# On integral representations by totally positive ternary quadratic forms

• Volume: 12, Issue: 1, page 147-164
• ISSN: 1246-7405

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## Abstract

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Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f\left({x}_{1},{x}_{2},{x}_{3}\right)$ be a totally definite ternary quadratic form with coefficients in $R$. We shall study representations of totally positive elements $N\in R$ by $f$. We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R\left[\sqrt{-{c}_{f}N}\right]$, where ${c}_{f}\in R$ is a constant depending only on $f$. We give an algebraic proof of a classical result of H. Maass on representations by sums of three squares over the integers in $ℚ\left(\sqrt{5}\right)$ and obtain an explicit dependence between the number of representations and the class number of the corresponding bi-quadratic field. We also give similar formulae for some quadratic forms arising from maximal quaternion orders, with class number one, over the integers in real quadratic number fields.

## How to cite

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Björkholdt, Elise. "On integral representations by totally positive ternary quadratic forms." Journal de théorie des nombres de Bordeaux 12.1 (2000): 147-164. <http://eudml.org/doc/248500>.

@article{Björkholdt2000,
abstract = {Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f(x_1, x_2, x_3)$ be a totally definite ternary quadratic form with coefficients in $R$. We shall study representations of totally positive elements $N \in R$ by $f$. We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R[\sqrt\{-c_f N\}]$, where $c_f \in R$ is a constant depending only on $f$. We give an algebraic proof of a classical result of H. Maass on representations by sums of three squares over the integers in $\mathbb \{Q\}(\sqrt\{5\})$ and obtain an explicit dependence between the number of representations and the class number of the corresponding bi-quadratic field. We also give similar formulae for some quadratic forms arising from maximal quaternion orders, with class number one, over the integers in real quadratic number fields.},
author = {Björkholdt, Elise},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {147-164},
publisher = {Université Bordeaux I},
title = {On integral representations by totally positive ternary quadratic forms},
url = {http://eudml.org/doc/248500},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Björkholdt, Elise
TI - On integral representations by totally positive ternary quadratic forms
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 147
EP - 164
AB - Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f(x_1, x_2, x_3)$ be a totally definite ternary quadratic form with coefficients in $R$. We shall study representations of totally positive elements $N \in R$ by $f$. We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R[\sqrt{-c_f N}]$, where $c_f \in R$ is a constant depending only on $f$. We give an algebraic proof of a classical result of H. Maass on representations by sums of three squares over the integers in $\mathbb {Q}(\sqrt{5})$ and obtain an explicit dependence between the number of representations and the class number of the corresponding bi-quadratic field. We also give similar formulae for some quadratic forms arising from maximal quaternion orders, with class number one, over the integers in real quadratic number fields.
LA - eng
UR - http://eudml.org/doc/248500
ER -

## References

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