### Abstract

We shall study some explicit connections between (1) the Vandiver conjecture on the class number of the real cyclotomic field Q(cos(2π/l)) and (2) the images of various Galois representations induced from the power series representation (constructed and studied by Ihara, Anderson, Coleman, etc.) of Gal(Q/Q(μ_{I∞})) which describes universally the Galois action on the Fermat curves of l-power degrees. One such connection was first discovered by Coleman. In the case of the original power series representation, we shall also describe the difference between the "expected image" and the actual Galois image in terms of a certain invariant of Iwasawa type.

Original language | English |
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Pages (from-to) | 312-334 |

Number of pages | 23 |

Journal | Journal of Number Theory |

Volume | 31 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1989 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

Ichimura, H., & Kaneko, M. (1989). On the universal power series for Jacobi sums and the vandiver conjecture.

*Journal of Number Theory*,*31*(3), 312-334. https://doi.org/10.1016/0022-314X(89)90076-0