## Journal of Mathematics of Kyoto University

### On the class number of the genus of $\mathbb{Z}$-maximal lattices with respect to quadratic form of the sum of squares

Takahiro Hiraoka

#### Abstract

In this paper we consider the quadratic form $\phi [x] = \sum _{i=1}^{n} x_{i}^{2}$ over the vector space $\mathbb{Q}_{n}^{1}$. We take a $\mathbb{Z}$-maximal lattice $L$ in $\mathbb{Q}_{n}^{1}$ with respect to $\phi$. Let $\{L^{(i)}\}_{i=1}^{k(n)}$ be a complete set of representatives for the classes belonging to the genus of $L$. Applying Shimura's mass formula, we determine these representatives $L^{(i)}$ explicitly for $n = 11, 13$, and $14$. Consequently we obtain class numbers $k(11) = 3$, $k(13) = 4$, and $k(14) = 4$.

#### Article information

Source
J. Math. Kyoto Univ., Volume 46, Number 2 (2006), 291-302.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250281778

Digital Object Identifier
doi:10.1215/kjm/1250281778

Mathematical Reviews number (MathSciNet)
MR2284345

Zentralblatt MATH identifier
1227.11060

Subjects
Primary: 11E41: Class numbers of quadratic and Hermitian forms
Secondary: 11E12: Quadratic forms over global rings and fields

#### Citation

Hiraoka, Takahiro. On the class number of the genus of $\mathbb{Z}$-maximal lattices with respect to quadratic form of the sum of squares. J. Math. Kyoto Univ. 46 (2006), no. 2, 291--302. doi:10.1215/kjm/1250281778. https://projecteuclid.org/euclid.kjm/1250281778