### Abstract

We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp(n, Z) which satisfy X^{2} + I = 0 (resp. X^{2} + X + I = 0) and the set of hermitian forms of rank n over Z[√-1] (resp. Z[(-1 + √-3)/2]) of determinant ±1. As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of m^{2} + 1 (resp. m + m + 1) can be represented by the quadratic form F(X, Y) = X^{2} + Y^{2} (resp. X^{2} + XY + Y^{2}) with relatively prime integers X, Y: Suppose n ≤ 3 (resp. n ≤ 5). Then S can be represented over Z by F^{⊗n} (n copies of F) if det S is represented by F as above. The proof is based on Siegel-Braun's Mass formula for hermitian forms.

Original language | English |
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Pages (from-to) | 102-110 |

Number of pages | 9 |

Journal | Journal of Number Theory |

Volume | 23 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1986 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms.** / Hashimoto, Kiichiro; Sibner, Robert J.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 23, no. 1, pp. 102-110. https://doi.org/10.1016/0022-314X(86)90007-7

}

TY - JOUR

T1 - Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms

AU - Hashimoto, Kiichiro

AU - Sibner, Robert J.

PY - 1986

Y1 - 1986

N2 - We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp(n, Z) which satisfy X2 + I = 0 (resp. X2 + X + I = 0) and the set of hermitian forms of rank n over Z[√-1] (resp. Z[(-1 + √-3)/2]) of determinant ±1. As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of m2 + 1 (resp. m + m + 1) can be represented by the quadratic form F(X, Y) = X2 + Y2 (resp. X2 + XY + Y2) with relatively prime integers X, Y: Suppose n ≤ 3 (resp. n ≤ 5). Then S can be represented over Z by F⊗n (n copies of F) if det S is represented by F as above. The proof is based on Siegel-Braun's Mass formula for hermitian forms.

AB - We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp(n, Z) which satisfy X2 + I = 0 (resp. X2 + X + I = 0) and the set of hermitian forms of rank n over Z[√-1] (resp. Z[(-1 + √-3)/2]) of determinant ±1. As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of m2 + 1 (resp. m + m + 1) can be represented by the quadratic form F(X, Y) = X2 + Y2 (resp. X2 + XY + Y2) with relatively prime integers X, Y: Suppose n ≤ 3 (resp. n ≤ 5). Then S can be represented over Z by F⊗n (n copies of F) if det S is represented by F as above. The proof is based on Siegel-Braun's Mass formula for hermitian forms.

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UR - http://www.scopus.com/inward/citedby.url?scp=38249039166&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(86)90007-7

DO - 10.1016/0022-314X(86)90007-7

M3 - Article

AN - SCOPUS:38249039166

VL - 23

SP - 102

EP - 110

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -