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Transactions of the American Mathematical Society

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Effective lower bounds for the norm of the Poincaré $ \Theta$-operator


Author: Mark Sheingorn
Journal: Trans. Amer. Math. Soc. 325 (1991), 453-463
MSC: Primary 11F12
DOI: https://doi.org/10.1090/S0002-9947-1991-1041053-9
MathSciNet review: 1041053
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Abstract: Motivated by McMullen's proof of Kra's conjecture that the norm of the Poincaré theta operator $ {\Theta _{q,\Gamma }}$ is less than $ 1$ for every $ q$ and $ \Gamma $ of finite volume, this paper provides explicit lower bounds for this norm. These bounds are sufficient to show that $ \left\Vert {{\Theta _{q,\Gamma }}} \right\Vert \to 1$ for fixed $ \Gamma $ as $ q \to \infty $. Here the difference from $ 1$ is less than $ O(\frac{{{{(2\pi e)}^{q - 2}}}}{{{q^{q - 2}}}})$. For $ \Gamma (N) \subseteq \Gamma \subseteq {\Gamma _0}(N)$, $ \left\Vert {{\Theta _{q,\Gamma }}} \right\Vert \to 1$ for fixed $ q$ as $ N \to \infty $. Here the difference from $ 1$ is $ O({N^{35 - q}})$. We prove these results by estimating $ \frac{{{{\left\Vert {{\Theta _{q,\Gamma }}({f_p})} \right\Vert}_{{A_q}(\Gamma)}}}} {{{{\left\Vert {{f_p}} \right\Vert}_{{A_q}}}}}$ where the $ {f_p}$ are cusp forms of weight $ p \leq q - 2$. (Thus such functions may in general tend to optimize $ {\Theta _{q,\Gamma }}$.) In the case of the congruence subgroups they are taken to be products of $ \Delta $ and Eisenstein series. Effective formulae are presented for all implied constants.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1041053-9
Keywords: Poincaré series, congruence subgroup, Eisenstein series
Article copyright: © Copyright 1991 American Mathematical Society

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