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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 1, Pages 185–224 (Mi izv233)  

This article is cited in 5 scientific papers (total in 5 papers)

On weights of the $l$-adic representation and arithmetic of Frobenius eigenvalues

S. G. Tankeev

Vladimir State University

Abstract: Let $J$ be an absolutely simple Abelian variety over a number field $k$, $[k:\mathbb Q]<\infty$. Assume that $\operatorname{Cent}(\operatorname{End}(J\otimes\overline k))=\mathbb Z$. If the division $\mathbb Q$-algebra $\operatorname{End}^0(J\otimes\overline k)$ splits at a prime number $l$, then the $l$-adic representation is defined by the miniscule weights (microweights) of simple classical Lie algebras of types $A_m$, $B_m$$C_m$ or $D_m$.
If $S$ is a K3 surface over a sufficiently large number field $k\subset\mathbb C$ and the Hodge group $\operatorname{Hg}(S\otimes_k\mathbb C)$ is semisimple, then $S$ has ordinary reduction at each non-Archimedean place of $k$ in some set of Dirichlet density 1.
If $J$ is an absolutely simple Abelian threefold of type IV in Albert's classification over a sufficiently large number field, then $J$ has ordinary reduction at each place in some set of Dirichlet density 1.

DOI: https://doi.org/10.4213/im233

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English version:
Izvestiya: Mathematics, 1999, 63:1, 181–218

Bibliographic databases:

MSC: 14K15
Received: 20.07.1997

Citation: S. G. Tankeev, “On weights of the $l$-adic representation and arithmetic of Frobenius eigenvalues”, Izv. RAN. Ser. Mat., 63:1 (1999), 185–224; Izv. Math., 63:1 (1999), 181–218

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Vasiu A., “Some cases of the Mumford-Tate conjecture and Shimura varieties”, Indiana University Mathematics Journal, 57:1 (2008), 1–75  crossref  mathscinet  zmath  isi  scopus
    2. Bogomolov F., Hassett B., Tschinkel Yu., “Constructing Rational Curves on K3 Surfaces”, Duke Math J, 157:3 (2011), 535–550  crossref  mathscinet  zmath  isi  scopus
    3. Yu J.-D., “Special Lifts of Ordinary K3 Surfaces and Applications”, Pure Appl Math Q, 8:3 (2012), 805–824  mathscinet  isi  elib
    4. Xue J., Yu Ch.-F., “Abelian Varieties Without a Prescribed Newton Polygon Reduction”, Proc. Amer. Math. Soc., 143:6 (2015), PII S0002-9939(2015)12483-5, 2339–2345  crossref  mathscinet  zmath  isi  scopus
    5. Bloom S., “The Square Sieve and a Lang-Trotter Question For Generic Abelian Varieties”, J. Number Theory, 191 (2018), 119–157  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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