Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2
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| Format: | Preprint |
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2023
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| _version_ | 1866917921035386880 |
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| author | Comtat, Félicien Marzec-Ballesteros, Jolanta Saha, Abhishek |
| author_facet | Comtat, Félicien Marzec-Ballesteros, Jolanta Saha, Abhishek |
| contents | Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound $|a(F,S)| \ll_ε\frac{k^{1/4+ε} (4π)^k}{Γ(k)} c(S)^{-\frac12} \det(S)^{\frac{k-1}2+ε}$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\|(\det Y)^{\frac{k}2}F\|_\infty \ll_εk^{\frac54+ε}.$ The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_07376 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2 Comtat, Félicien Marzec-Ballesteros, Jolanta Saha, Abhishek Number Theory Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound $|a(F,S)| \ll_ε\frac{k^{1/4+ε} (4π)^k}{Γ(k)} c(S)^{-\frac12} \det(S)^{\frac{k-1}2+ε}$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\|(\det Y)^{\frac{k}2}F\|_\infty \ll_εk^{\frac54+ε}.$ The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods. |
| title | Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2 |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.07376 |