Lassina Dembele

Explicit Computations of Hilbert Modular Forms on ℚ(√5)

This article presents an algorithm to compute Hilbert modular forms on the quadratic field ℚ(√5). It also provides a list of all modular abelian varieties defined over ℚ(√5) with prime level of norm less than 100 (up to ℚ-isogeny).

Explicit Methods for Hilbert Modular Forms

The study of modular forms remains a dominant theme in modern number theory, a consequence of their intrinsic appeal as well as their applications to a wide variety of mathematical problems. This subject has seen dramatic progress during the past half-century in an environment where both abstract theory and explicit computation have developed in parallel. Experiments will remain an essential tool in the years ahead, especially as we turn from classical contexts to less familiar terrain.

Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms

In this paper, we propose a generalization of the algorithm we developed previously. Along the way, we also develop a theory of quaternionic M-symbols whose definition bears some resemblance to the classical M-symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and we have illustrated it with several examples. Namely, we have computed all the newforms of prime levels of norm less than 100 over the quadratic fields Q(√29) and Q(√37), and whose Fourier coefficients are rational or are defined over a quadratic field.

Computing Hilbert modular forms over fields with nontrivial class group

We exhibit an algorithm for the computation of Hilbert modularforms over an arbitrary totally real number field of even degree,extending results of the first author. We present some new instancesof the conjectural Eichler-Shimura construction for totally real numberfields over the fields Q(√10) and Q(√85) and their Hilbert class fields,and in particular some new examples of modular abelian varieties witheverywhere good reduction over those fields.

Elliptic curves, Hilbert modular forms and Galois deformations

Part I: Galois Deformations.- On p-adic Galois Representations.- Deformations of Galois Representations.- Part II: Hilbert Modular Forms.- Arithmetic Aspects of Hilbert Modular Forms and Varieties.- Explicit Methods for Hilbert Modular Forms.- Part III: Elliptic Curves.- Notes on the Parity Conjecture.

Computing Genus-2 Hilbert–Siegel Modular Forms over via the Jacquet–Langlands Correspondence

In this paper we present an algorithm for computing Hecke eigensystems of Hilbert–Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field . In those examples, we identify Hilbert–Siegel eigenforms that are possible lifts from Hilbert eigenforms.

Nonsolvable number fields ramified only at 3 and 5

For p = 3 and p = 5, we exhibit a finite nonsolvable extension of Q which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

Serre weights and wild ramification in two-dimensional Galois representations

A generalization of Serres Conjecture asserts that if

Theta lifts of Bianchi modular forms and applications to paramodularity

F

On the computation of algebraic modular forms on compact inner forms of

is a totally real field, then certain characteristic