Fundamental Fourier coefficients of Siegel Modular Forms

Understanding the Fourier coefficients of Siegel modular forms which are `fundamental' i.e. indexed by matrices with fundamental discriminant, has many applications. Generalising A. Saha's work on this topic, we show that if F is a non-zero (possibly non-cuspidal) vector-valued level one Siegel modular form of any degree, then it has infinitely many non-zero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. As an application of a variant of our result and utilising the work of A. Pollack, this implies an unconditional proof of the standard analytic properties of the spinor L-function of a holomorphic cuspidal Siegel Hecke eigenform of degree 3 and level one. If time permits, we can also discuss a mod p version of this result.