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[abridged] We carry out numerical simulations to gauge the Gaia potential forprecision astrometry of exoplanets orbiting a sample of known dM stars within30 pc from the Sun. (1) It will be possible to accurately determine orbits andmasses for Jupiter-mass planets with orbital periods in the range 0.2<P<6.0 yrand with an astrometric signal-to-noise ratio > 10. Given present-day estimatesof the planet fraction f_p around M dwarfs, 100 giant planets could be found byGaia around the sample. Comprehensive screening by Gaia of the reservoir of4x10^5 M dwarfs within 100 pc could result in 2600 detections and as many as500 accurate orbit determinations. The value of f_p could then be determinedwith an accuracy of 2%, an improvement by over an order of magnitude withrespect to the most precise values available to-date; (2) in the same periodrange, inclination angles corresponding to quasi-edge-on configurations will bedetermined with enough precision (a few percent) so that it will be possible toidentify intermediate-separation planets which are potentially transitingwithin the errors. Gaia could alert us of the existence of 10 such systems.More than 250 candidates could be identified assuming solutions compatible withtransit configurations within 10% accuracy, although a large fraction of these(85%) could be false positives; (3) for well-sampled orbits, the uncertaintieson planetary ephemerides, separation and position angle, will degrade attypical rates of < 1 mas/yr and < 2 deg/yr, respectively; (4) planetary phaseswill be measured with typical uncertainties of several degrees, resulting(under the assumption of purely scattering atmospheres) in phase-averagederrors on the phase function of 0.05, and expected uncertainties in thedetermination of the emergent flux of intermediate-separation (0.3<a<2.0 AU)giant planets of 20%. [abridged]

Astrometric Detection of Giant Planets Around Nearby M Dwarfs: The Gaia Potential