By Shafarevich I.R.

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29 30 3. HEEGNER POINTS ON X0 (N ) If A has complex multiplication by O, the corresponding period lattice of A is a projective O-module of rank one, whose isomorphism class depends only on the isomorphism type of A. Conversely, if Λ ⊂ C is a projective O-module of rank one, the corresponding elliptic curve A = C/Λ has complex multiplication by O. Hence there is a bijection Elliptic curves with CM by O, up to isomorphism. −→ Rank one projective O-modules, up to isomorphism. The set on the right is called the Picard group, or the Class group, of O and is denoted Pic(O).

The group Kλ× can be viewed naturally as a subgroup of the group A× eles attached to K. Let ιλ (x) K,f of finite id` denote the id`ele attached to x ∈ Kλ× . On the global level, K (resp. K × ) can be viewed as a subring (resp. a subgroup) of AK,f (resp. A× K,f ) via the natural diagonal embedding. The group Pic(O) admits an adelic description, via the identification ˆ×, Pic(O) = A× /K × O K,f in which the class of the id`ele α corresponds to the homothety class of the lattice ˆ ∩ K ⊂ C. (α−1 O) The following is a special case of the main theorem of class field theory (cf.

19 is elementary and does not involve the notion of modularity, one knows at present of no method for tackling it directly without exploiting a connection between elliptic curves and automorphic forms. In fact, only in the rather limited number of cases where one can establish the analytic continuation and functional equation of L(E/K, χ, s)—by relating it to the L-series of an automorphic form on GL2 (F ), as in the case F = Q covered by Wiles’ theory—does one have any means at present of relating sign(E, K) to the behaviour of this associated L-series.

### Algebraic geometry by Shafarevich I.R.

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