The MELODIA Project

Methods for low-dimensional abelian varieties

Dimension tables for spaces of Bianchi modular forms

By Alexander D. Rahm, Mehmet Haluk Sengun and Panagiotis Tsaknias

- Download dimension tables for spaces of Bianchi modular forms from the bottom of this page

Spaces of Bianchi modular forms can be computed using the action of a Bianchi group on 3-dimensional hyperbolic space. The Bianchi diagram, which we print in the left margin of this page for the Bianchi group of discriminant -408, shows the bottom facets of a fundamental polyhedron for the action of a Bianchi group on hyperbolic 3-space.
From this page, you can download the dimension tables computed in an Irish Centre for High-End Computing (ICHEC) project by Alexander D. Rahm and Mehmet Haluk Sengun. In this project, we have calculated the dimensions of the spaces of cuspidal Bianchi modular forms, locating several of the very rare instances that are not lifts of elliptic modular forms. Elliptic modular forms are the subject of the now proven Taniyama--Shimura conjecture. A part of this proof has allowed to complete Fermat's Last Theorem, which has been a challenge to mathematics for over 300 years. The Taniyama--Shimura conjecture is nowadays called the modularity theorem, and attaches an elliptic curve to each classical modular form (so to call it an "elliptic" modular form), corresponding via the L-series. There are deep number-theoretical reasons for expecting that this kind of correspondence can extend to the Bianchi modular forms, attaching Abelian varieties to them (elliptic curves are one-dimensional Abelian varieties). Bianchi modular forms over an imaginary quadratic field K are automorphic forms of cohomological type associated to the Bianchi group of K, the latter being the rank two special linear group over the ring of integers in K. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. In this project, we have extended our published computations of cuspidal Bianchi modular forms (appeared in LMS J. Comp. & Math), that were carried out on "workstation" machines exclusively for the subgroup level 1 (i.e. the full Bianchi groups), to a large scope of congruence subgroups. For the subgroup level 1, we have already discovered several instances of particluar interest, namely Bianchi modular forms that are not lifts from the elliptic modular forms via the Langlands base change procedure. There are so far no theoretical predictions available on the occurrence of such non-lifted forms, so to make the numerical results very valuable. The L-functions and Modular Forms Data-Base (LMFDB) has accepted this data for publication. The LMFDB is open-access and guarantees a long-term high visibility of the results. In the meantime, you can download the dimension tables from this current page.

I.) The full cohomology spaces (cuspidal+Eisenstein) are collected in plain text files each for one imaginary quadratic field, specified by its discriminant. The rows of the tables specify
1) the discriminant
2) the Hermite Normal Form of the ideal which is the level of congruence,
3) the weight,
4) the computed dimension of the full cohomology space (cuspidal+Eisenstein).
For the precise setting, please look up our paper on level 1. For the computation, we have used the software to compute the necessary geometric-topological information about the whole Bianchi group, and then applied a MAGMA implementation by Haluk Sengun (for which we provide the algorithm here) to deduce the dimension of the relevant cohomology space for the congruence subgroup at the given level, passing by the Eckmann--Shapiro lemma.

II.) We have then substracted the Eisenstein series space to get the cuspidal cohomology space, which by the Eichler--Shimura(--Harder) isomorphism yields the cuspidal forms space. Then we did substract the oldforms using a well-known recursive formula, to get the dimensions of the newforms spaces. We provide the latter dimensions in MAGMA-readable files below.
Exception: For discriminants -3, -4, -8, -20 we provide the LMFDB raw file, which has columns "field_label" [the label of the imaginary quadratic field in the LMFDB], "weight(automorphic)" [automorphic weight k+2], "level(HNF)" [Hermite Normal Form], "cusp-dim" [dimension of the cuspidal space], "new-cusp-dim" [dimension of the space of cuspidal newforms].

III.) To the newforms dimensions, we have applied the formulas of our higher levels paper to subtract the dimensions of non-genuine subspaces (base-change, CM and twists of base-change). We provide the remaining genuine dimensions in human-readable format. By Theorem 2 of our higher levels paper, the entry "newCM" can be seen to be zero.

Caveat: While for (I.), we use the cohomological weight k (where k x k is the dimension of the coefficient module), we shift to automorphic weight k+2 for (II.) and (III.).
Discriminant -3 newforms
Discriminant -4 newforms
Discriminant -7 newforms genuine forms
Discriminant -8 newforms
Discriminant -11 newforms genuine forms
Discriminant -19 newforms genuine forms
Discriminant -20 newforms
Discriminant -43 newforms genuine forms
Discriminant -67 newforms genuine forms
Discriminant -163 : newforms , genuine forms
If you are interested in a Pari/GP script with which you can calculate more dimensions, as described in a Pari/GP workshop talk, please contact Alexander D. Rahm.

We acknowledge the funding of the principal investigators by the Irish Research Council (for Alexander D. Rahm) and by the Marie Curie program (for Mehmet Haluk Sengun).

And here are links back to the homepages of Alexander D. Rahm, M. Haluk Sengun and Panagiotis Tsaknias.