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Unformatted text preview: IDEAL CLASSES AND SL 2 KEITH CONRAD 1. Introduction A standard group action in complex analysis is the action of GL 2 ( C ) on the Riemann sphere C {} by linear fractional transformations (M obius transformations): (1.1) a b c d z = az + b cz + d . We need to allow the value since cz + d might be 0. (If that happens, az + b 6 = 0 since ( a b c d ) is invertible.) When z = , the value of (1.1) is a/c C {} . It is easy to see this action of GL 2 ( C ) on the Riemann sphere is transitive (that is, there is one orbit): for every z C , (1.2) a a 1 1 1 = a, so the orbit of passes through all points. In fact, since ( a a 1 1 1 ) has determinant 1, the action of SL 2 ( C ) (the 2 2 matrices with determinant 1) on C {} is transitive. However, the action of SL 2 ( R ) on the Riemann sphere is not transitive. The reason is the formula for imaginary parts under a real linear fractional transformation: Im az + b cz + d = ( ad bc ) Im( z )  cz + d  2 when ( a b c d ) GL 2 ( R ). Thus, z and ( a b c d ) z have the same imaginary part when ( a b c d ) has determinant 1. The action of SL 2 ( R ) on the Riemann sphere has three orbits: R {} , the upper halfplane h = { x + iy : y > } , and the lower halfplane. To see that the action of SL 2 ( R ) on h is transitive, pick x + iy with y > 0. Then y x/ y 1 / y i = x + iy, and the matrix here is in SL 2 ( R ). (This action of SL 2 ( R ) on the upper halfplane 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 GL 2 ( K ) on the set K {} . Just as over the complex numbers, the formula (1.2) shows the action of SL 2 ( K ) on K {} is transitive. Now take K to be a number field, and replace the group SL 2 ( K ) with the subgroup SL 2 ( O K ). We ask: how many orbits are there for the action of the group SL 2 ( O K ) on K {} ? Theorem 1.1. For a number field K , the number of orbits for SL 2 ( O K ) on K {} is the class number of K . 1 2 KEITH CONRAD Therefore there are finitely many orbits, and moreover this finiteness is a nontrivial statement! In Section 2, we will prove SL 2 ( O K ) acts transitively on K {} if and only if K has class number 1. This is the simplest case of Theorem 1.1. As preparation for the general case, in Section 3 we will change our language from K {} to the projective line over K , whose relevance (among other things) is that it removes the peculiar status of . (It seems useful to treat the special case of class number 1 without mentioning the projective line, if only to underscore what it is one is gaining by using the projective line in the general case.) In Section 4 we prove Theorem 1.1 in general by giving a bijection between the SL 2 ( O K )orbits and ideal classes in K ....
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 Fall '11
 Clark
 Geometry, Transformations

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