Revêtements tangentiels et tours infinies d'Artin-Schreier

Let (X,q) be a smooth marked curve of genus g>0, defined over an algebraic closed field of characteristic p≥0. We consider all generically étale covers π:Γ→X, marked at a subset D⊂π−1(q) of cardinality d≥0, satisfying a natural tangency condition inside JacΓ. We characterize the latter, so-called d-tangential covers, as zero-divisors of certain polynomials. We focus at last on some funny behaviour in positive characteristic. Namely, infinite towers of 1-tangential covers, étale over X∖{q}, but wildly ramified over q. The latter exist if and only if q∈X is a Cartier point.