Abstract. 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.

Abstract. In a previous article [6], the second author proved that the images of the Galois representations mod λ attached to a Hilbert modular form without Complex Multiplication are “large ” for all but finitely many primes λ. In this brief note, we give an explicit bound for this exceptional finite set of primes and determine the images in three different examples. Our examples are of Hilbert newforms on real quadratic fields, of parallel or non-parallel weight and of different levels. 1.

Abstract. Let F be a totally real field and p ≥ 3 a prime. If ρ: Gal(F/F) → GL2(Fp) is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem that verifies one half of the conjecture in many cases and use Dembélé’s computations of Hilbert modular forms over Q ( √ 5) to provide evidence in support of the conjecture. 1.

Abstract. We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations. Contents 1. Classical (elliptic) modular forms 2

In modern Number Theory, the theory of automorphic forms is deeply related to the understanding of the structure of the absolute Galois group GQ = Gal ( ¯ Q/Q), via its representations. The Theorem of Kronecker-Weber, which states that all abelian extensions

Abstract. We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms. The development and implementation of algorithms to compute with automorphic forms has emerged as a major topic in explicit arithmetic geometry. The first such computations were carried out for elliptic modular forms, and now very large and useful databases of such forms exist [2,13,14]. Recently, effective algorithms to compute with Hilbert modular forms over a totally real field F have been advanced. The first such method is due to Dembélé [4, 5], who worked initially under the assumption that F has even degree n = [F: Q] and strict class number 1. Exploiting the Jacquet-Langlands correspondence, systems of

only at 3 and 5

Abstract. For p = 3 and p = 5, we exhibit a finite nonsolvable extension of Q which is ramified only at p via explicit computations with Hilbert modular forms. The study of Galois number fields with prescribed ramification remains a central question in number theory. Class field theory, a triumph of early twentieth century algebraic number theory, provides a satisfactory way to understand solvable extensions of a number field. To investigate nonsolvable extensions, the use of the modern techniques of arithmetic geometry is essential. Implicit in his work on algebraic modular forms on groups of higher rank, Gross [17] proposed the following conjecture. Conjecture. For any prime p, there exists a nonsolvable Galois number field ramified only at p. In his analysis of Galois representations associated to classical cusp forms, Serre [32, 33] has shown that for each prime p ≥ 11, indeed there exists a nonsolvable Galois number field ramified only at p. Conversely, it is a consequence of the proof of Serre’s conjecture by Khare and Wintenberger [22] together with standard level lowering arguments that if p ≤ 7 then any odd representation of the absolute Galois group Gal(Q/Q) to GL2(Fp) is necessarily reducible. Therefore, to find nonsolvable number fields which are ramified only at a prime p ≤ 7 we are led to consider more general settings. Computations by Lansky and Pollack [24] predict the existence of a G2(F5)-extension of Q ramified only at 5; however, the general theory which would provably attach a Galois representation to such an automorphic form is absent (see [18]). Following a suggestion of Gross, the first author [9] has recently constructed a nonsolvable extension ramified only at p = 2 by instead enlarging the base field: he exhibits two Hilbert modular forms of level 1 and parallel weight 2 over the totally real subfield F = Q(ζ32) + of Q(ζ32) whose mod 2 Galois representations cut out a number field with Galois group 8·SL2(F 2 8) 2 ramified only at 2. (See also the introduction in his work [9] for a survey of the history of Gross ’ conjecture.)