ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

Let F be a Siegel cusp form of degree 2, even weight k >= 2, and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients a(F, S) of F at fundamental matrices S (i.e., with -4det(S) equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with det(S) (sic) X, the sequence a(F, S) has at least X(1-)(epsilon )sign changes and takes at least X-1-(epsilon) 'large values'. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan-Gross-Prasad conjecture, we prove the bound vertical bar a(F,S)vertical bar <<(F,epsilon) det(S)(k/2-1/2)/(log vertical bar det(S)vertical bar)(1/8-epsilon) for fundamental matrices S.