Representations of quaternion algebras over local fields and trace formulas of Hecke operators
For a nonarchimedean local field \(F\) and the division quaternion algebra \(B\) over \(F\) the authors investigate the decomposition of irreducible admissible representations of \(B^ \times\) upon restriction to the multiplicative group of a quadratic extension of \(F\) contained in \(B\). They give explicit formulae for the decomposition of these restrictions into characters. A similar result of Tunnell describes the decomposition of representations of \(\text{GL}_ 2\) upon restriction to Cartan subgroups using the \(\varepsilon\)-factors of base change lifts of the representations; the explicit formulae given here allow with the help of the correspondence of Jacquet/Langlands to calculate these factors. The authors then calculate the traces of the Hecke operators on the representation spaces and use their results to decompose spaces of cusp forms of prime power level. They give several explicit examples, among them the decomposition of the 1176-dimensional space of newforms of weight 2, level \(7^ 5\) and trivial character into subspaces of dimensions 13, 14, 15 (which are not necessarily irreducible).
automorphic forms on quaternion algebras
scientific article; zbMATH DE number 166064