Representation of integers by positive ternary quadratic forms

Let $m$ be a positive integer satisfying $m \equiv 1 (\mathrm{mod} \: 4)$ and $(\dfrac{m}{7}) = 1$. Then there exist integers $x, y, z \in \mathbb{Z}$ such that $m = x^2 + 7y^2 + 49z^2$. Recent work of J. Coates, Y. Li, Y. Tian and S. Zhai ([1]) shows that this conclusion is useful in the study of the arithmetic of elliptic curves. The above $m$ can be represented also by either $x^2 + 14y^2 + 28z^2 + 14yz$ or $2x^2 + 4y^2 + 49z^2 + 2xy$. Moreover, if $m \equiv 1 (\mathrm{mod} \: 8)$ and $(\dfrac{m}{7}) = 1$, then $m$ can be represented by $x^2 + 14y^2 + 28z^2 + 14yz$. The same is true for $2x^2 + 4y^2 + 49z^2 + 2xy$, provided that $m$ is not square. If we assume further that $m \equiv 1 (\mathrm{mod} \: 4)$ with $(\dfrac{m}{7}) = 1$ is not square, then $m$ can be represented by $8x^2 + 8y^2 + 9z^2 + 8yz + 8xz + 2xy$ and $2x^2 + 7y^2 + 25z^2 + 2xz$. Note that the genus of $x^2 + 7y^2 + 49z^2$ consists of exactly the above appeared five ternary quadratic forms.

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Supported by NSFC (Nos. 11571163, 11171141, 11471154, 11631009), NSFJ (Nos. BK2010007), PAPD and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No.708044).

Received 27 July 2015

Accepted 1 November 2015

Published 24 July 2017