Infinitesimal deformation of p-adic differential equations on Berkovich curves

We show that if a differential equation F over a quasi-smooth Berkovich curve X has a certain compatibility condition with respect to an automorphism sigma of X, then F acquires a semi-linear action of sigma (i.e. lifting that on X). The compatibility condition forces the automorphism sigma to be close to the identity of X, so the above construction applies to a certain class of automorphisms called infinitesimal. This generalizes Andre and Di Vizio (Asterisque 1(296): 55-111, 2004) and Pulita (Compos. Math. 144(4): 867-919, 2008). We also obtain an application to Morita's p-adic Gamma function, and to related values of p-adic L-functions.