If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n variables out of a linear combination of conjugacy classes of maximal orders in A.

When A has class number one, there is a single such conjugacy class. What's the corresponding modular form?


1 Answer 1


The Jacquet-Langlands correspondence in the case of a totally definite quaternion algebras over a strict class number one field, parallel weight $2$ gives a Hecke equivariant map from the space of (set theoretic) maps from the set of left ideal classes of an Eichler order $\mathcal{O}$ to $\mathbb{C}$ that are orthogonal to the constant map to some space of Hilbert modular forms. So in the case where the order $\mathcal{O}$ has class number one, this space is zero and there is no corresponding modular form.

If you really want to associate a form to the constant map, since the Hecke operator $T_{\mathfrak{p}}$ acts as $$T_{\mathfrak{p}}(f)([I]) = \sum_{J\subset I, [I:J]=\mathfrak{p}^2}f([J])\text{,}$$ the eigenvalue of the constant map is $N(\mathfrak{p})+1$ so the "corresponding" $f$ should be an Eisenstein series.

You may want to read http://arxiv.org/abs/1010.5727 : their goal is algorithmic so their presentation is very clear and concrete. In particular, you may want to look at Theorem 3.9 (Jacquet-Langlands correspondence).


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