Representations of finite number of quadratic forms with same rank

Abstract

Let $m,n$ be positive integers with $m\le n$. Let $\kappa (m,n)$ be the largest integer $k$ such that for any (positive definite and integral) quadratic forms $f_1,\dots ,f_k$ of rank $m$, there exists a quadratic form of rank $n$ that represents $f_i$ for any $i$ with $1\le i\le k$. In this article, we determine the number $\kappa (m,n)$ for any integer $m$ with $1\le m\le 8$, except for the cases when $(m,n)=(3,5)$ and $(4,6)$. In the exceptional cases, it will be proved that $1\le \kappa (3,5),\kappa (4,6)\le 2$. We also discuss some related topics.

13 pages