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An asymptotic formula for representations of integers by indefinite hermitian forms


Author: Emilio A. Lauret
Journal: Proc. Amer. Math. Soc. 142 (2014), 1-14
MSC (2010): Primary 11D45, 11E39; Secondary 58C40
DOI: https://doi.org/10.1090/S0002-9939-2013-11726-0
Published electronically: September 4, 2013
MathSciNet review: 3119175
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Abstract: We fix a maximal order $ \mathcal O$ in $ \mathbb{F}=\mathbb{R},\mathbb{C}$ or $ \mathbb{H}$, and an $ \mathbb{F}$-hermitian form $ Q$ of signature $ (n,1)$ with coefficients in $ \mathcal O$. Let $ k\in \mathbb{N}$. By applying a lattice point theorem on an $ n$-dimensional $ \mathbb{F}$-hyperbolic space, we give an asymptotic formula with an error term, as $ t\to +\infty $, for the number $ N_t(Q,-k)$ of integral solutions $ x\in \mathcal O^{n+1}$ of the equation $ Q[x]=-k$ satisfying $ \vert x_{n+1}\vert\leq t$.


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Additional Information

Emilio A. Lauret
Affiliation: FaMAF–CIEM, Universidad Nacional de Córdoba, X5000HUA–Córdoba, Argentina
Email: elauret@famaf.unc.edu.ar

DOI: https://doi.org/10.1090/S0002-9939-2013-11726-0
Keywords: Representation by hermitian forms, hyperbolic lattice point theorem
Received by editor(s): September 29, 2011
Received by editor(s) in revised form: February 22, 2012
Published electronically: September 4, 2013
Additional Notes: This work was supported by CONICET and Secyt-UNC
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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