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Explicit computations of Hilbert modular forms on $ℚ\left(\sqrt{5}\right)$. (English) Zbl 1152.11328

The author presents an algorithm that allows one to compute Hilbert modular forms of parallel weight 2 and level $𝔠$ on ${\text{GL}}_{2}\left(F\right)$, where $F=ℚ\left(\sqrt{5}\right)$ is the real quadratic field of smallest discriminant. Though the calculations are mainly focused on forms of parallel weight 2, it also works on forms of weight (2, 4) and the algorithm may easily be generalized to compute forms of arbitrary weights. The author indicates that there should be no major problem in generalizing this algorithm to compute forms of arbitrary weight and level over any totally real field of narrow class number 1.

However, the author concentrates on the simplest case of forms of parallel weight 2, the main reason being that, in this case, one knows where to look for some of the corresponding geometric objects (such as elliptic curves or hypergeometric abelian varieties studied by Darmon (2000) in connection with the equation ${x}^{n}+{y}^{n}={z}^{5}\right)$, at least conjecturally.

The method of computation is based on the Jacquet-Langlands correspondence (as has been done by A. Pizer, J. Algebra 64, 340–390 (1980; Zbl 0433.10012), C. Consani and J. Scholten, Int. J. Math. 12, No. 8, 943–972 (2001; Zbl 1111.11306), J. Socrates and D. Whitehouse, Pac. J. Math. 219, No. 2, 333–364 (2005; Zbl 1109.11029)).

The rest of the paper is as follows: Section 2 the author recalls preliminary results about automorphic forms on definite quaternion algebras together with the Jacquet-Langlands correspondence. In Section 3, the description of the algorithm is given. By direct investigations, one obtains a few of the elliptic curves corresponding to some of the forms computed. Their modularity is studied in Section 4 and also all modular elliptic curves over $ℚ\left(\sqrt{5}\right)$ of prime conductor of norm less than 100.

MSC:
 11F41 Hilbert modular forms and surfaces 11Y16 Algorithms; complexity (number theory)