IRREDUCIBLE SPACES OF MODULAR UNITS

Fix a prime p> 7, put G = PSL(2,Fp), and write U for the multiplicative group of modular units of level p. We shall determine the irreducible subspaces of the natural representation of G on U/Up. The outcome of the calculation can be de-scribed as follows: Every irreducible nontrivial representation of G over Fp occurs with multiplicity one in the maximal semisimple subspace of the “noncongruence part ” of U/Up (to be defined). Apart from the formulation and some slight differ-ences arising from the choice of group (PSL(2,Fp) instead of GL(2,Fp)/{±1}), the result is already in Gross [2]. Presumably one can give conditions as in [6] and [7] which ensure that the unit group remains large after descent and specialization to a number field, but this problem will not be addressed here. For the sake of perspective, it is useful to recall that the natural representation of G on the space of modular forms of weight 2 and level p was decomposed into irreducibles in two papers of Hecke [3], [4]. As one would expect, most of the work in these papers goes into decomposing the space of cusp forms, but it is actually the space of Eisenstein series – dealt with by Hecke in a few lines – which has som