Effective lower bounds for the norm of the Poincaré $\Theta$-operator
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- by Mark Sheingorn PDF
- Trans. Amer. Math. Soc. 325 (1991), 453-463 Request permission
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 $\left \| {{\Theta _{q,\Gamma }}} \right \| \to 1$ 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 $\Gamma (N) \subseteq \Gamma \subseteq {\Gamma _0}(N)$, $\left \| {{\Theta _{q,\Gamma }}} \right \| \to 1$ for fixed $q$ as $N \to \infty$. Here the difference from $1$ is $O({N^{35 - q}})$. We prove these results by estimating $\frac {{{{\left \| {{\Theta _{q,\Gamma }}({f_p})} \right \|}_{{A_q}(\Gamma )}}}} {{{{\left \| {{f_p}} \right \|}_{{A_q}}}}}$ 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.References
- Tom M. Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics, No. 41, Springer-Verlag, New York-Heidelberg, 1976. MR 0422157, DOI 10.1007/978-1-4684-9910-0 I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York, 1965.
- Tomio Kubota, Elementary theory of Eisenstein series, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. MR 0429749
- Curt McMullen, Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), no. 1, 95–127. MR 999314, DOI 10.1007/BF01850656 —, Letter to author (dated August 24, 1988).
- L. A. Parson and Mark Sheingorn, Bounding the norm of the Poincaré $\theta$-operator, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 422–441. MR 654543
- L. Alayne Parson, A lower bound for the norm of the theta operator, Math. Comp. 41 (1983), no. 164, 683–685. MR 717712, DOI 10.1090/S0025-5718-1983-0717712-0
- L. Alayne Parson, On the Poincaré theta operator, Complex Variables Theory Appl. 6 (1986), no. 2-4, 135–148. MR 871726, DOI 10.1080/17476938608814165
- Hideo Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33–71. MR 145106, DOI 10.2307/1970201
- Kurt Strebel, On lifts of extremal quasiconformal mappings, J. Analyse Math. 31 (1977), 191–203. MR 585316, DOI 10.1007/BF02813303
Additional Information
- © Copyright 1991 American Mathematical Society
- 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