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A note on the growth of nearly holomorphic vector-valued Siegel modular forms. (English) Zbl 1425.11095

Bruinier, Jan Hendrik (ed.) et al., L-functions and automorphic forms. LAF, Heidelberg, Germany, February 22–26, 2016. Cham: Springer. Contrib. Math. Comput. Sci. 10, 185-193 (2017).
Summary: Let \(F\) be a nearly holomorphic vector-valued Siegel modular form of weight \(\rho\) 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 \(\lambda_1 \ge \dots \ge \lambda_n\) is the highest weight of \(\rho\) and \(\mu_i(Y)\) are the eigenvalues of the matrix \(Y\).
For the entire collection see [Zbl 1391.11003].

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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