**GENERAL AND
NUMBER THEORY SEMINAR**

Class numbers of ray class fields of

imaginary quadratic fields

### by

#### ÖMER KÜÇÜKSAKALLI

(University
of Massachusetts)

__Abstract: __The class number is a powerful invariant in
algebraic number theory which can be used to investigate the integer solutions
of polynomials, such as Fermat's Equation. It can be computed for extensions
with small degree and

discriminant, however computations take a very long time for higher extensions.
In this talk, we will describe a heuristic method to compute the class numbers
of ray class fields of imaginary quadratic fields, the elliptic curve analogue
of real cyclotomic fields. We will use elliptic units analytically constructed
by Stark and the Galois action on them given by Shimura's reciprocity. In the
end we will give a counter example to Vandiver's conjecture in the elliptic
curve case.In my talk I am going to present the new trends in the area of
fractional calculus. Several recent applications of this type of calculus in economics,
bioengineering, physics and engineering will be illustrated.

Date : December 18, 2008 (Thursday)

Time : 15.40

Place: Mathematics Seminar Room SA141

__Tea
will be served before the seminar. __