Non-vanishing of theta components of Jacobi forms with level and an application

Abstract

We prove that a nonzero Jacobi form of level N (an odd integer) and square-free index m(1)m(2) with m1|N and (N, m(2)) = 1 has a nonzero theta component h mu with either (mu, 2m(1)m(2)) = 1 or (mu, 2m(1)m(2)) f (2)m(2). As an application, we prove that a nonzero Siegel cusp form F of degree 2 and an odd level N in the Atkin-Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of N.