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Let X be a smooth projective curve of positive genus defined over a numberfield K. Assume given a Galois covering map x from X to the projective lineover K and a place v of K. We introduce a local canonical height on the set ofK_v-valued points of X associated to x as an integral with logarithmicintegrand, generalizing Tate's local Neron function on an elliptic curve. Theresulting global height can be viewed as a 'Mahler measure' associated to x. Weprove that the local canonical height can be obtained by averaging, and takinga limit, over divisors of higher order Weierstrass points on X. Thisgeneralizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker. Ourconstruction of the local canonical height is an application of potentialtheory on Berkovich curves in the presence of a canonical measure.

Local heights on Galois covers of the projective line