Title: The eigencurve near the boundary of weight space for p=2 and p=3 Abstract: We will give a brief introduction to p-adic modular forms and define the eigencurve informally. Then, using techniques from Buzzard and Kilford's "The 2-adic eigencurve at the boundary of weight space," we will sketch the proof that for p=2 and p=3, the eigencurve breaks up as a countable disjoint union of annuli near the boundary of weight space.