The Hilbert Modular Forms Project

  • Motivation for the project
    Since the proof of the Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves over Q by Wiles in 95, which led to an affirmative solution to the famous Fermat's Last Theorem, there have been some tremendous progress on the Langlands correspondence for GL(2)/Q. Those progress culminated in the recent proofs of the Serre conjecture for mod p Galois representations, and the Shimura-Taniyama-Weil conjecture for abelian varieties of GL(2)-type over Q by Dieulefait, Khare and Wintenberger, and Kisin et al.

    It is fair to say that such theoretical progress would have been very difficult to achieve without our ability to compute and experiment on classical modular forms. Indeed, since the mid 60s, computers have been used by number theorists to accumulate databases on classical modular forms and related geometric objects. In fact, the above mentioned conjectures were motivated in part by such computations. This also includes the Birch and Swinnerton-Dyer conjecture which is among the Clay Mathematics Institute millennium problems.

    Currently, there are vast databases of classical modular forms and related objects as well as softwares which are used by many number theorists in order to test existing conjectures or experiment on new ones. The main contributors of those databases are J. Cremona and W. Stein.

    All the above mentioned conjectures have their natural generalizations to totally real number fields. Therefore, one hopes that by increasing our ability to experiment on Hilbert modular forms, one would get the same kind of insights as in the classical setting.

  • The goal of the project
    The goal of this project is to create a database of Hilbert modular forms that emulates the ones that exist for classical modular forms. This database will be used by the number theory community in order to experiment on the Shimura-Taniyama-Weil conjecture, the Serre conjecture, and the Birch and Swinnerton-Dyer conjecture for totally real number fields; and also, on the converse of the Shimura-Taniyama-Weil conjecture known as the Eichler-Shimura construction which is still an open conjecture in the totally real case.

  • Progress made so far: The Hilbert Modular Forms Package in Magma
     I was able to revisit the theory of Brandt matrices for totally real number fields of narrow class number one. This allowed me to obtain a more efficient algorithm for the computation of Hilbert modular forms. In a joint paper with Steve Donnelly, we were able to extend those results to fields with nontrivial class group. My algorithm has now been implemented into a Magma package together with another algorithm developed by John Voight and Matthew Greenberg, which computes spaces of Hilbert modular forms as Hecke modules by using the cohomology of Shimura curves. My code works best for fields with even degree whereas theirs is more suitable for fields of odd degree; in that sense, they are complimentary.

    Note: Recently, Paul Gunnells and Dan Yasaki also developed an algorithm for computing Hilbert modular forms over real quadratic fields via the cohomology of bundles over Hilbert modular surfaces.

  • Short term goal
    The short term goal is to create a database of Hilbert modular forms over real quadratic fields. I will start working on this before the completion of the implementation of the full algorithm. This initial database will include several quadratic fields of small discriminant.

  • Long term goal
    • After the implementation of the package is finished, I will start to expand the initial database by including more real quadratic fields as well as number fields of higher degree such as cubic and quartic fields. In this phase of the project, it will be helpful to have the contribution of other number theorists who work in the field of Hilbert modular forms.

    Once the infrastructures for number fields as well as quaternion algebras have been developed well enough in SAGE, it would be nice to have my code implemented there as well. I hope by then, I will have many collaborators who will be willing to take up this job.