Ternary quadratic forms over number fields with small class number

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

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Where are number fields with small class number?

Number Theory Carbondale 1979 - Lecture Notes in Mathematics ◽

10.1007/bfb0062711 ◽

1979 ◽

pp. 221-242 ◽

Cited By ~ 4

Author(s):

John Myron Masley

Keyword(s):

Class Number ◽

Number Fields ◽

Small Class

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The number of representations of integers by generalized Bell ternary quadratic forms

International Journal of Number Theory ◽

10.1142/s1793042121500135 ◽

2020 ◽

pp. 1-15

Author(s):

Kyoungmin Kim ◽

Yeong-Wook Kwon

Keyword(s):

Quadratic Form ◽

Class Number ◽

Quadratic Forms ◽

Positive Definite ◽

Cusp Forms ◽

Ternary Quadratic Form ◽

Closed Formula ◽

Integral Quadratic Form ◽

Ternary Quadratic Forms ◽

Representations Of Integers

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

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There are no universal ternary quadratic forms over biquadratic fields

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309152000022x ◽

2020 ◽

Vol 63 (3) ◽

pp. 861-912 ◽

Cited By ~ 1

Author(s):

Jakub Krásenský ◽

Magdaléna Tinková ◽

Kristýna Zemková

Keyword(s):

Quadratic Forms ◽

Number Fields ◽

Ring Of Integers ◽

Ternary Quadratic Forms ◽

Mild Conditions ◽

Positive Unit ◽

Positive Elements ◽

Totally Positive ◽

Biquadratic Fields ◽

Totally Real

AbstractWe study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

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