Genuine Bianchi modular forms of higher level at varying weight and discriminant (Q2199487)

Bianchi modular forms can be roughly defined as functions on the Lie group \(\mathrm{GL}_2(\mathbb C)\), with values in the irreducible representations of \(\mathrm{GL}_2(\mathbb C)\), which are invariant under the left-action of a congruence subgroup of a Bianchi group \(\mathrm{GL}_2(\mathcal O_K)\) with \(\mathcal O_K\) a ring of imaginary quadratic integers, equivariant under the right-action of the unitary group \(\mathrm U(2)\) and under the action of the center. For certain representations (and with a coherent choice of character for the center) they can be interpreted as cohomology classes on the quotient hyperbolic 3-orbifold associated with the arithmetic groups. This can be used to perform exact computations for the dimensions of these spaces, using a cellular structure for a deformation retract of the orbifold. Among these forms are the analogues of the Eisenstein series. The remainder, the so-called cuspidal forms which vanish to high order in the cusps of the orbifold, cannot be described uniformly, in contrast to the classical case (modular forms for \(\mathrm{PSL}_2(\mathbb Z)\) and its congruence subgroups) where an exact dimension formula is available. Still there are various explicit constructions for them, for instance the Langlands base-change operation associates Bianchi modular forms of weight \((k, k)\) to classical modular forms. This can also be twisted via characters of the class-group. Finally there is another explicit construction via power series (``CM-forms''). It has been remarked via explicit computation of homology groups on small examples that there may also occur ``genuine'' forms which do not arise from any of these constructions. Since there are formulas for the dimensions of the subspaces of base-change or CM-forms the dimension of these genuine forms can in principle be computed; the present paper details such an approach, with effective formulas that can be implemented. The authors also give the results of a systematic search for genuine forms for small discriminant, weights and level. These exhibit a relative paucity of such forms but they still find many new examples where they do appear.