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It is shown that for the modular representations associated to RationalConformal Field Theories, the kernel is a congruence subgroup whose levelequals the order of the Dehn-twist. An explicit algebraic characterization ofthe kernel is given. It is also shown that the conductor, i.e. the order of theDehn-twist is bounded by a function of the number of primary fields, allowingfor a systematic enumeration of the modular representations coming from RCFTs.Restrictions on the spectrum of the Dehn-twist and arithmetic properties ofmodular matrix elements are presented

The Kernel of the Modular Representation and the Galois Action in RCFT