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Tate Cycles on Abelian Varieties with Complex Multiplication

  • V. Kumar Murty,
    Department of Mathematics, University of Toronto, 40 St. George St., Toronto, CANADA M5S 2E4
  • Vijay M. Patankar,
    Indian Statistical Instiute, Chennai, India, 600113
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Abstract

We consider Tate cycles on an Abelian variety $A$ defined over a sufficiently large number field $K$ and having complex multiplication. We show that there is an effective bound $C = C(A,K)$ so that to check whether a given cohomology class is a Tate class on $A$, it suffices to check the action of Frobenius elements at primes $v$ of norm $ \leq C$. We also show that for a set of primes $v$ of $K$ of density $1$, the space of Tate cycles on the special fibre $A_v$ of the NĂ©ron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself.
Keywords: Abelian varieties, complex multiplication, Tate cycles Abelian varieties, complex multiplication, Tate cycles
MSC Classifications: 11G10, 14K22 show english descriptions Abelian varieties of dimension $> 1$ [See also 14Kxx]
Complex multiplication [See also 11G15]
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
14K22 - Complex multiplication [See also 11G15]
 

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