PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

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