Ramanujan–Petersson conjecture for Fourier–Jacobi coefficients of Siegel cusp forms

Kumar, Balesh; Paul, Biplab

doi:10.1112/blms.12419

Cited by 4 publications

(1 citation statement)

References 21 publications

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“…Kohnen and Sengupta [27] proved the above conjecture for Hecke eigenform F ∈ S k (Γ 2 ) which is a Saito-Kurokawa lift. Recently, Kumar and Paul [31] proved that the above conjecture is true on average for arbitrary degree. Assuming (2.4), one can clearly see that the Dirichlet series D F 1 ,F 2 (s) is absolutely convergent for Re(s) > k.…”

Section: Preliminariesmentioning

confidence: 91%

An asymptotic expansion for a Lambert series associated with Siegel cusp forms

Babita,

Jha,

Juyal

et al. 2024

Ramanujan J

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In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function |∆(x+iy)| 2 i.e., the Lambert series ∞ n=1 τ (n) 2 e −4πny can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study an asymptotic expansion of a generalized version of the aforementioned Lambert series associated to Siegel cusp forms.

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Section: Preliminariesmentioning

confidence: 91%

An asymptotic expansion for a Lambert series associated with Siegel cusp forms

Babita,

Jha,

Juyal

et al. 2024

Ramanujan J

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Additive twists of L2-norm of Fourier–Jacobi coefficients of Siegel cusp forms

Kumar

Paul

2023

Journal of Number Theory

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On Fourier Coefficients and Hecke Eigenvalues of Siegel Cusp Forms of Degree 2

Paul

Saha

2022

International Mathematics Research Notices

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We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k,$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito–Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negative Fourier coefficients in any “radial” sequence comprising of prime multiples of a fixed fundamental matrix. Finally, we consider the case when $F$ is of Saito–Kurokawa type, and in this case we prove the (essentially sharp) bound $| a(T) | ~\ll _{F, \epsilon }~ \big ( \det T \big )^{\frac {k-1}{2}+\epsilon }$ for the Fourier coefficients of $F$ whenever $\gcd (4 \det (T), N)$ is squarefree, confirming a conjecture made (in the case $N=1$) by Das and Kohnen.

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Ramanujan–Petersson conjecture for Fourier–Jacobi coefficients of Siegel cusp forms

Cited by 4 publications

References 21 publications

An asymptotic expansion for a Lambert series associated with Siegel cusp forms

An asymptotic expansion for a Lambert series associated with Siegel cusp forms

Additive twists of L2-norm of Fourier–Jacobi coefficients of Siegel cusp forms

On Fourier Coefficients and Hecke Eigenvalues of Siegel Cusp Forms of Degree 2

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