By Reinhold Hübl

ISBN-10: 3540509852

ISBN-13: 9783540509851

This monograph offers an advent to, in addition to a unification and extension of the broadcast paintings and a few unpublished principles of J. Lipman and E. Kunz approximately lines of differential kinds and their kin to duality concept for projective morphisms. The method makes use of Hochschild-homology, the definition of that is prolonged to the class of topological algebras. Many effects for Hochschild-homology of commutative algebras additionally carry for Hochschild-homology of topological algebras. specifically, after introducing a suitable inspiration completion of differential algebras, one will get a common transformation among differential varieties and Hochschild-homology of topological algebras. strains of differential types are of curiosity to all people operating with duality idea and residue symbols. Hochschild-homology is an invaluable instrument in lots of parts of k-theory. The therapy in all fairness basic and calls for in simple terms little wisdom in commutative algebra and algebraic geometry.

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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.

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