On the components of X0(pn

Abstract: We show he ordinary locus of X0(p n)(Cp) is normally the set of Cp-valued points on 2n affinoids which correspond to components of the stable model of X0(p n). We then show the points on Edixhoven’s “horizontal ” components of X0(p 2) correspond to elliptic curves which are p-isogenous to curves which Buzzard calls “too supersingular.” Fix a prime p. Suppose n ≥ 1. We first present a viewpoint of the ordinary locus of X0(pn), slightly different from that taken in Katz-Mazur and Edixhoven ([K-M], [E]) which allows one to see the stable structure of the ordinary locus. Next, we give a moduli-theoretic interpretation of Edixhoven’s (semi)stable model of X0(p2) [E]. Edixhoven found the p-adic stable model of X0(p2) by blowing up the Katz-Mazur-Edixhoven model ([K-M], [E]) at the supersingular points of the reduction. We interpret points on the components of this model in terms of the canonica