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There is an isomorphism (93) for K an imaginary quadratic field), then the endomorphisms of E are given by End(E) = OΛ , where OΛ is the order of Λ, (94) OΛ = {x ∈ K : xΛ ⊂ Λ}. Notice that the elliptic curves E = C/Λ, where Λ is a 1-dimensional K-lattice, have OΛ = O, the maximal order. The number of distinct isomorphism classes of elliptic curves E with End(E) = O is equal to the class number hK . All the other elliptic curves with complex multiplication by K are obtained from these by isogenies.

This last property, for this system, is then exactly as difficult as the original Hilbert 12th problem. This appears to show that the construction of “fabulous states” is just a reformulation of the original problem and need not simplify the task of obtaining explicit information about the generators of K ab and the Galois action. The point is precisely that such “trivial example” is only a reformulation of the Hilbert 12th problem, while Problem 4 allows for nontrivial constructions of quantum statistical mechanical systems, where one can use essentially the fact of working with algebras instead of fields.

Remark 10. Under the Gelfand–Naimark correspondence, unital C ∗ -algebras correspond to compact spaces. Thus, one can say that a noncommutative space X described by a C ∗ -algebra A is “Morita–compact” if the algebra A is Morita equivalent to a unital C ∗ -algebra. This provides us with a notion of “Morita–compactification” for noncommutative spaces. In the case of the GL2 -system, we can replace (58) by the noncommutative space Sh(nc) (H± , GL2 ) := GL2 (Q)\(M2 (Af )×P1 (C)) ∼ GL2 (Q)\M2 (A)· /C∗ , (60) where we use the notation M2 (A)· = M2 (Af ) × (M2 (R) {0}).

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2-affine complete algebras need not be affine complete by Aichinger E.


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