One of the main difficulties in studying the Galois groups of field extensions is that, in general, the set of roots of a polynomial need not have any additional algebraic structure. In 1965, Lubin and Tate showed that a certain type of power series f(x) over a p-adic field can be recognized as the "multiplication-by-pi" endomorphism of a formal group law with complex multiplication, or a larger-than-usual endomorphism ring. They used this to put a module structure on the roots of f(x) and to show that the splitting field of f(x) is Galois and abelian, culminating in a new proof of the main theorem of local class field theory. In this talk we will offer a new and more general construction of formal group laws with complex multiplication in higher dimensions and will examine how this differs from the one-dimensional case. Finally, we will once again see that the torsion points of such a formal group generate abelian extensions over p-adic fields.
This defense will be held via zoom, and all are welcome. For zoom link information, please email David Grant.
Everyone is welcome for the presentation, and will be asked to leave for the questioning and deliberations.
Higher dimensional formal group laws with complex multiplication
Tue, Apr. 7 11am (MATH 350)
TBA Number Theory thesis defenses today
Tue, Apr. 7 1pm (Zoom)
Daniel Martin (University of Colorado Boulder)
We will take a geometric perspective on Euclideaneity in imaginary quadratic fields. The result is a pseudo-Euclidean algorithm with applications to continued fractions, the class group, and the special linear group in two dimensions.
This defense will be held via zoom, and all are welcome:
Due to "zoombombing" concerns, please email Katherine Stange or Daniel Martin for link and password.
Everyone is welcome for the presentation, and will be asked to leave for the questioning and deliberations. After the defense, an announcement will be made and congratulations can be sent via email.
The geometry of imaginary quadratic fields
CANCELED Tue, Apr. 7
Bryan Gillespie (CSU) TBA Sponsored by the Meyer Fund