Ideal Classes and Sl 2
نویسنده
چکیده
and the matrix here is in SL2(R). (This action of SL2(R) on the upper half-plane is essentially one of the models for the isometries of the hyperbolic plane.) The action (1.1) makes sense with C replaced by any field K, and gives a transitive group action of GL2(K) on the set K ∪{∞}. Just as over the complex numbers, the formula (1.2) shows the action of SL2(K) on K ∪ {∞} is transitive. Now take K to be a number field, and replace the group SL2(K) with the subgroup SL2(OK). We ask: how many orbits are there for the action of the group SL2(OK) on K ∪ {∞}?
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