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Let k be an algebraically closed field of characteristic p>0 and C aconnected nonsingular projective curve over k with genus g>1. Let (C,G) be a"big action", i.e. a pair (C,G) where G is a p-subgroup of the k-automorphismgroup of C such that |G|/g > 2p / p-1. We first study finiteness results on thevalues taken by the quotient |G|/g^2 when (C,G) runs over the big actionssatisfying |G|/g^2 >M, for a given positive real M>0. Then, we exhibit aclassification and a parametrization of such big actions when M=4/ (p^2-1)^2.

Large p-groups actions with |G| /g^2 > 4/ (p^2-1)^2