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Mordell-Weil groups of generic Abelian varieties. (English) Zbl 0576.14020
The paper under review deals with the following question: Suppose $$X=D/\Gamma$$ is a Shimura-variety over $$\mathbb C$$, and $$A\to X$$ a family of abelian varieties derived from the interpretation of $$X$$ as a moduli space. What is the Mordell-Weil group $$A(\mathbb C(X))$$? (It is finitely generated.) The author shows that in many cases the group is finite. The details of the proofs are somehow intricate, but the main idea seems to be the following: (a) any rational section extends to a holomorphic section $$X\to A$$; (b) over $$X$$ we have an exact sequence $0\to\mathcal L\to\mathrm{Lie}(A)\to A\to 0,$ hence an exact sequence $H^ 0(X,\mathrm{Lie}(A))\to A(X)\to H^ 1(X,{\mathcal L})\to H^ 1(X,\mathrm{Lie}(A)).$ It suffices to show that the first term vanishes, and that the last arrow is injective up to torsion. If $$X$$ is compact we may apply vanishing theorems and Eichler integrals to achieve this. For noncompact $$X$$ one reduces to the case of the upper half-space.
Reviewer: Gerd Faltings

##### MSC:
 11G15 Complex multiplication and moduli of abelian varieties 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 14K15 Arithmetic ground fields for abelian varieties 11G18 Arithmetic aspects of modular and Shimura varieties 14K22 Complex multiplication and abelian varieties
##### Keywords:
Shimura variety; Mordell-Weil group
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##### References:
 [1] Gunning, R., Rossi, H.: Functions of several complex variables?Revised Edition. (in press) · Zbl 1204.01045 [2] Igusa, J.-I.: On the structure of a certain class of Kaehler varieties. Am. J. Math.76, 669-678 (1954) · Zbl 0058.37901 · doi:10.2307/2372709 [3] Matsushima, Y., Shimura, G.: On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes. Ann. Math.78, 417-449 (1963) · Zbl 0141.38704 · doi:10.2307/1970534 [4] Shimura, G.: Sur les integrales attachees aux formes automorphes. J. Math. Soc. Japan11, 291-311 (1959) · Zbl 0090.05503 · doi:10.2969/jmsj/01140291 [5] Shimura, G.: On analytic families of polarized abelian varieties and automorphic functions. Ann. Math.78, 149-192 (1963) · Zbl 0142.05402 · doi:10.2307/1970507 [6] Shimura, G.: On the field of definition for a field of automorphic functions. Ann. Math.80, 160-189 (1964) · Zbl 0196.53203 · doi:10.2307/1970497 [7] Shimura, G.: On the field of definition for a field of automorphic functions: II. Ann. Math.81, 124-165 (1965) · Zbl 0222.14026 · doi:10.2307/1970388 [8] Shimura, G.: Moduli and fibre systems of abelian varieties. Ann. Math.83, 294-338 (1966) · Zbl 0141.37503 · doi:10.2307/1970434 [9] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58-159 (1967) · Zbl 0204.07201 · doi:10.2307/1970526 [10] Shimura, G.: On canonical models of arithmetic quotients of bounded symmetric domains. Ann. of Math.91, 144-222 (1970) · Zbl 0237.14009 · doi:10.2307/1970604 [11] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Publ. Iwanami Shoten and Princeton University Press (1971) [12] Shimura, G., Taniyama, Y.: Complex multiplication of abelian varieties and its applications to number theory. Publ. Math. Soc. Japan,6, (1961) · Zbl 0112.03502 [13] Shioda, T.: On elliptic modular surfaces. J. Math. Soc. Japan24, 20-59 (1972) · Zbl 0226.14013 · doi:10.2969/jmsj/02410020 [14] Weil, A.: Foundations of algebraic geometry. Amer. Math. Soc. Coll. Publ.29 (2nd ed.) Providence (1962) · Zbl 0168.18701
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