## Abstract

The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L_{1} and the m-rank of some subgroup of the class group of its associated real cyclic function field L_{2} for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L_{2}, then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L_{2}.

Original language | English |
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Pages (from-to) | 408-414 |

Number of pages | 7 |

Journal | Journal of Number Theory |

Volume | 122 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2007 |

## Keywords

- Class groups
- Cyclic function fields
- Imaginary quadratic function fields
- Rank
- Real quadratic function fields
- Regulator
- Scholz theorem