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 Bianchi.gp 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.). |