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 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.

On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2

“…If N > 1, assume that F is an eigenform for the U (p) Hecke operator for the finitely many primes p|N ; we make no assumptions concerning whether F is a Hecke eigenform at primes not dividing the level N . Then by the results of [JLS21], one knows that given ǫ > 0, for all sufficiently large X there are ≥ X 1−ǫ distinct odd squarefree integers n i ∈ [X, 2X] satisfying gcd(n i , N ) = 1 and −n i ≡ 1 mod 4 (in particular −n i is a fundamental discriminant) such that for each n i as above there is a fundamental matrix T i with 4det T i = n i and a(T i ) = 0.…”

“…Further, it is not clear how to make sense of multiplicative properties of Fourier coefficients (which are indexed by matrices). Indeed, Fourier coefficients are closely related to central values of degree 8 L-functions and are more mysterious than the Hecke eigenvalues [DPSS20,JLS21,FM21]. For example, even though an analogue of Ramanujan's conjecture is known for the Hecke eigenvalues corresponding to an eigenform of degree n = 2, the analogous conjecture for Fourier coefficients (which is famously known as Resnikoff-Salda ña conjecture [RSn74]) is not known yet.…”

“…For example, even though an analogue of Ramanujan's conjecture is known for the Hecke eigenvalues corresponding to an eigenform of degree n = 2, the analogous conjecture for Fourier coefficients (which is famously known as Resnikoff-Salda ña conjecture [RSn74]) is not known yet. For recent progress towards the Resnikoff-Salda ña conjecture, see [JLS21]. In this paper we investigate the following questions for n = 2:…”

On the growth of Fourier coefficientsof Siegel cusp forms of degree two

“…One can hope to improve upon the bound (1.2) by using widely believed conjectures on bounds for L-functions. Indeed, assuming the generalized Riemann hypothesis (GRH) and the refined Gan-Gross-Prasad conjectures for Bessel periods, it was shown in [10] that (1.2) is true with the exponent on the right-hand side replaced by k/2 − 1/2 + .…”

On sign changes of primitive Fourier coefficients of Siegel cusp forms

“…In a recent article [2], Asaari, Lester and Saha discuss in detail the sign changes of the Fourier coefficients a F (T ) of Siegel cusp forms of degree 2 and odd, square-free level for T ∈ J 2 such that 4 det T is odd and square-free.…”