Abstract
Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ with respect to some congruence subgroup of \(\mathrm {Sp}_{2n}({{\mathbb Q}})\). In this note, we prove that the function on \(\mathrm {Sp}_{2n}({\mathbb R})\) obtained by lifting F has the moderate growth (or “slowly increasing”) property. This is a consequence of the following bound that we prove: \(\|\rho (Y^{1/2})F(Z) \| \ll \prod _{i=1}^n (\mu _i(Y)^{\lambda _1/2} + \mu _i(Y)^{-\lambda _1/2})\) where λ 1 ≥… ≥ λ n is the highest weight of ρ and μ i (Y ) are the eigenvalues of the matrix Y .
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Pitale, A., Saha, A., Schmidt, R. (2017). A Note on the Growth of Nearly Holomorphic Vector-Valued Siegel Modular Forms. In: Bruinier, J., Kohnen, W. (eds) L-Functions and Automorphic Forms. Contributions in Mathematical and Computational Sciences, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-69712-3_11
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DOI: https://doi.org/10.1007/978-3-319-69712-3_11
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