Well all credits should go to Shimura for proving all the results (but for elliptic curves I guess everything was already known by Deuring!), and Streng for a nice exposition.

@Will Sawin: you are in fact both correct, $K(j(E))$ is the subfield of the ray class field that corresponds to the ring class field; I provided more details in my answer below.

@Yves: thanks for the reference! There is another infinite example here mathoverflow.net/a/42184/26737

@Robinson: but $(\Z/2\Z) \times A_5$ is not isomorphic as a group to $SL(2,5)$, no?

Yes but as Will pointed out, connectivity is not a local property, so that's why I used locally noetherian instead. But this is a good point: if a local property fails for the global sections of a non affine scheme, is there any sort of additional global property that makes it work?

Yes you are right of course! I just wanted to add another example than reduced, that's why I gave the integrality example. One could correct this as follows: a noetherian ring whose stalks are integral is a product of domain. This is a local condition, and so if I am not mistaken a "locally integral" locally noetherian scheme has global sections a product of domains also.