On the average growth of fourier coefficients of siegel cusp forms of genus 2

Abstract

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp2(ℤ) of genus 2 and denote by a(T) (T ∈ Q(2,2), T > 0 half-integral) its Fourier coefficients. It is known (see Böcherer & Raghavan, 1988 and Fomenko, 1987) that (1) Σ {T>0, det(T)=N}/GL2(ℤ) |a(T)|2 ≪ε,F Nk-3/32+ε (ε > 0) where the sum is over GL2(ℤ)-classes of T > 0 with det(T) = N. In the present note we shall give a result on the average growth of |a(T)|2, where the average is taken w.r.t. the trace.