The Van Allen belts have very little to do with the earth's core as the other answer suggested.
They are believed to be created by the earth's magnetic field interacting with charged particles known as solar wind. To the extent that we believe the earth's core responsible for creating the earth's magnetic field, I guess the earth's core has some relevance.
As the solar winds encounter the earth's magnetic field, the radiation becomes trapped within the field. They are repelled from areas of stronger magnetic field where field lines converge.
Two alternative hypotheses have been proposed. Possibly the Van Allen Belts are lefor over from the aurora, and another argument is that they are the result of volcanic activity of the earth.
Some scientists believe that they may have a beneficial effect on the planet providing some protection from solar wind, and that weakening them would harm both electronics and living organisms.
There has been a proposal to reduce the charge of the belts, which are a hazard to satellites.