Effective methods for vanishing cycles of <f>p</f>-cyclic covers of the <f>p</f>-adic line

This paper studies the stable reduction of <f>p</f>-cyclic covers <f>X→PK1</f> of the projective line over <f>p</f>-adic fields. So far, an algorithm to effectively determine the stable reduction of such covers is only known under additional hypothesis on the branch locus of the cover. Here, rather than restricting the type of cover, we consider the general case and obtain results on the structure of the special fiber <f>Xk</f> of the stable reduction of <f>X</f>. Special attention is payed to making all constructions effective. The central result is a formula computing the number of vanishing cycles on <f>Xk</f>. In particular, we give criteria for the special fiber of the stable reduction to be tree-like and for when <f>X</f> is a Mumford curve. Refining the analysis of vanishing cycles, we describe an algorithm that computes all the components of positive <f>p</f>-rank in the stable model. [Copyright &y& Elsevier]