On fundamental Fourier coefficients of Siegel cusp forms of degree 2

Abstract

Let $F$ be a Siegel cusp form of degree 2, even weight $k\ge 2$ and odd squarefree 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 $det(S)\asymp X$, 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 $|a(F,S)|{\ll }_{F,\epsilon }\frac{\det (S{)}^{\frac{k}{2}-\frac{1}{2}}}{(\log |\det (S)|{)}^{\frac{1}{8}-\epsilon }}$ for fundamental matrices $S$.

40 pages; to appear in J. Inst. Math. Jussieu

References in corpus (1)

• Large Fourier coefficients of half-integer weight modular forms

Cited by in corpus (1)

• On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture