Computing ideal classes representatives in quaternion algebras

Abstract

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q[\sqrt{5}]$.Comment: Corrected version. Section 4 has been rewritten from scratc

Preimages for the Shimura map on Hilbert modular forms

“…the algebra generated by 1, i, j, k, where i 2 = j 2 = −1, ij = k = −ji. If R is an Eichler order of discriminant c in B, then the theory of Jacquet-Langlands asserts that there exists v ∈ S(R) which is an eigenvector for T 0 with the same eigenvalues as g. Using the algorithm presented in [PS12], with the aid of SAGE ([S11]), we obtain the desired order, which is given by This order has class number equal to 2, and hence there is no need to compute the Hecke operators, since S(R) is 1-dimensional. A set of representatives for the set of R-ideal classes is given by R and the ideal I given by Let f = θ(v).…”

“…Using the algorithm presented in [PS12], with the aid of SAGE ([S11]), we obtain the desired order, which is given by For this zeros Theorem 5.6 is easy to verify. The local-global principle for quadratic forms implies that the non existence of points x ∈ L R ∪ L I with −∆(x) = ξ is equivallent to the equality ǫ ξ (c) = −1, so in this case both sides of (5.7) vanish trivially.…”

“…The preimages are obtained by considering certain ternary theta series associated to ideals in quaternion algebras. The problem of computing these ideals is thus crucial for our method, and has been studied in [DV10] and [PS12].…”

On the change of root numbers under twisting and applications

“…To see whether we are in the Principal Series case or in the Supercuspidal one, we can search for the curve in the quaternion algebra ramified at 2 and infinity. This can be done by choosing the correct order in such algebra (see [HPS89]) and constructing ideal representatives for it in order to compute the Brandt matrices (see [PS10] for an effective way to construct the ideals). It turns out that all four curves appear in such algebra (although it is clear that if one does the others do as well), hence the component at 2 of all of them is supercuspidal.…”