On integral representations by totally positive ternary quadratic forms

Elise Björkholdt

Journal de théorie des nombres de Bordeaux (2000)

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

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 ( 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 ∈ 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 [ - c f N ] , where c f ∈ 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 ( 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.

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},
keywords = {ternary quadratic form; quadratic number field; quaternion order},
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
KW - ternary quadratic form; quadratic number field; quaternion order
UR - http://eudml.org/doc/248500
ER -

References

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