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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 6, Pages 1231–1251 (Mi izv2069)  

This article is cited in 2 scientific papers (total in 2 papers)

On homomorphisms of Abelian schemes. II

S. G. Tankeev


Abstract: Let $k$ be a field of algebraic functions of one variable over the field $\mathbf C$ of complex numbers, let $S$ be the complete smooth model of $k$ over $\mathbf C$, and let $\mathscr I_i\to S$ ($i=1,2$) be the Néron models of Abelian varieties $I_i$ over $k$. Suppose that one of the following conditions holds:
1) The minimal models $\mathscr I_i\to S$ admit compactifications whose degenerate fibers are unions of normally crossing smooth irreducible components, and
$$ H^0(S,\mathscr Lie_S(\mathscr I_1)\otimes_{\mathscr O_S}\mathscr Lie_S(\mathscr I_2))=(0). $$

2) The Abelian variety $I_1$ has totally degenerate reduction at a point $v$ of $k$, i.e. the algebraic group $\mathscr I_{1v}$ is an extension of a finite group by a torus.
Then for every prime number $l$ the canonical map
$$ \operatorname{Hom}_k(I_1,I_2)\otimes_\mathbf Z\mathbf Z_l\to\operatorname{Hom}_{\operatorname{Gal}(\bar k/k)}(T_l(I_1),T_l(I_2)) $$
is an isomorphism.
Bibliography: 17 titles.

Full text: PDF file (1797 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:6, 1175–1194

Bibliographic databases:

UDC: 513.6
MSC: Primary 14K05, 14G13, 14F30; Secondary 14K10, 14K30, 14H40, 14D10
Received: 18.11.1976

Citation: S. G. Tankeev, “On homomorphisms of Abelian schemes. II”, Izv. Akad. Nauk SSSR Ser. Mat., 41:6 (1977), 1231–1251; Math. USSR-Izv., 11:6 (1977), 1175–1194

Citation in format AMSBIB
\Bibitem{Tan77}
\by S.~G.~Tankeev
\paper On homomorphisms of Abelian schemes.~II
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 6
\pages 1231--1251
\mathnet{http://mi.mathnet.ru/izv2069}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=480536}
\zmath{https://zbmath.org/?q=an:0368.14015|0399.14007}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 6
\pages 1175--1194
\crossref{https://doi.org/10.1070/IM1977v011n06ABEH001765}


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    This publication is cited in the following articles:
    1. S. G. Tankeev, “On algebraic cycles on Abelian varieties”, Math. USSR-Izv., 12:3 (1978), 617–643  mathnet  crossref  mathscinet  zmath
    2. S. G. Tankeev, “On algebraic cycles on surfaces and Abelian varieties”, Math. USSR-Izv., 18:2 (1982), 349–380  mathnet  crossref  mathscinet  zmath
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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