Abstract
This paper is a continuation of Kim et al. (J Inst Math Jussieu, 2018). We supplement four results on a family of holomorphic Siegel cusp forms for \(GSp_4/\mathbb {Q}\). First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of \(GSp_4\). Second, we prove simultaneous vertical Sato–Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many primes. Third, we compute the n-level density of degree 4 spinor L-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove the conjecture on root numbers. Main tools are the equidistribution theorem in our previous work and Shin–Templier’s (Compos Math 150(12):2003–2053, 2014) work.
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Acknowledgements
We would like to thank K. Morimoto, R. Schmidt and S-W. Shin for helpful discussions. We thank the referee and P. Michel for their help.
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Henry H. Kim is partially supported by NSERC #482564. Satoshi Wakatsuki is partially supported by JSPS Grant-in-Aid for Scientific Research (Nos. 26800006, 25247001, 15K04795). Takuya Yamauchi is partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 15K04787.
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Kim, H.H., Wakatsuki, S. & Yamauchi, T. Equidistribution theorems for holomorphic Siegel modular forms for \(GSp_4\); Hecke fields and n-level density. Math. Z. 295, 917–943 (2020). https://doi.org/10.1007/s00209-019-02378-7
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DOI: https://doi.org/10.1007/s00209-019-02378-7