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Discrete versions of the Painleve equations (dPII and qPII) over finitefields are studied. We first show that they are well defined by extending thedomain according to the theory of the space of initial conditions, taking thedPII equation as an example. Then we define them over the field of p-adicnumbers and see that they have a property that is similar to the good reductionof dynamical systems modulo a prime. This property is called 'almost goodreduction'. We study the q-discrete analogue of the Painleve II equation inthis paper, following the method in our previous work (arXiv: 1206.4456), inwhich the discrete Painleve II equation has been treated. We can consideralmost good reduction as an arithmetic analogue of the singularity confinementtest. We can also obtain special solutions over finite fields from thosedefined over fields of characteristic zero.  (v2: excluded the review of arXiv: 1206.4456) (v3: several typos arecorrected) (v4: final version to appear in J. Nonlin. Math. Phys.)

The space of initial conditions and the property of an almost good reduction in discrete Painleve II equations over finite fields