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 -4 (S) equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with (S) X , the sequence $a(F,S)$ has at least X1-ε sign changes and takes at least X1-ε ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound a(F,S) F, ε (S)^ k2 - 1/2 (log (S) )18 - ε for fundamental matrices S .