§ 61.31(d)(2) would be the controlling rule determining pilot in command privilages at night :
Per § 61.31(d)(2), the rule states:
(d) Aircraft category, class, and type ratings: Limitations on operating an aircraft as the pilot in command. To serve as the pilot in command of an aircraft, a person must-
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(2) Have received training required by this part that is appropriate to the pilot certification level, aircraft category, class, and type rating (if a class or type rating is required) for the aircraft to be flown, and have received an endorsement for solo flight in that aircraft from an authorized instructor.
Again, I am assuming this training will be in an airplane.
In order for a holder of a Private Pilot Certificate with an Airplane Single Engine or Airplane Multiengine rating to be allowed to fly solo at night, that person must have received night training from an authorized flight instructor on at least the following:
[For Single Engine Airplane Aeronautical Experience at Night]
(2) Except as provided in § 61.110 of this part, 3 hours of night flight training in a single-engine airplane that includes-
(i) One cross-country flight of over 100 nautical miles total distance; and
(ii) 10 takeoffs and 10 landings to a full stop (with each landing involving a flight in the traffic pattern) at an airport.
or
[For Multiengine Airplane Aeronautical Experience at Night]
(2) Except as provided in § 61.110 of this part, 3 hours of night flight training in a multiengine airplane that includes-
(i) One cross-country flight of over 100 nautical miles total distance; and
(ii) 10 takeoffs and 10 landings to a full stop (with each landing involving a flight in the traffic pattern) at an airport.
They would add the Multi to your FAA if you verified it during the verification process.