For more information about this meeting, contact Robert Vaughan.

Title: | Small Height and Infinite Non-abelian Extensions |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Philipp Habegger, IAS, Princeton |

Abstract: |

The absolute, logarithmic Weil height is non-negative and vanishes precisely at 0 and at the roots of unity. Moreover, when restricted to a number field there are no arbitrarily small, positive heights. Amoroso, Bombieri, David, Dvornicich, Schinzel, Zannier and others exhibited many infinite extensions of the rationals with a height gap. For example, the maximal abelian extension of any number field has this property. To see a non-abelian example, let E be an elliptic curve defined over the rationals without complex multiplication. The field K generated by all complex points of E with finite order is an infinite extension of the rationals. Its Galois group contains no commutative subgroup of finite index. In the talk, I will sketch a proof that K contains no elements of sufficiently small, positive height. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 04 / 04 / 2013 |

Time: | 11:15am - 12:05pm |