Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms

Das, Soumya

doi:10.1007/978-981-15-8719-1_5

Cited by 8 publications

(6 citation statements)

References 19 publications

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“…We note that c( f i ) is closely related to the adjoint L-value L(1, Ad( f i )), and there is the principle: "the prime factors of the denominators of the adjoint L-value give the congruence primes of f i ". This is proved in the seminal papers [11,12,20]. Let us put…”

Section: (Non)-congruences Algebraic Waymentioning

confidence: 86%

On Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ with applications to Siegel modular forms

Böcherer

Das

2021

manuscripta math.

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We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms mod , partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish mod . We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients mod of a Siegel modular form with integral algebraic Fourier coefficients provided is large enough. We also make some efforts to make this "largeness" of effective.

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Section: (Non)-congruences Algebraic Waymentioning

confidence: 86%

On Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ with applications to Siegel modular forms

Böcherer

Das

2021

manuscripta math.

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“…It is also interesting to ask for Omega results in the context of Fourier coefficients; this has recently been addressed in [4]. It is not immediately clear how the results of this article influence those of [4] and vice-versa.…”

Section: Introductionmentioning

confidence: 95%

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

Anamby,

Das,

Pal

2020

Preprint

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We prove a result on the distribution of Hecke eigenvalues, µ F (p r ) (for r = 1, 2 or 3) of a non Saito-Kurokawa lift F of degree 2. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an F, which is the best possible in terms of orders of magnitude. k subspace of S 2 k orthogonal to S 2, * k . Our main result Theorem 1 implies (via Theorem 2) in particular that the Ramanujan-Petersson conjecture for eigenforms in S 2,⊥ k is optimal, in a sense described below.

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“…We emphasize that while these kinds of questions have been previously studied for the full sequence ( ) of Fourier coefficients attached to F (see [12, 22, 24] for results on sign changes and [13] for an -result), there appears to be virtually no previous work in the more subtle setting where one restricts to fundamental Fourier coefficients. There has also been a fair bit of work on sign changes of Hecke eigenvalues of Siegel cusp forms [14, 36, 45, 53], which can be combined with the Hecke relations [1] to deduce sign changes among the with , where d is a fixed fundamental discriminant and m varies.…”

Section: Introductionmentioning

confidence: 99%

On Fundamental Fourier Coefficients of Siegel Cusp Forms of Degree 2

Jääsaari¹,

Lester

Saha³

2021

J. Inst. Math. Jussieu

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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 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.

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Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms

Cited by 8 publications

References 19 publications

On Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ with applications to Siegel modular forms

On Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ with applications to Siegel modular forms

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

On Fundamental Fourier Coefficients of Siegel Cusp Forms of Degree 2

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