Adele Lopez and Dr. Paul Jenkins, Department of Mathematics

We prove that there are innitely many imaginary quadratic fields with 2-class group isomorphic to C2kC2 for any given positive integer k. This work is based on a related result for cyclic 2-class groups by Dominguez, Miller and Wong, and our proof proceeds similarly. However, we make some improvements in some of the techniques used. In particular, we provide a more general way of introducing congruence conditions into Perelli’s version of the Goldbach conjecture for polynomials.

My original project concerning Sturm bounds turned out to already be known, so my advisor and I decided to switch directions and look at 2 class groups.

Since the time of Gauss, mathematicians have been interested in imaginary quadratic fields and their ideal class groups. Gauss himself provided much of the framework for such studies, with the development of his genus theory for binary quadratic forms. Later developments by Redei and others have allowed us to be able to easily calculate the 2-class group from the discriminant of such a fields.

However, not much work has been done in the converse direction. Recently, Dominguez, Miller and Wong proved that there are innitely many imaginary quadratic fields with any given cyclic 2-class group [1]. They proved this by determining a set of criteria that the discriminant of such a field would have to satisfy, and then using the circle method, show that there are innitely many integers satisfying those criteria using the circle method.

In their paper, Dominguez, Miller and Wong asked if similar results could be found for other types of groups. We use the same technique to prove that for 2-class groups of the form C_2^k x C₂ for any given positive integer k, innitely many imaginary quadratic elds with such a 2-class group exist.

Additionally, Steven Miller asked in a personal correspondence if there was a more general way to introduce congruence conditions into the circle method portion of the proof. We have found such a method, based on Perelli’s arguement [2] which also works for the circle method portion of their original result.

We prove the following theorems.

References

1. Carlos Dominguez, Steven J. Miller, and Siman Wong. Quadratic fields with cyclic 2-class groups. Journal of Number Theory (to appear).
2. Alberto Perelli. Goldbach numbers represented by polynomials. Rev. Mat. Iberoamericana, 12(2):477{490, 1996.