By Goro Shimura (auth.)

In 1996 the AMS provided Goro Shimura the Steele Prize for Lifetime success for his "important and vast paintings on arithmetical geometry and automorphic forms." His seminal paintings has led to the "many notations in quantity idea that hold his identify and that experience lengthy been widespread to employees within the field." those five volumes include 103 of his most crucial papers, starting in 1954 and carrying on with up during the current.

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Extra resources for Collected Papers: Volume II 1967–1977

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The algebra B has an involution x f-+ x' which is uniquely determined by the property that x + x' is the reduced trace of x to the center F. We put NB1P(x) = xx' , NB/O(X) = NpIO(NBIP(X») , TrBIP(x) = x + x' TrBIQ(x) = TrF/o(TrBIP(x») (XE B) . Let 0 be a maximal order in Band \:} a right o-ideal. ) We denote by NB/P(\:}) the ideal in Fgenerated over Ip by the elements NBIP(X) for all x E \:}. We say that \:} is prime to an integral ideal c in F if \:}p = op for every prime factor p of c (cf. 6).

X A [2]. 26). : Fibre varieties over a symmetric space whose fibres are abelian varieties. Lecture note, University of Chicago 1963-64 (1965). : Families of abelian varieties, Proceedings of Symposia in Pure Mathematics, vol. 9, Algebraic Groups, Amer. Math. Soc. (1966). : Symplectic representations of algebraic groups satisfying a certain analylicity condition. To appear. : On the theory of automorphic functions. Ann. Math. 70, 101-144 (1959). [5] -- On analytic families of polarized abelian varieties and automorphic functions.

Is not necessarily abelian over F, but a class field over a certain number field K'. However, the class field CF is the intersection of the maximal unramified abelian extensions CK' for all these K'. Thus the main point of the proof in the case r = 1 lies in the procedure of constructing from these (W, ofr) a model (V, cp) defined over CF• In each family :SK' the class field over S' mentioned in (D) is shown to be contained in the composite CJ{CK'. The property (3) of cp can be established by means of the fact that CJ{ is the intersection of the CJ{·C K , for all K'.

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