### Abstract

In this article we make an explicit approach to the problem: "For a given CM field M, construct its maximal unramified abelian extension C(M) by the adjunction of special values of certain modular functions" in some restricted cases with [M: Q] ≥ 4. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F. We determine the modular function which gives the canonical model for all B's coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group δ(3, 3, 5). By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.

Original language | English |
---|---|

Pages (from-to) | 408-430 |

Number of pages | 23 |

Journal | Journal of Number Theory |

Volume | 165 |

DOIs | |

Publication status | Published - 2016 Aug 1 |

### Fingerprint

### Keywords

- Complex multiplication
- Hilbert class field
- Hypergeometric functions
- Moduli of abelian varieties
- Theta functions

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*165*, 408-430. https://doi.org/10.1016/j.jnt.2016.01.016

**To the Hilbert class field from the hypergeometric modular function.** / Nagano, A.; Shiga, H.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 165, pp. 408-430. https://doi.org/10.1016/j.jnt.2016.01.016

}

TY - JOUR

T1 - To the Hilbert class field from the hypergeometric modular function

AU - Nagano, A.

AU - Shiga, H.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - In this article we make an explicit approach to the problem: "For a given CM field M, construct its maximal unramified abelian extension C(M) by the adjunction of special values of certain modular functions" in some restricted cases with [M: Q] ≥ 4. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F. We determine the modular function which gives the canonical model for all B's coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group δ(3, 3, 5). By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.

AB - In this article we make an explicit approach to the problem: "For a given CM field M, construct its maximal unramified abelian extension C(M) by the adjunction of special values of certain modular functions" in some restricted cases with [M: Q] ≥ 4. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F. We determine the modular function which gives the canonical model for all B's coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group δ(3, 3, 5). By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.

KW - Complex multiplication

KW - Hilbert class field

KW - Hypergeometric functions

KW - Moduli of abelian varieties

KW - Theta functions

UR - http://www.scopus.com/inward/record.url?scp=84962510284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962510284&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2016.01.016

DO - 10.1016/j.jnt.2016.01.016

M3 - Article

AN - SCOPUS:84962510284

VL - 165

SP - 408

EP - 430

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -