Abstract
AbstractLetFbe a Siegel cusp form of degree$2$, even weight$k \ge 2$, and odd square-free levelN. We undertake a detailed study of the analytic properties of Fourier coefficients$a(F,S)$ofFat fundamental matricesS(i.e., with$-4\det (S)$equal to a fundamental discriminant). We prove that asSvaries 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 matricesS.
Highlights
Let Sk(Γ(02)(N )) denote the space of Siegel cusp forms of degree 2 and weight k with respect to the congruence subgroup Γ(02)(N ) ⊆ Sp4(Z) of level N
We prove a lower bound for many fundamental Fourier coefficients with an exponent of the same strength
We show in Proposition 5.9 that a variant of equation (6) where the equality is replaced by an inequality holds in a more general setup
Summary
Let Sk(Γ(02)(N )) denote the space of Siegel cusp forms of degree 2 and weight k with respect to the congruence subgroup Γ(02)(N ) ⊆ Sp4(Z) of level N. For F as before with real Fourier coefficients, one can fix M such that given ε > 0 and sufficiently large X, there are ≥ X1−ε distinct odd square-free integers ni ∈ [X,M X] and associated fundamental matrices Si ∈ Λ2 with |disc(Si)| = ni, such that with the ni ordered in increasing manner, we have a(F,Si)a(F,Si+1) < 0. This improves the exponent of the nonvanishing results of [56, 58] mentioned earlier, where it was proved that there are ε X5/8−ε nonvanishing fundamental Fourier coefficients of discriminant up to X Another question left unanswered in all previous works is that of lower bounds for |a(F,S)| with S fundamental. We note that a bound similar to that obtained in Theorem C has been recently proved in the special case where F is a Yoshida lift by Blomer and Brumley [5, Corollary 4]
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