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ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2
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Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic… Click to show full abstract
Let F be a Siegel cusp form of degree
$2$
, even weight
$k \ge 2$
, 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
$-4\det (S)$
equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with
$\det (S) \asymp X$
, the sequence
$a(F,S)$
has at least
$X^{1-\varepsilon }$
sign changes and takes at least
$X^{1-\varepsilon }$
‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound
$\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$
for fundamental matrices S.
