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|
|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
|Date:||04 / 04 / 2013|
|Time:||11:15am - 12:05pm|