zbMATH — the first resource for mathematics

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

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
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.