Context : The space experiment CoRoT ( Convection , Rotation and Transits ) has recently detected a transiting hot Jupiter in orbit around a moderately active F-type main-sequence star ( CoRoT-Exo-4a ) .
This planetary system is of particular interest because it has an orbital period of 9.202 days , the second longest one among the transiting planets known to date .

Aims : We study the surface rotation and the activity of the host star during an uninterrupted sequence of optical observations of 58 days .

Methods : Our approach is based on a maximum entropy spot modelling technique extensively tested by modelling the variation of the total solar irradiance .
Recently , it has been successfully applied to model the light curve of another active star with a transiting planet observed by CoRoT , i.e. , CoRoT-Exo-2a .
It assumes that stellar active regions consist of cool spots and bright faculae , analogous to sunspots and solar photospheric faculae , whose visibility is modulated by stellar rotation .

Results : The modelling of the light curve of CoRoT-Exo-4a reveals three main active longitudes with lifetimes between \sim 30 and \sim 60 days that rotate quasi-synchronously with the orbital motion of the planet .
The different rotation rates of the active longitudes are interpreted in terms of surface differential rotation and a lower limit of 0.057 \pm 0.015 is derived for its relative amplitude .
The enhancement of activity observed close to the subplanetary longitude suggests a magnetic star-planet interaction , although the short duration of the time series prevents us from drawing definite conclusions .

Conclusions : The present work confirms the quasi-synchronicity between stellar rotation and planetary orbital motion in the CoRoT-Exo-4 system and provides for the first time a lower limit for the surface differential rotation of the star .
This information can be important in trying to understand the formation and evolution of this highly interesting planetary system .
Moreover , there is an indication for a possible star-planet magnetic interaction that needs to be confirmed by future studies .