On signs of Fourier coefficients of Hecke-Maass cusp forms on 𝐺𝐿₃

Jääsaari, Jesse

doi:10.1090/tran/9012

On signs of Fourier coefficients of Hecke-Maass cusp forms on $\mathrm {GL}_3$
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by Jesse Jääsaari;

Trans. Amer. Math. Soc. 376 (2023), 8193-8223

DOI: https://doi.org/10.1090/tran/9012

Published electronically: August 14, 2023

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Abstract:

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\mathrm {SL}_3(\mathbb Z)$. When the underlying form is self-dual, we show that there are $\gg _\varepsilon X^{5/6-\varepsilon }$ sign changes among the coefficients $\{A(m,1)\}_{m\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\mathrm {GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\mathrm {GL}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.

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