Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On indefinite $k$-universal integral quadratic forms over number fields   Let $k\ge 1$ be an integer. An integral quadratic form is called
$k$-universal if it represents all integral quadratic forms of dimension $k$.
We prove that the $k$-universal property satisfies the local-global principle
over number fields for $k\geq 3$. We also show that a number field $F$ admits
an integral quadratic form which is locally $2$-universal but not globally if
and only if the class number of $F$ is even. When it is the case, there are
only finitely many classes of such forms over $F$. A variant for classic
ternary forms is also discussed. In particular, we classify all the quadratic
fields over which there exist classic ternary quadratic forms that are locally
universal but not globally.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs