \} attached to them are linearly independent for weights k k in infinitely many arithmetic progressions. We also give a quite explicit description of those arithmetic progressions for all even degrees, when the matrices T T have no improper automorphisms and their level is an odd prime. Our main tools are theta series with simple harmonic polynomials as coefficients and techniques familiar from the theory of modular forms mod p p .